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Discriminating States: The Quantum Chernoff Bound

2007-04-17
Katrien Audenaert , John Calsamiglia , R. Muñoz-Tapia +4 more
We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the minimal probability of error. This leads to the identification of … We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the minimal probability of error. This leads to the identification of the quantum Chernoff bound, thereby solving a long-standing open problem. The bound reduces to the classical Chernoff bound when the quantum states under consideration commute. The quantum Chernoff bound is the natural symmetric distance measure between quantum states because of its clear operational meaning and because it does not seem to share some of the undesirable features of other distance measures.
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Relations between Coherence and Path Information

2016-04-22
E. Bagán , János A. Bergou , Seth Cottrell +1 more
We find two relations between coherence and path information in a multipath interferometer. The first builds on earlier results for the two-path interferometer, which used minimum-error state discrimination between detector … We find two relations between coherence and path information in a multipath interferometer. The first builds on earlier results for the two-path interferometer, which used minimum-error state discrimination between detector states to provide the path information. For visibility, which was used in the two-path case, we substitute a recently defined ${l}_{1}$ measure of quantum coherence. The second is an entropic relation in which the path information is characterized by the mutual information between the detector states and the outcome of the measurement performed on them, and the coherence measure is one based on relative entropy.
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Quantum Chernoff bound as a measure of distinguishability between density matrices: Application to qubit and Gaussian states

2008-03-07
John Calsamiglia , R. Muñoz-Tapia , Lluís Masanes +2 more
Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability … Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability when discriminating two hypotheses given a large number of observations. Recently the combined work of Audenaert et al. [Phys. Rev. Lett. 98, 160501 (2007)] and Nussbaum and Szkola [e-print arXiv:quant-ph/0607216] has proved the quantum analog of this bound, which applies when the hypotheses correspond to two quantum states. Based on this quantum Chernoff bound, we define a physically meaningful distinguishability measure and its corresponding metric in the space of states; the latter is shown to coincide with the Wigner-Yanase metric. Along the same lines, we define a second, more easily implementable, distinguishability measure based on the error probability of discrimination when the same local measurement is performed on every copy. We study some general properties of these measures, including the probability distribution of density matrices, defined via the volume element induced by the metric. It is shown that the Bures and the local-measurement based metrics are always proportional. Finally, we illustrate their use in the paradigmatic cases of qubits and Gaussian infinite-dimensional states.
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Radiative corrections to the decay B→πeν and the heavy quark limit

1998-01-01
E. Bagán , Patricia Ball , V. M. Braun
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Quantum-limited metrology with product states

2008-01-15
Sergio Boixo , Animesh Datta , Steven T. Flammia +3 more
We study the performance of initial product states of $n$-body systems in generalized quantum metrology protocols that involve estimating an unknown coupling constant in a nonlinear $k$-body $(k\ensuremath{\ll}n)$ Hamiltonian. We … We study the performance of initial product states of $n$-body systems in generalized quantum metrology protocols that involve estimating an unknown coupling constant in a nonlinear $k$-body $(k\ensuremath{\ll}n)$ Hamiltonian. We obtain the theoretical lower bound on the uncertainty in the estimate of the parameter. For arbitrary initial states, the lower bound scales as $1/{n}^{k}$, and for initial product states, it scales as $1/{n}^{k\ensuremath{-}1/2}$. We show that the latter scaling can be achieved using simple, separable measurements. We analyze in detail the case of a quadratic Hamiltonian $(k=2)$, implementable with Bose-Einstein condensates. We formulate a simple model, based on the evolution of angular-momentum coherent states, which explains the $O({n}^{\ensuremath{-}3/2})$ scaling for $k=2$; the model shows that the entanglement generated by the quadratic Hamiltonian does not play a role in the enhanced sensitivity scaling. We show that phase decoherence does not affect the $O({n}^{\ensuremath{-}3/2})$ sensitivity scaling for initial product states.
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Optimal full estimation of qubit mixed states

2006-03-02
E. Bagán , Manuel A. Ballester , Richard D. Gill +2 more
We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the … We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary N, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large N. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.
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Multiple-copy two-state discrimination with individual measurements

2005-03-23
Antonio Acín , E. Bagán , M. Baig +2 more
We address the problem of non-orthogonal two-state discrimination when multiple copies of the unknown state are available. We give the optimal strategy when only fixed individual measurements are allowed and … We address the problem of non-orthogonal two-state discrimination when multiple copies of the unknown state are available. We give the optimal strategy when only fixed individual measurements are allowed and show that its error probability saturates the collective (lower) bound asymptotically. We also give the optimal strategy when adaptivity of individual von Neumann measurements is allowed (which requires classical communication), and show that the corresponding error probability is exactly equal to the collective one for any number of copies. We show that this strategy can be regarded as Bayesian updating.
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Charges from Dressed Matter: Construction

2000-06-01
E. Bagán , Martin Lavelle , David McMullan
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Phase estimation for thermal Gaussian states

2009-03-23
M. Aspachs , John Calsamiglia , R. Muñoz-Tapia +1 more
We give the optimal bounds on the phase-estimation precision for mixed Gaussian states in the single-copy and many-copy regimes. Specifically, we focus on displaced thermal and squeezed thermal states. We … We give the optimal bounds on the phase-estimation precision for mixed Gaussian states in the single-copy and many-copy regimes. Specifically, we focus on displaced thermal and squeezed thermal states. We find that while for displaced thermal states an increase in temperature reduces the estimation fidelity, for squeezed thermal states a larger temperature can enhance the estimation fidelity. The many-copy optimal bounds are compared with the minimum variance achieved by three important single-shot measurement strategies. We show that the single-copy canonical phase measurement does not always attain the optimal bounds in the many-copy scenario. Adaptive homodyning schemes do attain the bounds for displaced thermal states, but for squeezed states they yield fidelities that are insensitive to temperature variations and are, therefore, suboptimal. Finally, we find that heterodyne measurements perform very poorly for pure states but can attain the optimal bounds for sufficiently mixed states. We apply our results to investigate the influence of losses in an optical metrology experiment. In the presence of losses squeezed states cease to provide the Heisenberg limited precision, and their performance is close to that of coherent states with the same mean photon number.
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Local Discrimination of Mixed States

2010-08-18
John Calsamiglia , Julio I. de Vicente , R. Muñoz-Tapia +1 more
We provide rigorous, efficiently computable and tight bounds on the average error probability of multiple-copy discrimination between qubit mixed states by local operations assisted with classical communication (LOCC). In contrast … We provide rigorous, efficiently computable and tight bounds on the average error probability of multiple-copy discrimination between qubit mixed states by local operations assisted with classical communication (LOCC). In contrast with the pure-state case, these experimentally feasible protocols perform strictly worse than the general collective ones. Our numerical results indicate that the gap between LOCC and collective error rates persists in the asymptotic limit. In order for LOCC and collective protocols to achieve the same accuracy, the former can require up to twice the number of copies of the latter. Our techniques can be used to bound the power of LOCC strategies in other similar settings, which is still one of the most elusive questions in quantum communication.
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Next-to-leading order radiative corrections to the decay b → ccs

1995-06-01
E. Bagán , Patricia Ball , Bartomeu Fiol +1 more
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Aligning Reference Frames with Quantum States

2001-11-01
E. Bagán , M. Baig , R. Muñoz-Tapia
We analyze the problem of sending, in a single transmission, the information required to specify an orthogonal trihedron or reference frame through a quantum channel made out of N elementary … We analyze the problem of sending, in a single transmission, the information required to specify an orthogonal trihedron or reference frame through a quantum channel made out of N elementary spins. We analytically obtain the optimal strategy, i.e., the best encoding state and the best measurement. For large N, we show that the average error goes to zero linearly in 1/N. Finally, we discuss the construction of finite optimal measurements.
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Quantum reverse engineering and reference-frame alignment without nonlocal correlations

2004-09-07
E. Bagán , M. Baig , R. Muñoz-Tapia
The estimation of unknown qubit elementary gates and the alignment of reference frames are formally the same problem. Using quantum states made out of $N$ qubits, we show that the … The estimation of unknown qubit elementary gates and the alignment of reference frames are formally the same problem. Using quantum states made out of $N$ qubits, we show that the theoretical precision limit for both problems, which behaves as $1∕{N}^{2}$, can be asymptotically attained with a covariant protocol that exploits the quantum correlation of internal degrees of freedom instead of the more fragile entanglement between distant parties. This cuts by half the number of qubits needed to achieve the precision of the dense covariant coding protocol.
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Theoretical update of the semileptonic branching ratio of B mesons

1995-01-01
E. Bagán , Patricia Ball , V. M. Braun +1 more
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Optimal discrimination of quantum states with a fixed rate of inconclusive outcomes

2012-10-15
E. Bagán , R. Muñoz-Tapia , G. A. Olivares-Rentería +1 more
In this paper we present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate … In this paper we present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between Unambiguous and Minimum Error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function of the rate of inconclusive answers. A critical value of this rate is identified that coincides with the minimum failure probability in the cases where unambiguous discrimination is possible and provides a natural generalization of it when states cannot be unambiguously discriminated. The method is illustrated on two explicit examples: discrimination of two pure states with arbitrary prior probabilities and discrimination of trine states.
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Optimal encoding and decoding of a spin direction

2001-04-17
E. Bagán , M. Baig , A. Brey +2 more
For a system of N spins 1/2 there are quantum states that can encode a direction in an intrinsic way. Information on this direction can later be decoded by means … For a system of N spins 1/2 there are quantum states that can encode a direction in an intrinsic way. Information on this direction can later be decoded by means of a quantum measurement. We present here the optimal encoding and decoding procedure using the fidelity as a figure of merit. We compute the maximal fidelity and prove that it is directly related to the largest zeroes of the Legendre and Jacobi polynomials. We show that this maximal fidelity approaches unity quadratically in 1/N. We also discuss this result in terms of the dimension of the encoding Hilbert space.
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Hadrons with charm and beauty

1994-03-01
E. Bagán , H. G. Dosch , P. Gosdzinsky +2 more
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Quantum learning without quantum memory

2012-10-05
Gael Sentís , John Calsamiglia , R. Muñoz-Tapia +1 more
A quantum learning machine for binary classification of qubit states that does not require quantum memory is introduced and shown to perform with the minimum error rate allowed by quantum … A quantum learning machine for binary classification of qubit states that does not require quantum memory is introduced and shown to perform with the minimum error rate allowed by quantum mechanics for any size of the training set. This result is shown to be robust under (an arbitrary amount of) noise and under (statistical) variations in the composition of the training set, provided it is large enough. This machine can be used an arbitrary number of times without retraining. Its required classical memory grows only logarithmically with the number of training qubits, while its excess risk decreases as the inverse of this number and twice as fast as the excess risk of an "estimate-and-discriminate" machine, which estimates the states of the training qubits and classifies the data qubit with a discrimination protocol tailored to the obtained estimates.
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Optimal Strategies for Sending Information through A Quantum Channel

2000-12-11
E. Bagán , M. Baig , A. Brey +2 more
Quantum states can be used to encode the information contained in a direction, i.e., in a unit vector. We present the best encoding procedure when the quantum state is made … Quantum states can be used to encode the information contained in a direction, i.e., in a unit vector. We present the best encoding procedure when the quantum state is made up of N spins (qubits). We find that the quality of this optimal procedure, which we quantify in terms of the fidelity, depends solely on the dimension of the encoding space. We also investigate the use of spatial rotations on a quantum state, which provide a natural and less demanding encoding. In this case we prove that the fidelity is directly related to the largest zeros of the Legendre and Jacobi polynomials. We also discuss our results in terms of the information gain.
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Charm quark mass dependence of QCD corrections to nonleptonic inclusive B decays

1994-12-01
E. Bagán , Patricia Ball , V. M. Braun +1 more
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Duality Games and Operational Duality Relations

2018-01-30
E. Bagán , John Calsamiglia , János A. Bergou +1 more
We give operational meaning to wave-particle duality in terms of discrimination games. Duality arises as a constraint on the probability of winning these games. The games are played with the … We give operational meaning to wave-particle duality in terms of discrimination games. Duality arises as a constraint on the probability of winning these games. The games are played with the aid of an n-port interferometer, and involve 3 parties, Alice and Bob, who cooperate, and the House, who supervises the game. In one game called ways they attempt to determine the path of a particle in the interferometer. In another, called phases, they attempt to determine which set of known phases have been applied to the different paths. The House determines which game is to be played by flipping a coin. We find a tight wave-particle duality relation that allows us to relate the probabilities of winning these games, and use it to find an upper bound on the probability of winning the combined game. This procedure allows us to express wave-particle duality in terms of discrimination probabilities.
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Separable Measurement Estimation of Density Matrices and its Fidelity Gap with Collective Protocols

2006-09-25
E. Bagán , Manuel A. Ballester , Richard D. Gill +2 more
We show that there exists a gap between the performance of separable and collective measurements in qubit mixed-state estimation that persists in the large sample limit. We characterize such gap … We show that there exists a gap between the performance of separable and collective measurements in qubit mixed-state estimation that persists in the large sample limit. We characterize such gap in terms of the corresponding bounds on the mean fidelity. We present an adaptive protocol that attains the separable-measurement bound. This (optimal separable) protocol uses von Neumann measurements and can be easily implemented with current technology.
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Charges from Dressed Matter: Physics and Renormalisation

2000-06-01
E. Bagán , Martin Lavelle , David McMullan
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The Isgur-Wise function to O(αs) from sum rules in the heavy quark effective theory

1993-03-01
E. Bagán , Patricia Ball , P. Gosdzinsky
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Collective versus local measurements in a qubit mixed-state estimation

2004-01-23
E. Bagán , M. Baig , R. Muñoz-Tapia +1 more
We discuss the problem of estimating a qubit mixed state. We give the optimal estimation that can be inferred from any given set of measurements. For collective measurements and for … We discuss the problem of estimating a qubit mixed state. We give the optimal estimation that can be inferred from any given set of measurements. For collective measurements and for a large number N of copies, we show that the error in the estimation varies as $1/N.$ For local measurements, we focus on the simpler case of states lying on the equatorial plane of the Bloch sphere. We show that the error using plain tomography varies as ${1/N}^{1/4},$ while our approach leads to an error proportional to ${1/N}^{3/4}.$
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Quantum Change Point

2016-10-07
Gael Sentís , E. Bagán , John Calsamiglia +2 more
Sudden changes are ubiquitous in nature. Identifying them is of crucial importance for a number of applications in medicine, biology, geophysics, and social sciences. Here we investigate the problem in … Sudden changes are ubiquitous in nature. Identifying them is of crucial importance for a number of applications in medicine, biology, geophysics, and social sciences. Here we investigate the problem in the quantum domain, considering a source that emits particles in a default state, until a point where it switches to another state. Given a sequence of particles emitted by the source, the problem is to find out where the change occurred. For large sequences, we obtain an analytical expression for the maximum probability of correctly identifying the change point when joint measurements on the whole sequence are allowed. We also construct strategies that measure the particles individually and provide an online answer as soon as a new particle is emitted by the source. We show that these strategies substantially underperform the optimal strategy, indicating that quantum sudden changes, although happening locally, are better detected globally.
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Optimal Scheme for Estimating a Pure Qubit State via Local Measurements

2002-12-19
E. Bagán , M. Baig , R. Muñoz-Tapia
We present the optimal scheme for estimating a pure qubit state by means of local measurements on N identical copies. We give explicit examples for low N. For large N, … We present the optimal scheme for estimating a pure qubit state by means of local measurements on N identical copies. We give explicit examples for low N. For large N, we show that the fidelity saturates the collective measurement bound up to order 1/N. When the signal state lays on a meridian of the Bloch sphere, we show that this can be achieved without classical communication.
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Comprehensive analysis of quantum pure-state estimation for two-level systems

2005-06-15
E. Bagán , Alex Monràs , R. Muñoz-Tapia
We analyze the estimation of a qubit pure state by means of local measurements on $N$ identical copies and compare its averaged fidelity for an isotropic prior probability distribution to … We analyze the estimation of a qubit pure state by means of local measurements on $N$ identical copies and compare its averaged fidelity for an isotropic prior probability distribution to the absolute upper bound given by collective measurements. We discuss two situations: the first one, where the state is restricted to lie on the equator of the Bloch sphere, is formally equivalent to phase estimation; the second one, where there is no constrain on the state, can also be regarded as the estimation of a direction in space using a quantum arrow made out of $N$ parallel spins. We discuss various schemes with and without classical communication and compare their efficiency. We show that the fidelity of the most general collective measurement can always be achieved asymptotically with local measurements and no classical communication.
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Unsupervised Classification of Quantum Data

2019-11-08
Gael Sentís , Alex Monràs , R. Muñoz-Tapia +2 more
We introduce the problem of unsupervised classification of quantum data, namely, of systems whose quantum states are unknown. We derive the optimal single-shot protocol for the binary case, where the … We introduce the problem of unsupervised classification of quantum data, namely, of systems whose quantum states are unknown. We derive the optimal single-shot protocol for the binary case, where the states in a disordered input array are of two types. Our protocol is universal and able to automatically sort the input under minimal assumptions, yet partially preserving information contained in the states. We quantify analytically its performance for arbitrary size and dimension of the data. We contrast it with the performance of its classical counterpart, which clusters data that has been sampled from two unknown probability distributions. We find that the quantum protocol fully exploits the dimensionality of the quantum data to achieve a much higher performance, provided data is at least three-dimensional. For the sake of comparison, we discuss the optimal protocol when the classical and quantum states are known.
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Secrecy properties of quantum channels

2006-01-20
Antonio Acín , Joonwoo Bae , E. Bagán +3 more
We identify those properties of a quantum channel that are relevant for cryptography. We focus on general key distribution protocols that use prepare and measure schemes and the existing classical … We identify those properties of a quantum channel that are relevant for cryptography. We focus on general key distribution protocols that use prepare and measure schemes and the existing classical reconciliation techniques, as these are the protocols feasible with current technology. Given a channel, we derive an easily computable necessary condition of security for such protocols. In spite of its simplicity, this condition is shown to be tight for the Bennett-Brassard 1984 and six-state protocols. We show that the condition becomes also sufficient in the event of a so-called collective attack.
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Quantum Metrology Assisted by Abstention

2013-03-04
B. Gendra , E. Ronco-Bonvehi , John Calsamiglia +2 more
The main goal of quantum metrology is to obtain accurate values of physical parameters using quantum probes. In this context, we show that abstention, i.e., the possibility of getting an … The main goal of quantum metrology is to obtain accurate values of physical parameters using quantum probes. In this context, we show that abstention, i.e., the possibility of getting an inconclusive answer at readout, can drastically improve the measurement precision and even lead to a change in its asymptotic behavior, from the shot-noise to the Heisenberg scaling. We focus on phase estimation and quantify the required amount of abstention for a given precision. We also develop analytical tools to obtain the asymptotic behavior of the precision and required rate of abstention for arbitrary pure states.
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Phase-covariant quantum benchmarks

2009-05-08
John Calsamiglia , M. Aspachs , R. Muñoz-Tapia +1 more
We give a quantum benchmark for teleportation and quantum storage experiments suited for pure and mixed test states. The benchmark is based on the average fidelity over a family of … We give a quantum benchmark for teleportation and quantum storage experiments suited for pure and mixed test states. The benchmark is based on the average fidelity over a family of phase-covariant states and certifies that an experiment cannot be emulated by a classical setup, i.e., by a measure-and-prepare scheme. We give an analytical solution for qubits, which shows important differences with standard state estimation approach, and compute the value of the benchmark for coherent and squeezed states, both pure and mixed.
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Entanglement-assisted alignment of reference frames using a dense covariant coding

2004-05-14
E. Bagán , M. Baig , R. Muñoz-Tapia
We present a procedure inspired by dense coding, which enables a highly efficient transmission of information of a continuous nature. The procedure requires the sender and the recipient to share … We present a procedure inspired by dense coding, which enables a highly efficient transmission of information of a continuous nature. The procedure requires the sender and the recipient to share a maximally entangled state. We deal with the concrete problem of aligning reference frames or trihedra by means of a quantum system. We find the optimal covariant measurement and compute the corresponding average error, which has a remarkably simple close form. The connection of this procedure with that of estimating unitary transformations on qubits is briefly discussed.
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Optimal signal states for quantum detectors

2011-07-22
Ognyan Oreshkov , John Calsamiglia , R. Muñoz-Tapia +1 more
Quantum detectors provide information about quantum systems by establishing correlations between certain properties of those systems and a set of macroscopically distinct states of the corresponding measurement devices. A natural … Quantum detectors provide information about quantum systems by establishing correlations between certain properties of those systems and a set of macroscopically distinct states of the corresponding measurement devices. A natural question of fundamental significance is how much information a quantum detector can extract from the quantum system it is applied to. In the present paper we address this question within a precise framework: given a quantum detector implementing a specific generalized quantum measurement, what is the optimal performance achievable with it for a concrete information readout task, and what is the optimal way to encode information in the quantum system in order to achieve this performance? We consider some of the most common information transmission tasks - the Bayes cost problem (of which minimal error discrimination is a special case), unambiguous message discrimination, and the maximal mutual information. We provide general solutions to the Bayesian and unambiguous discrimination problems. We also show that the maximal mutual information has an interpretation of a capacity of the measurement, and derive various properties that it satisfies, including its relation to the accessible information of an ensemble of states, and its form in the case of a group-covariant measurement. We illustrate our results with the example of a noisy two-level symmetric informationally complete measurement, for whose capacity we give analytical proofs of optimality. The framework presented here provides a natural way to characterize generalized quantum measurements in terms of their information readout capabilities.
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Minimal measurements of the gate fidelity of a qudit map

2003-01-23
E. Bagán , M. Baig , R. Muñoz-Tapia
We obtain a simple formula for the average gate fidelity of a linear map acting on qudits. It is given in terms of minimal sets of pure state preparations alone, … We obtain a simple formula for the average gate fidelity of a linear map acting on qudits. It is given in terms of minimal sets of pure state preparations alone, which may be interesting from the experimental point of view. These preparations can be seen as the outcomes of certain minimal positive operator valued measures. The connection of our results with these generalized measurements is briefly discussed.
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Multicopy programmable discrimination of general qubit states

2010-10-15
Gael Sentís , E. Bagán , John Calsamiglia +1 more
Quantum state discrimination is a fundamental primitive in quantum statistics where one has to correctly identify the state of a system that is in one of two possible known states. … Quantum state discrimination is a fundamental primitive in quantum statistics where one has to correctly identify the state of a system that is in one of two possible known states. A programmable discrimination machine performs this task when the pair of possible states is not a priori known but instead the two possible states are provided through two respective program ports. We study optimal programmable discrimination machines for general qubit states when several copies of states are available in the data or program ports. Two scenarios are considered: One in which the purity of the possible states is a priori known, and the fully universal one where the machine operates over generic mixed states of unknown purity. We find analytical results for both the unambiguous and minimum error discrimination strategies. This allows us to calculate the asymptotic performance of programmable discrimination machines when a large number of copies are provided and to recover the standard state discrimination and state comparison values as different limiting cases.
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Purity Estimation with Separable Measurements

2005-09-09
E. Bagán , Manuel A. Ballester , R. Muñoz-Tapia +1 more
Given a large number N of copies of a qubit state of which we wish to estimate its purity, we prove that separable-measurement protocols can be as efficient as the … Given a large number N of copies of a qubit state of which we wish to estimate its purity, we prove that separable-measurement protocols can be as efficient as the optimal joint-measurement one if classical communication is used. This shows that the optimal estimation of the entanglement of a two-qubit state can also be achieved asymptotically with fully separable measurements. Thus, quantum memories provide no advantage in this situation. The relationship between our global Bayesian approach and the quantum Cramér-Rao bound is discussed.
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Communication of spin directions with product states and finite measurements

2001-07-05
E. Bagán , M. Baig , R. Muñoz-Tapia
Total spin eigenstates can be used to intrinsically encode a direction, which can later be decoded by means of a quantum measurement. We study the optimal strategy that can be … Total spin eigenstates can be used to intrinsically encode a direction, which can later be decoded by means of a quantum measurement. We study the optimal strategy that can be adopted if, as is likely in practical applications, only product states of N spins are available. We obtain the asymptotic behavior of the average fidelity, which provides a proof that the optimal states must be entangled. We also give a prescription for constructing finite measurements for general encoding eigenstates.
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Infrared finite electron propagator

1997-09-15
E. Bagán , Martin Lavelle , David McMullan
We investigate the properties of a dressed electron which reduces, in a particular class of gauges, to the usual fermion. A one loop calculation of the propagator is presented. We … We investigate the properties of a dressed electron which reduces, in a particular class of gauges, to the usual fermion. A one loop calculation of the propagator is presented. We show explicitly that an infra-red finite, multiplicative, mass shell renormalisation is possible for this dressed electron, or, equivalently, for the usual fermion in the abovementioned gauges. The results are in complete accord with previous conjectures.
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Beating noise with abstention in state estimation

2012-10-16
B. Gendra , E. Ronco-Bonvehi , John Calsamiglia +2 more
We address the problem of estimating pure qubit states with non-ideal (noisy) measurements in the multiple-copy scenario, where the data consists of a number N of identically prepared qubits. We … We address the problem of estimating pure qubit states with non-ideal (noisy) measurements in the multiple-copy scenario, where the data consists of a number N of identically prepared qubits. We show that the average fidelity of the estimates can increase significantly if the estimation protocol allows for inconclusive answers, or abstentions. We present the optimal such protocol and compute its fidelity for a given probability of abstention. The improvement over standard estimation, without abstention, can be viewed as an effective noise reduction. These and other results are exemplified for small values of N. For asymptotically large N, we derive analytical expressions of the fidelity and the probability of abstention, and show that for a fixed fidelity gain the latter decreases with N at an exponential rate given by a Kulback-Leibler (relative) entropy. As a byproduct, we obtain an asymptotic expression in terms of this very entropy of the probability that a system of N qubits, all prepared in the same state, has a given total angular momentum. We also discuss an extreme situation where noise increases with N and where estimation with abstention provides a most significant improvement as compared to the standard approach.
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Effective electroweak chiral Lagrangian: The matter sector

1999-11-12
E. Bagán , D. Espriu , J. Manzano
We parametrize in a model-independent way possible departures from the minimal standard model predictions in the matter sector. We only assume the symmetry breaking pattern of the standard model and … We parametrize in a model-independent way possible departures from the minimal standard model predictions in the matter sector. We only assume the symmetry breaking pattern of the standard model and that new particles are sufficiently heavy so that the symmetry is nonlinearly realized. Models with dynamical symmetry breaking are generically of this type. We review in effective theory language to what extent the simplest models of dynamical breaking are actually constrained and the assumptions going into the comparison with experiment. Dynamical symmetry breaking models can be approximated at intermediate energies by four-fermion operators. We present a complete classification of the latter when new particles appear in the usual representations of the $\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(3)}_{c}$ group as well as a partial classification in the general case. We discuss the accuracy of the four-fermion description by matching to a simple ``fundamental'' theory. The coefficients of the effective Lagrangian in the matter sector for dynamical symmetry breaking models (expressed in terms of the coefficients of the four-quark operators) are then compared to those of models with elementary scalars (such as the minimal standard model). Contrary to a somewhat widespread belief, we see that the sign of the vertex corrections is not fixed in dynamical symmetry breaking models. This work provides the theoretical tools required to analyze, in a rather general setting, constraints on the matter sector of the standard model.
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Scavenging quantum information: Multiple observations of quantum systems

2011-09-19
Peter Rapčan , John Calsamiglia , R. Muñoz-Tapia +2 more
Given an unknown state of a qudit that has already been measured optimally, can one still extract any information about the original unknown state? Clearly, after a maximally informative measurement, … Given an unknown state of a qudit that has already been measured optimally, can one still extract any information about the original unknown state? Clearly, after a maximally informative measurement, the state of the system collapses into a postmeasurement state from which the same observer cannot obtain further information about the original state of the system. However, the system still encodes a significant amount of information about the original preparation for a second observer who is unaware of the actions of the first one. We study how a series of independent observers can obtain, or can scavenge, information about the unknown state of a system (quantified by the fidelity) when they sequentially measure it. We give closed-form expressions for the estimation fidelity when one or several qudits are available to carry information about the single-qudit state, and we study the classical limit when an arbitrarily large number of observers can obtain (nearly) complete information on the system. In addition to the case where all observers perform most informative measurements, we study the scenario where a finite number of observers estimates the state with equal fidelity, regardless of their position in the measurement sequence and the scenario where all observers use identical measurement apparatuses (up to a mutually unknown orientation) chosen so that a particular observer's estimation fidelity is maximized.
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Programmable discrimination with an error margin

2013-11-07
Gael Sentís , E. Bagán , John Calsamiglia +1 more
The problem of optimally discriminating between two completely unknown qubit states is generalized by allowing an error margin. It is visualized as a device---the programmable discriminator---with one data and two … The problem of optimally discriminating between two completely unknown qubit states is generalized by allowing an error margin. It is visualized as a device---the programmable discriminator---with one data and two program ports, each fed with a number of identically prepared qubits---the data and the programs. The device aims at correctly identifying the data state with one of the two program states. This scheme has the unambiguous and the minimum-error schemes as extremal cases, when the error margin is set to zero or it is sufficiently large, respectively. Analytical results are given in the two situations where the margin is imposed on the average error probability---weak condition---or it is imposed separately on the two probabilities of assigning the state of the data to the wrong program---strong condition. It is a general feature of our scheme that the success probability rises sharply as soon as a small error margin is allowed, thus providing a significant gain over the unambiguous scheme while still having high confidence results.
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Soft dynamics and gauge theories

1998-04-15
E. Bagán , Martin Lavelle , David McMullan
Infrared divergences obscure the underlying soft dynamics in gauge theories. They remove the pole structures associated with particle propagation in the various Green's functions of gauge theories. Here we present … Infrared divergences obscure the underlying soft dynamics in gauge theories. They remove the pole structures associated with particle propagation in the various Green's functions of gauge theories. Here we present a solution to this problem. We give two equations which describe how charged particles must be dressed by gauge degrees of freedom. One follows from gauge invariance, the other, which is new, from velocity superselection rules familiar from the heavy quark effective theory. The solution to these equations in the Abelian theory is proven to lead to on-shell Green's functions that are free of soft divergences at all orders in perturbation theory.
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Relative states, quantum axes, and quantum references

2006-02-24
E. Bagán , S. Iblisdir , R. Muñoz-Tapia
We address the problem of measuring the relative angle between two ``quantum axes'' made out of ${N}_{1}$ and ${N}_{2}$ spins. Closed forms of our fidelitylike figure of merit are obtained … We address the problem of measuring the relative angle between two ``quantum axes'' made out of ${N}_{1}$ and ${N}_{2}$ spins. Closed forms of our fidelitylike figure of merit are obtained for an arbitrary number of parallel spins. The asymptotic regimes of large ${N}_{1}$ and/or ${N}_{2}$ are discussed in detail. The extension of the concept ``quantum axis'' to more general situations is addressed. We give optimal strategies when the first quantum axis is made out of parallel spins whereas the second is a general state made out of two spins.
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Optimal parameter estimation with a fixed rate of abstention

2013-07-29
B. Gendra , E. Ronco-Bonvehi , John Calsamiglia +2 more
The problems of optimally estimating a phase, a direction, and the orientation of a Cartesian frame (or trihedron) with general pure states are addressed. Special emphasis is put on estimation … The problems of optimally estimating a phase, a direction, and the orientation of a Cartesian frame (or trihedron) with general pure states are addressed. Special emphasis is put on estimation schemes that allow for inconclusive answers or abstention. It is shown that such schemes enable drastic improvements, up to the extent of attaining the Heisenberg limit in some cases, and the required amount of abstention is quantified. A general mathematical framework to deal with the asymptotic limit of many qubits or large angular momentum is introduced and used to obtain analytical results for all the relevant cases under consideration. Parameter estimation with abstention is also formulated as a semidefinite programming problem, for which very efficient numerical optimization techniques exist.
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Wave-particle-duality relations based on entropic bounds for which-way information

2020-08-24
E. Bagán , János A. Bergou , Mark Hillery
We present wave-particle duality relations involving the relative entropy coherence measure, which plays a prominent role in the resource theory of coherence. The main input in these relations is an … We present wave-particle duality relations involving the relative entropy coherence measure, which plays a prominent role in the resource theory of coherence. The main input in these relations is an entropic bound for the which-way information, which we derive in this letter. We show that this latter crucially depends on the choice of the measurement strategy to obtain the path information. In particular, we present results for two strategies: zero-error identification of the path-detector states, which never produces an error but sometimes fails to return a conclusive answer, and a mixed strategy where both errors and failure are allowed.
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Phase variance of squeezed vacuum states

2008-10-31
E. Bagán , Alex Monràs , R. Muñoz-Tapia
We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number $N$ of … We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number $N$ of uncorrelated copies. We find that it scales with the mean photon number $n$, as dictated by the Heisenberg limit, i.e., as ${n}^{\ensuremath{-}2}$, only for $N>4$. For $N\ensuremath{\leqslant}4$ this fundamental scaling breaks down and it becomes ${n}^{\ensuremath{-}N∕2}$. Thus, a single squeezed vacuum state performs worse than a single coherent state with the same energy. We find the optimal splitting of a fixed given energy among various copies. We also compute the variance for repeated individual measurements (without classical communication or adaptivity) and find that the standard Heisenberg-limited scaling ${n}^{\ensuremath{-}2}$ is recovered for large samples.
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Renormalizability of the heavy quark effective theory

1994-01-01
E. Bagán , P. Gosdzinsky
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The structure of the QCD potential in 2+1 dimensions

2000-03-01
E. Bagán , Martin Lavelle , D. Michael McMullan
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Quantum Edge Detection

2024-05-18
Santiago Llorens , Walther González , Gael Sentís +3 more
This paper introduces quantum edge detection, aimed at locating boundaries of quantum domains where all particles share the same pure state. Focusing on the 1D scenario of a string of … This paper introduces quantum edge detection, aimed at locating boundaries of quantum domains where all particles share the same pure state. Focusing on the 1D scenario of a string of particles, we develop an optimal protocol for quantum edge detection, efficiently computing its success probability through Schur-Weyl duality and semidefinite programming techniques. We analyze the behavior of the success probability as a function of the string length and local dimension, with emphasis in the limit of long strings. We present a protocol based on square root measurement, which proves asymptotically optimal. Additionally, we explore a mixed quantum change point detection scenario where the state of particles transitions from known to unknown, which may find practical applications in detecting malfunctions in quantum devices
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Wave-particle-duality relations based on entropic bounds for which-way information

2020-08-24
E. Bagán , János A. Bergou , Mark Hillery
We present wave-particle duality relations involving the relative entropy coherence measure, which plays a prominent role in the resource theory of coherence. The main input in these relations is an … We present wave-particle duality relations involving the relative entropy coherence measure, which plays a prominent role in the resource theory of coherence. The main input in these relations is an entropic bound for the which-way information, which we derive in this letter. We show that this latter crucially depends on the choice of the measurement strategy to obtain the path information. In particular, we present results for two strategies: zero-error identification of the path-detector states, which never produces an error but sometimes fails to return a conclusive answer, and a mixed strategy where both errors and failure are allowed.
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Unsupervised Classification of Quantum Data

2019-11-08
Gael Sentís , Alex Monràs , R. Muñoz-Tapia +2 more
We introduce the problem of unsupervised classification of quantum data, namely, of systems whose quantum states are unknown. We derive the optimal single-shot protocol for the binary case, where the … We introduce the problem of unsupervised classification of quantum data, namely, of systems whose quantum states are unknown. We derive the optimal single-shot protocol for the binary case, where the states in a disordered input array are of two types. Our protocol is universal and able to automatically sort the input under minimal assumptions, yet partially preserving information contained in the states. We quantify analytically its performance for arbitrary size and dimension of the data. We contrast it with the performance of its classical counterpart, which clusters data that has been sampled from two unknown probability distributions. We find that the quantum protocol fully exploits the dimensionality of the quantum data to achieve a much higher performance, provided data is at least three-dimensional. For the sake of comparison, we discuss the optimal protocol when the classical and quantum states are known.
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A generalized wave-particle duality relation for finite groups

2018-04-03
E. Bagán , John Calsamiglia , János A. Bergou +1 more
Wave-particle duality relations express the fact that knowledge about the path a particle took suppresses information about its wave-like properties, in particular, its ability to generate an interference pattern. Recently, … Wave-particle duality relations express the fact that knowledge about the path a particle took suppresses information about its wave-like properties, in particular, its ability to generate an interference pattern. Recently, duality relations in which the wave-like properties are quantified by using measures of quantum coherence have been proposed. Quantum coherence can be generalized to a property called group asymmetry. Here we derive a generalized duality relation involving group asymmetry, which is closely related to the success probability of discriminating between the actions of the elements of a group. The second quantity in the duality relation, the one generalizing which-path information, is related to information about the irreducible representations that make up the group representation.
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Duality Games and Operational Duality Relations

2018-01-30
E. Bagán , John Calsamiglia , János A. Bergou +1 more
We give operational meaning to wave-particle duality in terms of discrimination games. Duality arises as a constraint on the probability of winning these games. The games are played with the … We give operational meaning to wave-particle duality in terms of discrimination games. Duality arises as a constraint on the probability of winning these games. The games are played with the aid of an n-port interferometer, and involve 3 parties, Alice and Bob, who cooperate, and the House, who supervises the game. In one game called ways they attempt to determine the path of a particle in the interferometer. In another, called phases, they attempt to determine which set of known phases have been applied to the different paths. The House determines which game is to be played by flipping a coin. We find a tight wave-particle duality relation that allows us to relate the probabilities of winning these games, and use it to find an upper bound on the probability of winning the combined game. This procedure allows us to express wave-particle duality in terms of discrimination probabilities.
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Discrimination Power of a Quantum Detector

2017-04-17
Christoph Hirche , Masahito Hayashi , E. Bagán +1 more
We investigate the ability of a quantum measurement device to discriminate two states or, generically, two hypotheses. In full generality, the measurement can be performed a number $n$ of times, … We investigate the ability of a quantum measurement device to discriminate two states or, generically, two hypotheses. In full generality, the measurement can be performed a number $n$ of times, and arbitrary preprocessing of the states and postprocessing of the obtained data are allowed. There is an intrinsic error associated with the measurement device, which we aim to quantify, that limits its discrimination power. We minimize various error probabilities (averaged or constrained) over all pairs of $n$-partite input states. These probabilities, or their exponential rates of decrease in the case of large $n$, give measures of the discrimination power of the device. For the asymptotic rate of the averaged error probability, we obtain a Chernoff-type bound, dual to the standard Chernoff bound for which the state pair is fixed and the optimization is over all measurements. The key point in the derivation is that identical copies of input states become optimal in asymptotic settings. Optimal asymptotic rates are also obtained for constrained error probabilities, dual to Stein's lemma and Hoeffding's bound. We further show that adaptive protocols where the state preparer gets feedback from the measurer do not improve the asymptotic rates. These rates thus quantify the ultimate discrimination power of a measurement device.
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Probabilistic metrology or how some measurement outcomes render ultra-precise estimates

2016-10-25
John Calsamiglia , B. Gendra , R. Muñoz-Tapia +1 more
We show on theoretical grounds that, even in the presence of noise, probabilistic measurement strategies (which have a certain probability of failure or abstention) can provide, upon a heralded successful … We show on theoretical grounds that, even in the presence of noise, probabilistic measurement strategies (which have a certain probability of failure or abstention) can provide, upon a heralded successful outcome, estimates with a precision that exceeds the deterministic bounds for the average precision. This establishes a new ultimate bound on the phase estimation precision of particular measurement outcomes (or sequence of outcomes). For probe systems subject to local dephasing, we quantify such precision limit as a function of the probability of failure that can be tolerated. Our results show that the possibility of abstaining can set back the detrimental effects of noise.
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Quantum Change Point

2016-10-07
Gael Sentís , E. Bagán , John Calsamiglia +2 more
Sudden changes are ubiquitous in nature. Identifying them is of crucial importance for a number of applications in medicine, biology, geophysics, and social sciences. Here we investigate the problem in … Sudden changes are ubiquitous in nature. Identifying them is of crucial importance for a number of applications in medicine, biology, geophysics, and social sciences. Here we investigate the problem in the quantum domain, considering a source that emits particles in a default state, until a point where it switches to another state. Given a sequence of particles emitted by the source, the problem is to find out where the change occurred. For large sequences, we obtain an analytical expression for the maximum probability of correctly identifying the change point when joint measurements on the whole sequence are allowed. We also construct strategies that measure the particles individually and provide an online answer as soon as a new particle is emitted by the source. We show that these strategies substantially underperform the optimal strategy, indicating that quantum sudden changes, although happening locally, are better detected globally.
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Probabilistically Perfect Cloning of Two Pure States: Geometric Approach

2016-05-20
V. Yerokhin , Andi Shehu , É. P. Feľdman +2 more
We solve the long-standing problem of making n perfect clones from m copies of one of two known pure states with minimum failure probability in the general case where the … We solve the long-standing problem of making n perfect clones from m copies of one of two known pure states with minimum failure probability in the general case where the known states have arbitrary a priori probabilities. The solution emerges from a geometric formulation of the problem. This formulation reveals that cloning converges to state discrimination followed by state preparation as the number of clones goes to infinity. The convergence exhibits a phenomenon analogous to a second-order symmetry-breaking phase transition.
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Relations between Coherence and Path Information

2016-04-22
E. Bagán , János A. Bergou , Seth Cottrell +1 more
We find two relations between coherence and path information in a multipath interferometer. The first builds on earlier results for the two-path interferometer, which used minimum-error state discrimination between detector … We find two relations between coherence and path information in a multipath interferometer. The first builds on earlier results for the two-path interferometer, which used minimum-error state discrimination between detector states to provide the path information. For visibility, which was used in the two-path case, we substitute a recently defined ${l}_{1}$ measure of quantum coherence. The second is an entropic relation in which the path information is characterized by the mutual information between the detector states and the outcome of the measurement performed on them, and the coherence measure is one based on relative entropy.
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Optimal Cloning of Quantum States with a Fixed Failure Rate

2016-01-01
E. Bagán , V. Yerokhin , Andi Shehu +2 more
Perfect cloning of a known set of states with arbitrary prior probabilities is possible if we allow the cloner to sometimes fail completely. In the optimal case the probability of … Perfect cloning of a known set of states with arbitrary prior probabilities is possible if we allow the cloner to sometimes fail completely. In the optimal case the probability of failure is at its minimum allowed by the laws of quantum mechanics. Here we show that it is possible to lower the failure rate below that of the perfect probabilistic cloner but the price to pay is that the clones are not perfect; the global fidelity is less than one. We determine the optimal fidelity of a cloner with a Fixed Failure Rate (FFR cloner) in the case of a pair of known states. Optimality is shown to be attainable by a measure-and-prepare protocol in the limit of infinitely many clones. The optimal protocol consists of discrimination with a fixed rate of inconclusive outcome followed by preparation of the appropriate clones. The convergence shows a symmetry-breaking second-order phase transition in the fidelity of the approximate infinite clones.

A geometric approach to quantum state separation

2015-12-10
E. Bagán , V. Yerokhin , Andi Shehu +2 more
Probabilistic quantum state transformations can be characterized by the degree of state separation they provide. This, in turn, sets limits on the success rate of these transformations. We consider optimum … Probabilistic quantum state transformations can be characterized by the degree of state separation they provide. This, in turn, sets limits on the success rate of these transformations. We consider optimum state separation of two known pure states in the general case where the known states have arbitrary a priori probabilities. The problem is formulated from a geometric perspective and shown to be equivalent to the problem of finding tangent curves within two families of conics that represent the unitarity constraints and the objective functions to be optimized, respectively. We present the corresponding analytical solutions in various forms. In the limit of perfect state separation, which is equivalent to unambiguous state discrimination, the solution exhibits a phenomenon analogous to a second order symmetry breaking phase transition. We also propose a linear optics implementation of separation which is based on the dual rail representation of qubits and single-photon multiport interferometry.
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Wave-particle duality in multi-path interferometers: A relation between quantum coherence and path information

2015-09-15
E. Bagán , János A. Bergou , Mark Hillery

Probabilistic Metrology Attains Macroscopic Cloning of Quantum Clocks

2014-12-31
B. Gendra , John Calsamiglia , R. Muñoz-Tapia +2 more
It has recently been shown that probabilistic protocols based on postselection boost the performances of the replication of quantum clocks and phase estimation. Here we demonstrate that the improvements in … It has recently been shown that probabilistic protocols based on postselection boost the performances of the replication of quantum clocks and phase estimation. Here we demonstrate that the improvements in these two tasks have to match exactly in the macroscopic limit where the number of clones grows to infinity, preserving the equivalence between asymptotic cloning and state estimation for arbitrary values of the success probability. Remarkably, the cloning fidelity depends critically on the number of rationally independent eigenvalues of the clock Hamiltonian. We also prove that probabilistic metrology can simulate cloning in the macroscopic limit for arbitrary sets of states when the performance of the simulation is measured by testing small groups of clones.
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Probabilistic metrology defeats ultimate deterministic bound

2014-01-01
John Calsamiglia , B. Gendra , R. Muñoz-Tapia +1 more
Quantum-enhanced measurements exploit quantum mechanical effects to provide ultra-precise estimates of physical variables for use in advanced technologies, such as frequency calibration of atomic clocks, gravitational waves detection, and biosensing. … Quantum-enhanced measurements exploit quantum mechanical effects to provide ultra-precise estimates of physical variables for use in advanced technologies, such as frequency calibration of atomic clocks, gravitational waves detection, and biosensing. Quantum metrology studies the fundamental limits in the estimation precision given a certain amount of resources (e.g. the number of probe systems) and restrictions (e.g. limited interaction time, or coping with unavoidable presence of noise). Here we show that, even in the presence of noise, probabilistic measurement strategies (which have a certain probability of failure or abstention) can provide, upon a heralded successful outcome, estimates with a precision that violates the deterministic bounds. This establishes a new ultimate quantum metrology limit. For probe systems subject to local dephasing, we quantify such precision limit as a function of the probability of failure that can be tolerated. We show that the possibility of abstaining can substantially set back the detrimental effects of noise.

Programmable discrimination with an error margin

2013-11-07
Gael Sentís , E. Bagán , John Calsamiglia +1 more
The problem of optimally discriminating between two completely unknown qubit states is generalized by allowing an error margin. It is visualized as a device---the programmable discriminator---with one data and two … The problem of optimally discriminating between two completely unknown qubit states is generalized by allowing an error margin. It is visualized as a device---the programmable discriminator---with one data and two program ports, each fed with a number of identically prepared qubits---the data and the programs. The device aims at correctly identifying the data state with one of the two program states. This scheme has the unambiguous and the minimum-error schemes as extremal cases, when the error margin is set to zero or it is sufficiently large, respectively. Analytical results are given in the two situations where the margin is imposed on the average error probability---weak condition---or it is imposed separately on the two probabilities of assigning the state of the data to the wrong program---strong condition. It is a general feature of our scheme that the success probability rises sharply as soon as a small error margin is allowed, thus providing a significant gain over the unambiguous scheme while still having high confidence results.
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Optimal parameter estimation with a fixed rate of abstention

2013-07-29
B. Gendra , E. Ronco-Bonvehi , John Calsamiglia +2 more
The problems of optimally estimating a phase, a direction, and the orientation of a Cartesian frame (or trihedron) with general pure states are addressed. Special emphasis is put on estimation … The problems of optimally estimating a phase, a direction, and the orientation of a Cartesian frame (or trihedron) with general pure states are addressed. Special emphasis is put on estimation schemes that allow for inconclusive answers or abstention. It is shown that such schemes enable drastic improvements, up to the extent of attaining the Heisenberg limit in some cases, and the required amount of abstention is quantified. A general mathematical framework to deal with the asymptotic limit of many qubits or large angular momentum is introduced and used to obtain analytical results for all the relevant cases under consideration. Parameter estimation with abstention is also formulated as a semidefinite programming problem, for which very efficient numerical optimization techniques exist.
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Quantum Metrology Assisted by Abstention

2013-03-04
B. Gendra , E. Ronco-Bonvehi , John Calsamiglia +2 more
The main goal of quantum metrology is to obtain accurate values of physical parameters using quantum probes. In this context, we show that abstention, i.e., the possibility of getting an … The main goal of quantum metrology is to obtain accurate values of physical parameters using quantum probes. In this context, we show that abstention, i.e., the possibility of getting an inconclusive answer at readout, can drastically improve the measurement precision and even lead to a change in its asymptotic behavior, from the shot-noise to the Heisenberg scaling. We focus on phase estimation and quantify the required amount of abstention for a given precision. We also develop analytical tools to obtain the asymptotic behavior of the precision and required rate of abstention for arbitrary pure states.
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Beating noise with abstention in state estimation

2012-10-16
B. Gendra , E. Ronco-Bonvehi , John Calsamiglia +2 more
We address the problem of estimating pure qubit states with non-ideal (noisy) measurements in the multiple-copy scenario, where the data consists of a number N of identically prepared qubits. We … We address the problem of estimating pure qubit states with non-ideal (noisy) measurements in the multiple-copy scenario, where the data consists of a number N of identically prepared qubits. We show that the average fidelity of the estimates can increase significantly if the estimation protocol allows for inconclusive answers, or abstentions. We present the optimal such protocol and compute its fidelity for a given probability of abstention. The improvement over standard estimation, without abstention, can be viewed as an effective noise reduction. These and other results are exemplified for small values of N. For asymptotically large N, we derive analytical expressions of the fidelity and the probability of abstention, and show that for a fixed fidelity gain the latter decreases with N at an exponential rate given by a Kulback-Leibler (relative) entropy. As a byproduct, we obtain an asymptotic expression in terms of this very entropy of the probability that a system of N qubits, all prepared in the same state, has a given total angular momentum. We also discuss an extreme situation where noise increases with N and where estimation with abstention provides a most significant improvement as compared to the standard approach.
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Optimal discrimination of quantum states with a fixed rate of inconclusive outcomes

2012-10-15
E. Bagán , R. Muñoz-Tapia , G. A. Olivares-Rentería +1 more
In this paper we present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate … In this paper we present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between Unambiguous and Minimum Error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function of the rate of inconclusive answers. A critical value of this rate is identified that coincides with the minimum failure probability in the cases where unambiguous discrimination is possible and provides a natural generalization of it when states cannot be unambiguously discriminated. The method is illustrated on two explicit examples: discrimination of two pure states with arbitrary prior probabilities and discrimination of trine states.
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Quantum learning without quantum memory

2012-10-05
Gael Sentís , John Calsamiglia , R. Muñoz-Tapia +1 more
A quantum learning machine for binary classification of qubit states that does not require quantum memory is introduced and shown to perform with the minimum error rate allowed by quantum … A quantum learning machine for binary classification of qubit states that does not require quantum memory is introduced and shown to perform with the minimum error rate allowed by quantum mechanics for any size of the training set. This result is shown to be robust under (an arbitrary amount of) noise and under (statistical) variations in the composition of the training set, provided it is large enough. This machine can be used an arbitrary number of times without retraining. Its required classical memory grows only logarithmically with the number of training qubits, while its excess risk decreases as the inverse of this number and twice as fast as the excess risk of an "estimate-and-discriminate" machine, which estimates the states of the training qubits and classifies the data qubit with a discrimination protocol tailored to the obtained estimates.
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Robust optimal quantum learning without quantum memory

2012-01-01
Gael Sentís , John Calsamiglia , R. Muñoz-Tapia +1 more
A quantum learning machine for binary classification of qubit states that does not require quantum memory is introduced and shown to perform with the minimum error rate allowed by quantum … A quantum learning machine for binary classification of qubit states that does not require quantum memory is introduced and shown to perform with the minimum error rate allowed by quantum mechanics for any size of the training set. This result is shown to be robust under (an arbitrary amount of) noise and under (statistical) variations in the composition of the training set, provided it is large enough. This machine can be used an arbitrary number of times without retraining. Its required classical memory grows only logarithmically with the number of training qubits, while its excess risk decreases as the inverse of this number, and twice as fast as the excess risk of an estimate-and-discriminate machine, which estimates the states of the training qubits and classifies the data qubit with a discrimination protocol tailored to the obtained estimates.
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Scavenging quantum information: Multiple observations of quantum systems

2011-09-19
Peter Rapčan , John Calsamiglia , R. Muñoz-Tapia +2 more
Given an unknown state of a qudit that has already been measured optimally, can one still extract any information about the original unknown state? Clearly, after a maximally informative measurement, … Given an unknown state of a qudit that has already been measured optimally, can one still extract any information about the original unknown state? Clearly, after a maximally informative measurement, the state of the system collapses into a postmeasurement state from which the same observer cannot obtain further information about the original state of the system. However, the system still encodes a significant amount of information about the original preparation for a second observer who is unaware of the actions of the first one. We study how a series of independent observers can obtain, or can scavenge, information about the unknown state of a system (quantified by the fidelity) when they sequentially measure it. We give closed-form expressions for the estimation fidelity when one or several qudits are available to carry information about the single-qudit state, and we study the classical limit when an arbitrarily large number of observers can obtain (nearly) complete information on the system. In addition to the case where all observers perform most informative measurements, we study the scenario where a finite number of observers estimates the state with equal fidelity, regardless of their position in the measurement sequence and the scenario where all observers use identical measurement apparatuses (up to a mutually unknown orientation) chosen so that a particular observer's estimation fidelity is maximized.
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Optimal signal states for quantum detectors

2011-07-22
Ognyan Oreshkov , John Calsamiglia , R. Muñoz-Tapia +1 more
Quantum detectors provide information about quantum systems by establishing correlations between certain properties of those systems and a set of macroscopically distinct states of the corresponding measurement devices. A natural … Quantum detectors provide information about quantum systems by establishing correlations between certain properties of those systems and a set of macroscopically distinct states of the corresponding measurement devices. A natural question of fundamental significance is how much information a quantum detector can extract from the quantum system it is applied to. In the present paper we address this question within a precise framework: given a quantum detector implementing a specific generalized quantum measurement, what is the optimal performance achievable with it for a concrete information readout task, and what is the optimal way to encode information in the quantum system in order to achieve this performance? We consider some of the most common information transmission tasks - the Bayes cost problem (of which minimal error discrimination is a special case), unambiguous message discrimination, and the maximal mutual information. We provide general solutions to the Bayesian and unambiguous discrimination problems. We also show that the maximal mutual information has an interpretation of a capacity of the measurement, and derive various properties that it satisfies, including its relation to the accessible information of an ensemble of states, and its form in the case of a group-covariant measurement. We illustrate our results with the example of a noisy two-level symmetric informationally complete measurement, for whose capacity we give analytical proofs of optimality. The framework presented here provides a natural way to characterize generalized quantum measurements in terms of their information readout capabilities.
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Quantum learning without quantum memory

2011-01-01
Gael Sentís , John Calsamiglia , R. Muñoz-Tapia +1 more
A quantum learning machine for binary classification of qubit states that does not require quantum memory is introduced and shown to perform with the very same error rate as the … A quantum learning machine for binary classification of qubit states that does not require quantum memory is introduced and shown to perform with the very same error rate as the optimal (programmable) discrimination machine for any size of the training set. At variance with the latter, this machine can be used an arbitrary number of times without retraining. Its required (classical) memory grows only logarithmically with the number of training qubits, while (asymptotically) its excess risk decreases as the inverse of this number, and twice as fast as the excess risk of an "estimate-and-discriminate" machine, which estimates the states of the training qubits and classifies the data qubit with a discrimination protocol tailored to the obtained estimates.
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Multicopy programmable discrimination of general qubit states

2010-10-15
Gael Sentís , E. Bagán , John Calsamiglia +1 more
Quantum state discrimination is a fundamental primitive in quantum statistics where one has to correctly identify the state of a system that is in one of two possible known states. … Quantum state discrimination is a fundamental primitive in quantum statistics where one has to correctly identify the state of a system that is in one of two possible known states. A programmable discrimination machine performs this task when the pair of possible states is not a priori known but instead the two possible states are provided through two respective program ports. We study optimal programmable discrimination machines for general qubit states when several copies of states are available in the data or program ports. Two scenarios are considered: One in which the purity of the possible states is a priori known, and the fully universal one where the machine operates over generic mixed states of unknown purity. We find analytical results for both the unambiguous and minimum error discrimination strategies. This allows us to calculate the asymptotic performance of programmable discrimination machines when a large number of copies are provided and to recover the standard state discrimination and state comparison values as different limiting cases.
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Recycling of qubits

2010-09-01
Peter Rapčan , John Calsamiglia , R. Muñoz-Tapia +2 more
Given a finite number of copies of an unknown qubit state that have already been measured optimally, can one still extract any information about the original unknown state?We give a … Given a finite number of copies of an unknown qubit state that have already been measured optimally, can one still extract any information about the original unknown state?We give a positive answer to this question and quantify the information obtainable by a given observer as a function of the number of copies in the ensemble, and of the number of independent observers that, one after the other, have independently measured the same ensemble of qubits before him.The optimality of the protocol is proven and extensions to other states and encodings are also studied.According to the general lore, the state after a measurement has no information about the state before the measurement.Our results manifestly show that this statement has to be taken with a grain of salt, specially in situations where the quantum states encode confidential information.
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Local Discrimination of Mixed States

2010-08-18
John Calsamiglia , Julio I. de Vicente , R. Muñoz-Tapia +1 more
We provide rigorous, efficiently computable and tight bounds on the average error probability of multiple-copy discrimination between qubit mixed states by local operations assisted with classical communication (LOCC). In contrast … We provide rigorous, efficiently computable and tight bounds on the average error probability of multiple-copy discrimination between qubit mixed states by local operations assisted with classical communication (LOCC). In contrast with the pure-state case, these experimentally feasible protocols perform strictly worse than the general collective ones. Our numerical results indicate that the gap between LOCC and collective error rates persists in the asymptotic limit. In order for LOCC and collective protocols to achieve the same accuracy, the former can require up to twice the number of copies of the latter. Our techniques can be used to bound the power of LOCC strategies in other similar settings, which is still one of the most elusive questions in quantum communication.
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Estimation of quantum finite mixtures

2010-01-29
Julio I. de Vicente , John Calsamiglia , R. Muñoz-Tapia +1 more
We consider the problem of determining the weights of a quantum ensemble. That is to say, given a quantum system that is in a set of possible known states according … We consider the problem of determining the weights of a quantum ensemble. That is to say, given a quantum system that is in a set of possible known states according to an unknown probability law, we give strategies to estimate the individual probabilities, weights, or mixing proportions. Such strategies can be used to estimate the frequencies at which different independent signals are emitted by a source. They can also be used to estimate the weights of particular terms in a canonical decomposition of a quantum channel. The quality of these strategies is quantified by a covariance-type error matrix. According with this cost function, we give optimal strategies in both the single-shot and multiple-copy scenarios. The latter is also analyzed in the asymptotic limit of large number of copies. We give closed expressions of the error matrix for two-component quantum mixtures of qubit systems. The Fisher information plays an unusual role in the problem at hand, providing exact expressions of the minimum covariance matrix for any number of copies.
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Phase-covariant quantum benchmarks

2009-05-08
John Calsamiglia , M. Aspachs , R. Muñoz-Tapia +1 more
We give a quantum benchmark for teleportation and quantum storage experiments suited for pure and mixed test states. The benchmark is based on the average fidelity over a family of … We give a quantum benchmark for teleportation and quantum storage experiments suited for pure and mixed test states. The benchmark is based on the average fidelity over a family of phase-covariant states and certifies that an experiment cannot be emulated by a classical setup, i.e., by a measure-and-prepare scheme. We give an analytical solution for qubits, which shows important differences with standard state estimation approach, and compute the value of the benchmark for coherent and squeezed states, both pure and mixed.
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Phase estimation for thermal Gaussian states

2009-03-23
M. Aspachs , John Calsamiglia , R. Muñoz-Tapia +1 more
We give the optimal bounds on the phase-estimation precision for mixed Gaussian states in the single-copy and many-copy regimes. Specifically, we focus on displaced thermal and squeezed thermal states. We … We give the optimal bounds on the phase-estimation precision for mixed Gaussian states in the single-copy and many-copy regimes. Specifically, we focus on displaced thermal and squeezed thermal states. We find that while for displaced thermal states an increase in temperature reduces the estimation fidelity, for squeezed thermal states a larger temperature can enhance the estimation fidelity. The many-copy optimal bounds are compared with the minimum variance achieved by three important single-shot measurement strategies. We show that the single-copy canonical phase measurement does not always attain the optimal bounds in the many-copy scenario. Adaptive homodyning schemes do attain the bounds for displaced thermal states, but for squeezed states they yield fidelities that are insensitive to temperature variations and are, therefore, suboptimal. Finally, we find that heterodyne measurements perform very poorly for pure states but can attain the optimal bounds for sufficiently mixed states. We apply our results to investigate the influence of losses in an optical metrology experiment. In the presence of losses squeezed states cease to provide the Heisenberg limited precision, and their performance is close to that of coherent states with the same mean photon number.
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Phase variance of squeezed vacuum states

2008-10-31
E. Bagán , Alex Monràs , R. Muñoz-Tapia
We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number $N$ of … We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number $N$ of uncorrelated copies. We find that it scales with the mean photon number $n$, as dictated by the Heisenberg limit, i.e., as ${n}^{\ensuremath{-}2}$, only for $N>4$. For $N\ensuremath{\leqslant}4$ this fundamental scaling breaks down and it becomes ${n}^{\ensuremath{-}N∕2}$. Thus, a single squeezed vacuum state performs worse than a single coherent state with the same energy. We find the optimal splitting of a fixed given energy among various copies. We also compute the variance for repeated individual measurements (without classical communication or adaptivity) and find that the standard Heisenberg-limited scaling ${n}^{\ensuremath{-}2}$ is recovered for large samples.
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Quantum Chernoff bound as a measure of distinguishability between density matrices: Application to qubit and Gaussian states

2008-03-07
John Calsamiglia , R. Muñoz-Tapia , Lluís Masanes +2 more
Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability … Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability when discriminating two hypotheses given a large number of observations. Recently the combined work of Audenaert et al. [Phys. Rev. Lett. 98, 160501 (2007)] and Nussbaum and Szkola [e-print arXiv:quant-ph/0607216] has proved the quantum analog of this bound, which applies when the hypotheses correspond to two quantum states. Based on this quantum Chernoff bound, we define a physically meaningful distinguishability measure and its corresponding metric in the space of states; the latter is shown to coincide with the Wigner-Yanase metric. Along the same lines, we define a second, more easily implementable, distinguishability measure based on the error probability of discrimination when the same local measurement is performed on every copy. We study some general properties of these measures, including the probability distribution of density matrices, defined via the volume element induced by the metric. It is shown that the Bures and the local-measurement based metrics are always proportional. Finally, we illustrate their use in the paradigmatic cases of qubits and Gaussian infinite-dimensional states.
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Quantum-limited metrology with product states

2008-01-15
Sergio Boixo , Animesh Datta , Steven T. Flammia +3 more
We study the performance of initial product states of $n$-body systems in generalized quantum metrology protocols that involve estimating an unknown coupling constant in a nonlinear $k$-body $(k\ensuremath{\ll}n)$ Hamiltonian. We … We study the performance of initial product states of $n$-body systems in generalized quantum metrology protocols that involve estimating an unknown coupling constant in a nonlinear $k$-body $(k\ensuremath{\ll}n)$ Hamiltonian. We obtain the theoretical lower bound on the uncertainty in the estimate of the parameter. For arbitrary initial states, the lower bound scales as $1/{n}^{k}$, and for initial product states, it scales as $1/{n}^{k\ensuremath{-}1/2}$. We show that the latter scaling can be achieved using simple, separable measurements. We analyze in detail the case of a quadratic Hamiltonian $(k=2)$, implementable with Bose-Einstein condensates. We formulate a simple model, based on the evolution of angular-momentum coherent states, which explains the $O({n}^{\ensuremath{-}3/2})$ scaling for $k=2$; the model shows that the entanglement generated by the quadratic Hamiltonian does not play a role in the enhanced sensitivity scaling. We show that phase decoherence does not affect the $O({n}^{\ensuremath{-}3/2})$ sensitivity scaling for initial product states.
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Recycling of quantum information: Multiple observations of quantum systems

2007-08-08
Peter Rapčan , John Calsamiglia , R. Muñoz-Tapia +2 more
Given a finite number of copies of an unknown qubit state that have already been measured optimally, can one still extract any information about the original unknown state? We give … Given a finite number of copies of an unknown qubit state that have already been measured optimally, can one still extract any information about the original unknown state? We give a positive answer to this question and quantify the information obtainable by a given observer as a function of the number of copies in the ensemble, and of the number of independent observers that, one after the other, have independently measured the same ensemble of qubits before him. The optimality of the protocol is proven and extensions to other states and encodings are also studied. According to the general lore, the state after a measurement has no information about the state before the measurement. Our results manifestly show that this statement has to be taken with a grain of salt, specially in situations where the quantum states encode confidential information.
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How to hide a secret direction

2007-08-01
E. Bagán , John Calsamiglia , Rafał Demkowicz-Dobrzański +1 more
We present a procedure that uses a multipartite quantum state to communicate a secret spatial direction when there is no shared reference frame between the preparer and the group of … We present a procedure that uses a multipartite quantum state to communicate a secret spatial direction when there is no shared reference frame between the preparer and the group of recipients. The procedure guarantees that the recipients can determine the direction if they perform joint measurements on the state, but fail to do so if they restrict themselves to local operations and classical communication (LOCC). We calculate the fidelity for joint measurements, give bounds on the fidelity achievable by LOCC, and prove that there is a non-vanishing gap between the two of them, even in the limit of infinitely many copies. The robustness of the procedure under particle loss is also studied. Additionally, we find bounds on the probability of discriminating by LOCC between the invariant subspaces of total angular momentum N/2 and N/2−1 in a system of N elementary spins.
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Discriminating States: The Quantum Chernoff Bound

2007-04-17
Katrien Audenaert , John Calsamiglia , R. Muñoz-Tapia +4 more
We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the minimal probability of error. This leads to the identification of … We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the minimal probability of error. This leads to the identification of the quantum Chernoff bound, thereby solving a long-standing open problem. The bound reduces to the classical Chernoff bound when the quantum states under consideration commute. The quantum Chernoff bound is the natural symmetric distance measure between quantum states because of its clear operational meaning and because it does not seem to share some of the undesirable features of other distance measures.
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Separable Measurement Estimation of Density Matrices and its Fidelity Gap with Collective Protocols

2006-09-25
E. Bagán , Manuel A. Ballester , Richard D. Gill +2 more
We show that there exists a gap between the performance of separable and collective measurements in qubit mixed-state estimation that persists in the large sample limit. We characterize such gap … We show that there exists a gap between the performance of separable and collective measurements in qubit mixed-state estimation that persists in the large sample limit. We characterize such gap in terms of the corresponding bounds on the mean fidelity. We present an adaptive protocol that attains the separable-measurement bound. This (optimal separable) protocol uses von Neumann measurements and can be easily implemented with current technology.
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Optimal full estimation of qubit mixed states

2006-03-02
E. Bagán , Manuel A. Ballester , Richard D. Gill +2 more
We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the … We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary N, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large N. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.
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Relative states, quantum axes, and quantum references

2006-02-24
E. Bagán , S. Iblisdir , R. Muñoz-Tapia
We address the problem of measuring the relative angle between two ``quantum axes'' made out of ${N}_{1}$ and ${N}_{2}$ spins. Closed forms of our fidelitylike figure of merit are obtained … We address the problem of measuring the relative angle between two ``quantum axes'' made out of ${N}_{1}$ and ${N}_{2}$ spins. Closed forms of our fidelitylike figure of merit are obtained for an arbitrary number of parallel spins. The asymptotic regimes of large ${N}_{1}$ and/or ${N}_{2}$ are discussed in detail. The extension of the concept ``quantum axis'' to more general situations is addressed. We give optimal strategies when the first quantum axis is made out of parallel spins whereas the second is a general state made out of two spins.
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ALIGNING SPATIAL FRAMES THROUGH QUANTUM CHANNELS

2006-02-01
E. Bagán , R. Muñoz-Tapia
We review the optimal protocols for aligning spatial frames using quantum systems. The communication problem addressed here concerns a type of information that cannot be digitalized. Asher Peres referred to … We review the optimal protocols for aligning spatial frames using quantum systems. The communication problem addressed here concerns a type of information that cannot be digitalized. Asher Peres referred to it as "unspeakable information." We comment on his contribution to this subject and give a brief account of his scientific interaction with the authors.
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Secrecy properties of quantum channels

2006-01-20
Antonio Acín , Joonwoo Bae , E. Bagán +3 more
We identify those properties of a quantum channel that are relevant for cryptography. We focus on general key distribution protocols that use prepare and measure schemes and the existing classical … We identify those properties of a quantum channel that are relevant for cryptography. We focus on general key distribution protocols that use prepare and measure schemes and the existing classical reconciliation techniques, as these are the protocols feasible with current technology. Given a channel, we derive an easily computable necessary condition of security for such protocols. In spite of its simplicity, this condition is shown to be tight for the Bennett-Brassard 1984 and six-state protocols. We show that the condition becomes also sufficient in the event of a so-called collective attack.
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Optimal discrimination of mixed states: the quantum Chernoff bound

2006-01-01
John Calsamiglia , Lluís Masanes , R. Muñoz-Tapia +2 more
This paper has been withdrawn by the authors, due to a flaw in the proof of Theorem 1. This preprint is superseded by quant-ph/0610027, where a correct proof can be … This paper has been withdrawn by the authors, due to a flaw in the proof of Theorem 1. This preprint is superseded by quant-ph/0610027, where a correct proof can be found. Thanks to Rainer Siegmund-Schultze for spotting the error.

Optimal estimation of qubit mixed states with local measurements

2005-12-21
Oriol Romero‐Isart , E. Bagán , Manuel A. Ballester +2 more
We show that there exists a gap between the performance of separable and collective measurements in qubit mixed-state estimation that persists in the large sample limit. We characterize such gap … We show that there exists a gap between the performance of separable and collective measurements in qubit mixed-state estimation that persists in the large sample limit. We characterize such gap in terms of the corresponding bounds on the mean fidelity. We present an adaptive protocol that attains the separable-measurement bound. This (optimal separable) protocol uses von Neumann measurements and can be easily implemented with current technology.

Anti-screening by quarks and the structure of the inter-quark potential

2005-11-23
E. Bagán , Martin Lavelle , David McMullan
The inter-quark potential is dominated by anti-screening effects which underly asymptotic freedom. We calculate the order g^6 anti-screening contribution from light fermions and demonstrate that these effects introduce a non-local … The inter-quark potential is dominated by anti-screening effects which underly asymptotic freedom. We calculate the order g^6 anti-screening contribution from light fermions and demonstrate that these effects introduce a non-local divergence. These divergences are shown to make it impossible to define a coupling renormalisation scheme that renormalises this minimal, anti-screening potential. Hence the beta function cannot be divided into screening and anti-screening parts beyond lowest order. However, we then demonstrate that renormalisation can be carried out in terms of the anti-screening potential.
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Purity Estimation with Separable Measurements

2005-09-09
E. Bagán , Manuel A. Ballester , R. Muñoz-Tapia +1 more
Given a large number N of copies of a qubit state of which we wish to estimate its purity, we prove that separable-measurement protocols can be as efficient as the … Given a large number N of copies of a qubit state of which we wish to estimate its purity, we prove that separable-measurement protocols can be as efficient as the optimal joint-measurement one if classical communication is used. This shows that the optimal estimation of the entanglement of a two-qubit state can also be achieved asymptotically with fully separable measurements. Thus, quantum memories provide no advantage in this situation. The relationship between our global Bayesian approach and the quantum Cramér-Rao bound is discussed.
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Comprehensive analysis of quantum pure-state estimation for two-level systems

2005-06-15
E. Bagán , Alex Monràs , R. Muñoz-Tapia
We analyze the estimation of a qubit pure state by means of local measurements on $N$ identical copies and compare its averaged fidelity for an isotropic prior probability distribution to … We analyze the estimation of a qubit pure state by means of local measurements on $N$ identical copies and compare its averaged fidelity for an isotropic prior probability distribution to the absolute upper bound given by collective measurements. We discuss two situations: the first one, where the state is restricted to lie on the equator of the Bloch sphere, is formally equivalent to phase estimation; the second one, where there is no constrain on the state, can also be regarded as the estimation of a direction in space using a quantum arrow made out of $N$ parallel spins. We discuss various schemes with and without classical communication and compare their efficiency. We show that the fidelity of the most general collective measurement can always be achieved asymptotically with local measurements and no classical communication.
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Multiple-copy two-state discrimination with individual measurements

2005-03-23
Antonio Acín , E. Bagán , M. Baig +2 more
We address the problem of non-orthogonal two-state discrimination when multiple copies of the unknown state are available. We give the optimal strategy when only fixed individual measurements are allowed and … We address the problem of non-orthogonal two-state discrimination when multiple copies of the unknown state are available. We give the optimal strategy when only fixed individual measurements are allowed and show that its error probability saturates the collective (lower) bound asymptotically. We also give the optimal strategy when adaptivity of individual von Neumann measurements is allowed (which requires classical communication), and show that the corresponding error probability is exactly equal to the collective one for any number of copies. We show that this strategy can be regarded as Bayesian updating.
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A Quantum Field Theory Twist to Photon Angular Momentum

2005-01-01
E. Bagán
A quantum field theory approach is put forward to generalize the concept of classical spatial light beams carrying orbital angular momentum to the single-photon level. This quantization framework is carried … A quantum field theory approach is put forward to generalize the concept of classical spatial light beams carrying orbital angular momentum to the single-photon level. This quantization framework is carried out both in the paraxial and nonparaxial regimes. Upon extension to the optical phase space, closed-form expressions are found for a photon Wigner representation describing transformations on the orbital Poincare sphere of unitarily related families of paraxial spatial modes.

Optimal full estimation of qubit mixed states

2005-01-01
E. Bagán , MA Ballester Sánchez , RD Richard Gill
We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the … We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary N , we give a prior-independent scheme which becomes optimal in the asymptotic limit of large N . We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.
Coauthor Papers Together
R. Muñoz-Tapia 52
John Calsamiglia 29
Martin Lavelle 22
David McMullan 20
M. Baig 15
János A. Bergou 9
Gael Sentís 8
B. Gendra 6
Lluís Masanes 6
P. Gosdzinsky 6
Antonio Acín 6
Bartomeu Fiol 5
Robin Horan 5
Alex Monràs 5
Mark Hillery 5
Manuel A. Ballester 4
V. Yerokhin 3
V. M. Braun 3
Andi Shehu 3
Peter Rapčan 3
Nandan Roy 3
Richard D. Gill 3
Oriol Romero‐Isart 3
Vladimír Bužek 3
E. Ronco-Bonvehi 3
Patricia Ball 3
D. Michael McMullan 2
Julio I. de Vicente 2
A. Brey 2
M. Aspachs 2
Patricia Ball 2
R. Tarrach 2
Giulio Chiribella 2
É. P. Feľdman 1
Ognyan Oreshkov 1
D. Espriu 1
S. Iblisdir 1
Katrien Audenaert 1
Jean-Marc Richard 1
Animesh Datta 1
Sergio Boixo 1
Eugene Feldman 1
Stéphan Narison 1
MA Ballester Sánchez 1
Joonwoo Bae 1
Anil Shaji 1
Carlton M. Caves 1
Alberto Rodríguez 1
Edgar Feldman 1
RD Richard Gill 1

Optimal Strategies for Sending Information through A Quantum Channel

2000-12-11
E. Bagán , M. Baig , A. Brey +2 more
Quantum states can be used to encode the information contained in a direction, i.e., in a unit vector. We present the best encoding procedure when the quantum state is made … Quantum states can be used to encode the information contained in a direction, i.e., in a unit vector. We present the best encoding procedure when the quantum state is made up of N spins (qubits). We find that the quality of this optimal procedure, which we quantify in terms of the fidelity, depends solely on the dimension of the encoding space. We also investigate the use of spatial rotations on a quantum state, which provide a natural and less demanding encoding. In this case we prove that the fidelity is directly related to the largest zeros of the Legendre and Jacobi polynomials. We also discuss our results in terms of the information gain.
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Universal Algorithm for Optimal Estimation of Quantum States from Finite Ensembles via Realizable Generalized Measurement

1998-02-23
R. Derka , Vladimír Bužek , Artur Ekert
We present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system. The algorithm specifies a physically realizable (i.e., finite) positive operator valued measurement on … We present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system. The algorithm specifies a physically realizable (i.e., finite) positive operator valued measurement on a finite number of identically prepared systems. We illustrate the general formalism by applying it to different scenarios of the state estimation of $N$ independent and identically prepared two-level systems (qubits).
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Spin Flips and Quantum Information for Antiparallel Spins

1999-07-12
Nicolas Gisin , Sandu Popescu
We consider two different ways to encode quantum information, by parallel or antiparallel pairs of spins. We find that there is more information in the antiparallel ones. This purely quantum … We consider two different ways to encode quantum information, by parallel or antiparallel pairs of spins. We find that there is more information in the antiparallel ones. This purely quantum mechanical effect is due to entanglement, not of the states but occurring in the course of the measuring process. We also introduce a range of quantum information processing machines, such as spin flip and anticloning.
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Optimal encoding and decoding of a spin direction

2001-04-17
E. Bagán , M. Baig , A. Brey +2 more
For a system of N spins 1/2 there are quantum states that can encode a direction in an intrinsic way. Information on this direction can later be decoded by means … For a system of N spins 1/2 there are quantum states that can encode a direction in an intrinsic way. Information on this direction can later be decoded by means of a quantum measurement. We present here the optimal encoding and decoding procedure using the fidelity as a figure of merit. We compute the maximal fidelity and prove that it is directly related to the largest zeroes of the Legendre and Jacobi polynomials. We show that this maximal fidelity approaches unity quadratically in 1/N. We also discuss this result in terms of the dimension of the encoding Hilbert space.
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State estimation for large ensembles

2000-03-17
Richard D. Gill , Serge Massar
We consider the problem of estimating the state of a large but finite number N of identical quantum systems. As N becomes large the problem simplifies dramatically. The only relevant … We consider the problem of estimating the state of a large but finite number N of identical quantum systems. As N becomes large the problem simplifies dramatically. The only relevant measure of the quality of estimation becomes the mean quadratic error matrix. Here we present a bound on this quantity: a quantum Cram\'er-Rao inequality. This bound succinctly expresses how in the quantum case one can trade information about one parameter for information about another. The bound holds for arbitrary measurements on pure states, but only for separable measurements on mixed states---a striking example of nonlocality without entanglement for mixed but not for pure states. Cram\'er-Rao bounds are generally only derived for unbiased estimators. Here we give a version of our bound for biased estimators, and a simple asymptotic version for large N. Finally we prove that when the unknown state belongs to a two-dimensional Hilbert space our quantum Cram\'er-Rao bound can always be attained, and we provide an explicit measurement strategy that attains it. Thus we have a complete solution to the problem of estimating as efficiently as possible the unknown state of a large ensemble of qubits in the same pure state. The same is true for qubits in the same mixed state if one restricts oneself to separable measurements, but nonseparable measurements allow dramatic increase of efficiency. Exactly how much increase is possible is a major open problem.
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Aligning Reference Frames with Quantum States

2001-11-01
E. Bagán , M. Baig , R. Muñoz-Tapia
We analyze the problem of sending, in a single transmission, the information required to specify an orthogonal trihedron or reference frame through a quantum channel made out of N elementary … We analyze the problem of sending, in a single transmission, the information required to specify an orthogonal trihedron or reference frame through a quantum channel made out of N elementary spins. We analytically obtain the optimal strategy, i.e., the best encoding state and the best measurement. For large N, we show that the average error goes to zero linearly in 1/N. Finally, we discuss the construction of finite optimal measurements.
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Statistical distance and the geometry of quantum states

1994-05-30
Samuel L. Braunstein , Carlton M. Caves
By finding measurements that optimally resolve neighboring quantum states, we use statistical distinguishability to define a natural Riemannian metric on the space of quantum-mechanical density operators and to formulate uncertainty … By finding measurements that optimally resolve neighboring quantum states, we use statistical distinguishability to define a natural Riemannian metric on the space of quantum-mechanical density operators and to formulate uncertainty principles that are more general and more stringent than standard uncertainty principles.

Entangled Quantum States as Direction Indicators

2001-04-30
Asher Peres , Petra F. Scudo
We consider the use of N spin-1/2 particles for indicating a direction in space. If N>2, their optimal state is entangled. For large N, the mean square error decreases as … We consider the use of N spin-1/2 particles for indicating a direction in space. If N>2, their optimal state is entangled. For large N, the mean square error decreases as N-2 (rather than N-1 for parallel spins).
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Minimal Optimal Generalized Quantum Measurements

1998-08-17
José I. Latorre , P. Pascual , R. Tarrach
Optimal and finite positive operator valued measurements on a finite number $N$ of identically prepared systems have recently been presented. With physical realization in mind, we propose here optimal and … Optimal and finite positive operator valued measurements on a finite number $N$ of identically prepared systems have recently been presented. With physical realization in mind, we propose here optimal and minimal generalized quantum measurements for two-level systems. We explicitly construct them up to $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}7$ and verify that they are minimal up to $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5$.
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Optimal Scheme for Estimating a Pure Qubit State via Local Measurements

2002-12-19
E. Bagán , M. Baig , R. Muñoz-Tapia
We present the optimal scheme for estimating a pure qubit state by means of local measurements on N identical copies. We give explicit examples for low N. For large N, … We present the optimal scheme for estimating a pure qubit state by means of local measurements on N identical copies. We give explicit examples for low N. For large N, we show that the fidelity saturates the collective measurement bound up to order 1/N. When the signal state lays on a meridian of the Bloch sphere, we show that this can be achieved without classical communication.
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Transmission of a Cartesian Frame by a Quantum System

2001-09-26
Asher Peres , Petra F. Scudo
A single quantum system, such as a hydrogen atom, can transmit a Cartesian coordinate frame (three axes). For this it has to be prepared in a superposition of states belonging … A single quantum system, such as a hydrogen atom, can transmit a Cartesian coordinate frame (three axes). For this it has to be prepared in a superposition of states belonging to different irreducible representations of the rotation group. The algorithm for decoding such a state is presented, and the fidelity of transmission is evaluated.
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Comprehensive analysis of quantum pure-state estimation for two-level systems

2005-06-15
E. Bagán , Alex Monràs , R. Muñoz-Tapia
We analyze the estimation of a qubit pure state by means of local measurements on $N$ identical copies and compare its averaged fidelity for an isotropic prior probability distribution to … We analyze the estimation of a qubit pure state by means of local measurements on $N$ identical copies and compare its averaged fidelity for an isotropic prior probability distribution to the absolute upper bound given by collective measurements. We discuss two situations: the first one, where the state is restricted to lie on the equator of the Bloch sphere, is formally equivalent to phase estimation; the second one, where there is no constrain on the state, can also be regarded as the estimation of a direction in space using a quantum arrow made out of $N$ parallel spins. We discuss various schemes with and without classical communication and compare their efficiency. We show that the fidelity of the most general collective measurement can always be achieved asymptotically with local measurements and no classical communication.
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Constituent quarks from QCD

1997-01-01
Martin Lavelle , David McMullan
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Optimal full estimation of qubit mixed states

2006-03-02
E. Bagán , Manuel A. Ballester , Richard D. Gill +2 more
We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the … We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary N, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large N. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.
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Handbook of Mathematical Functions

1966-02-01
Donald A. McQuarrie
First Page First Page

Communication of spin directions with product states and finite measurements

2001-07-05
E. Bagán , M. Baig , R. Muñoz-Tapia
Total spin eigenstates can be used to intrinsically encode a direction, which can later be decoded by means of a quantum measurement. We study the optimal strategy that can be … Total spin eigenstates can be used to intrinsically encode a direction, which can later be decoded by means of a quantum measurement. We study the optimal strategy that can be adopted if, as is likely in practical applications, only product states of N spins are available. We obtain the asymptotic behavior of the average fidelity, which provides a proof that the optimal states must be entangled. We also give a prescription for constructing finite measurements for general encoding eigenstates.
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Optimal discrimination of mixed quantum states involving inconclusive results

2003-01-29
Jaromı́r Fiurášek , Miroslav Ježek
We propose a generalized discrimination scheme for mixed quantum states. In the present scenario we allow for certain fixed fraction of inconclusive results and we maximize the success rate of … We propose a generalized discrimination scheme for mixed quantum states. In the present scenario we allow for certain fixed fraction of inconclusive results and we maximize the success rate of the quantum-state discrimination. This protocol interpolates between the Ivanovic-Dieks-Peres scheme and the Helstrom one. We formulate the extremal equations for the optimal positive operator valued measure describing the discrimination device and establish a criterion for its optimality. We also devise a numerical method for efficient solving of these extremal equations.
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Quantum cryptography

2002-03-08
Nicolas Gisin , G. Ribordy , Wolfgang Tittel +1 more
Quantum cryptography could well be the first application of quantum mechanics at the single-quantum level. The rapid progress in both theory and experiment in recent years is reviewed, with emphasis … Quantum cryptography could well be the first application of quantum mechanics at the single-quantum level. The rapid progress in both theory and experiment in recent years is reviewed, with emphasis on open questions and technological issues.
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Quantum State Estimation

2004-01-01
Christopher A. Fuchs , Ruediger Schack
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Optimal Purification of Single Qubits

1999-05-24
J. I. Cirac , Artur Ekert , Chiara Macchiavello
We introduce a new decomposition of the multiqubit states of the form $\rho^{\otimes N}$ and employ it to construct the optimal single qubit purification procedure. The same decomposition allows us … We introduce a new decomposition of the multiqubit states of the form $\rho^{\otimes N}$ and employ it to construct the optimal single qubit purification procedure. The same decomposition allows us to study optimal quantum cloning and state estimation of mixed states.
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Collective versus local measurements on two parallel or antiparallel spins

2000-09-19
Serge Massar
We give a complete analysis of covariant measurements on two spins. We consider the cases of two parallel and two antiparallel spins, and we consider both collective measurements on the … We give a complete analysis of covariant measurements on two spins. We consider the cases of two parallel and two antiparallel spins, and we consider both collective measurements on the two spins and measurements that require only local quantum operations and classical communication (LOCC). In all cases we obtain the optimal measurements for arbitrary fidelities. In particular, we show that if the aim is to determine as accurately as possible the direction in which the spins are pointing, it is best to carry out measurements on antiparallel spins (as already shown by Gisin and Popescu), second best to carry out measurements on parallel spins, and worst to be restricted to LOCC measurements. If the aim is to determine as accurately as possible a direction orthogonal to that in which the spins are pointing, it is best to carry out measurements on parallel spins, whereas measurements on antiparallel spins and LOCC measurements are both less suitable but equivalent.
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Discrimination with error margin between two states: Case of general occurrence probabilities

2009-11-17
H. Sugimoto , T. Hashimoto , M. Horibe +1 more
We investigate a state discrimination problem which interpolates minimum-error and unambiguous discrimination by introducing a margin for the probability of error. We closely analyze discrimination of two pure states with … We investigate a state discrimination problem which interpolates minimum-error and unambiguous discrimination by introducing a margin for the probability of error. We closely analyze discrimination of two pure states with general occurrence probabilities. The optimal measurements are classified into three types. One of the three types of measurement is optimal depending on parameters (occurrence probabilities and error margin). We determine the three domains in the parameter space and the optimal discrimination success probability in each domain in a fully analytic form. It is also shown that when the states to be discriminated are multipartite, the optimal success probability can be attained by local operations and classical communication. For discrimination of two mixed states, an upper bound of the optimal success probability is obtained.
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Discriminating States: The Quantum Chernoff Bound

2007-04-17
Katrien Audenaert , John Calsamiglia , R. Muñoz-Tapia +4 more
We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the minimal probability of error. This leads to the identification of … We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the minimal probability of error. This leads to the identification of the quantum Chernoff bound, thereby solving a long-standing open problem. The bound reduces to the classical Chernoff bound when the quantum states under consideration commute. The quantum Chernoff bound is the natural symmetric distance measure between quantum states because of its clear operational meaning and because it does not seem to share some of the undesirable features of other distance measures.
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Quantum Metrology

2006-01-03
Vittorio Giovannetti , Seth Lloyd , Lorenzo Maccone
We point out a general framework that encompasses most cases in which quantum effects enable an increase in precision when estimating a parameter (quantum metrology). The typical quantum precision enhancement … We point out a general framework that encompasses most cases in which quantum effects enable an increase in precision when estimating a parameter (quantum metrology). The typical quantum precision enhancement is of the order of the square root of the number of times the system is sampled. We prove that this is optimal, and we point out the different strategies (classical and quantum) that permit one to attain this bound.
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Charges in gauge theories

1998-09-01
Robin Horan , Martin Lavelle , David McMullan
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Optimal probabilistic estimation of quantum states

2006-09-14
Jaromı́r Fiurášek
We extend the concept of probabilistic unambiguous discrimination of quantum states to quantum state estimation. We consider a scenario where the measurement device can output either an estimate of the … We extend the concept of probabilistic unambiguous discrimination of quantum states to quantum state estimation. We consider a scenario where the measurement device can output either an estimate of the unknown input state or an inconclusive result. We present a general method how to evaluate the maximum fidelity achievable by the probabilistic estimation strategy. We illustrate our method on two explicit examples: estimation of a qudit from a pair of conjugate qudits and phase covariant estimation of a qubit from N copies. We show that by allowing for inconclusive results it is possible to reach estimation fidelity higher than that achievable by the best deterministic estimation strategy.
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Colour charges and the anti-screening contribution to the interquark potential

1998-09-01
Martin Lavelle , David McMullan
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Quantum Metrology Assisted by Abstention

2013-03-04
B. Gendra , E. Ronco-Bonvehi , John Calsamiglia +2 more
The main goal of quantum metrology is to obtain accurate values of physical parameters using quantum probes. In this context, we show that abstention, i.e., the possibility of getting an … The main goal of quantum metrology is to obtain accurate values of physical parameters using quantum probes. In this context, we show that abstention, i.e., the possibility of getting an inconclusive answer at readout, can drastically improve the measurement precision and even lead to a change in its asymptotic behavior, from the shot-noise to the Heisenberg scaling. We focus on phase estimation and quantify the required amount of abstention for a given precision. We also develop analytical tools to obtain the asymptotic behavior of the precision and required rate of abstention for arbitrary pure states.
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Optimal discrimination of quantum states with a fixed rate of inconclusive outcomes

2012-10-15
E. Bagán , R. Muñoz-Tapia , G. A. Olivares-Rentería +1 more
In this paper we present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate … In this paper we present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between Unambiguous and Minimum Error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function of the rate of inconclusive answers. A critical value of this rate is identified that coincides with the minimum failure probability in the cases where unambiguous discrimination is possible and provides a natural generalization of it when states cannot be unambiguously discriminated. The method is illustrated on two explicit examples: discrimination of two pure states with arbitrary prior probabilities and discrimination of trine states.
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State discrimination with error margin and its locality

2008-07-18
A. Hayashi , T. Hashimoto , M. Horibe
There are two common settings in a quantum-state discrimination problem. One is minimum-error discrimination where a wrong guess (error) is allowed and the discrimination success probability is maximized. The other … There are two common settings in a quantum-state discrimination problem. One is minimum-error discrimination where a wrong guess (error) is allowed and the discrimination success probability is maximized. The other is unambiguous discrimination where errors are not allowed but the inconclusive result ``I do not know'' is possible. We investigate a discrimination problem with a finite margin imposed on the error probability. The two common settings correspond to the error margins 1 and 0. For arbitrary error margin, we determine the optimal discrimination probability for two pure states with equal occurrence probabilities. We also consider the case where the states to be discriminated are multipartite and show that the optimal discrimination probability can be achieved by local operations and classical communication.
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Multiple-copy two-state discrimination with individual measurements

2005-03-23
Antonio Acín , E. Bagán , M. Baig +2 more
We address the problem of non-orthogonal two-state discrimination when multiple copies of the unknown state are available. We give the optimal strategy when only fixed individual measurements are allowed and … We address the problem of non-orthogonal two-state discrimination when multiple copies of the unknown state are available. We give the optimal strategy when only fixed individual measurements are allowed and show that its error probability saturates the collective (lower) bound asymptotically. We also give the optimal strategy when adaptivity of individual von Neumann measurements is allowed (which requires classical communication), and show that the corresponding error probability is exactly equal to the collective one for any number of copies. We show that this strategy can be regarded as Bayesian updating.
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Quantum replication at the Heisenberg limit

2013-12-05
Giulio Chiribella , Yuxiang Yang , Andrew Chi-Chih Yao
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Local Discrimination of Mixed States

2010-08-18
John Calsamiglia , Julio I. de Vicente , R. Muñoz-Tapia +1 more
We provide rigorous, efficiently computable and tight bounds on the average error probability of multiple-copy discrimination between qubit mixed states by local operations assisted with classical communication (LOCC). In contrast … We provide rigorous, efficiently computable and tight bounds on the average error probability of multiple-copy discrimination between qubit mixed states by local operations assisted with classical communication (LOCC). In contrast with the pure-state case, these experimentally feasible protocols perform strictly worse than the general collective ones. Our numerical results indicate that the gap between LOCC and collective error rates persists in the asymptotic limit. In order for LOCC and collective protocols to achieve the same accuracy, the former can require up to twice the number of copies of the latter. Our techniques can be used to bound the power of LOCC strategies in other similar settings, which is still one of the most elusive questions in quantum communication.
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Quantum Chernoff bound as a measure of distinguishability between density matrices: Application to qubit and Gaussian states

2008-03-07
John Calsamiglia , R. Muñoz-Tapia , Lluís Masanes +2 more
Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability … Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability when discriminating two hypotheses given a large number of observations. Recently the combined work of Audenaert et al. [Phys. Rev. Lett. 98, 160501 (2007)] and Nussbaum and Szkola [e-print arXiv:quant-ph/0607216] has proved the quantum analog of this bound, which applies when the hypotheses correspond to two quantum states. Based on this quantum Chernoff bound, we define a physically meaningful distinguishability measure and its corresponding metric in the space of states; the latter is shown to coincide with the Wigner-Yanase metric. Along the same lines, we define a second, more easily implementable, distinguishability measure based on the error probability of discrimination when the same local measurement is performed on every copy. We study some general properties of these measures, including the probability distribution of density matrices, defined via the volume element induced by the metric. It is shown that the Bures and the local-measurement based metrics are always proportional. Finally, we illustrate their use in the paradigmatic cases of qubits and Gaussian infinite-dimensional states.
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Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms

2006-10-27
Dave Bacon , Isaac L. Chuang , Aram W. Harrow
The Schur basis on n d-dimensional quantum systems is a generalization of the total angular momentum basis that is useful for exploiting symmetry under permutations or collective unitary rotations. We … The Schur basis on n d-dimensional quantum systems is a generalization of the total angular momentum basis that is useful for exploiting symmetry under permutations or collective unitary rotations. We present efficient {size poly[n,d,log(1/epsilon)] for accuracy epsilon} quantum circuits for the Schur transform, which is the change of basis between the computational and the Schur bases. Our circuits provide explicit efficient methods for solving such diverse problems as estimating the spectrum of a density operator, quantum hypothesis testing, and communicating without a shared reference frame. We thus render tractable a large series of methods for extracting resources from quantum systems and for numerous quantum information protocols.
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Explicit computation of the Bures distance for density matrices

1992-03-01
Matthias Hübner

Improvement of Frequency Standards with Quantum Entanglement

1997-11-17
Susana F. Huelga , Chiara Macchiavello , T. Pellizzari +3 more
The optimal precision of frequency measurements in the presence of decoherence is discussed. We analyze different preparations of $n$ two-level systems as well as different measurement procedures. We show that … The optimal precision of frequency measurements in the presence of decoherence is discussed. We analyze different preparations of $n$ two-level systems as well as different measurement procedures. We show that standard Ramsey spectroscopy on uncorrelated atoms and optimal measurements on maximally entangled states provide the same resolution. The best resolution is achieved using partially entangled preparations with a high degree of symmetry.
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Quantum-state estimation

1997-03-01
Z. Hradil
New algorithm for quantum state estimation based on the maximum likelihood estimation is proposed. Existing techniques for state reconstruction based on the inversion of measured data are shown to be … New algorithm for quantum state estimation based on the maximum likelihood estimation is proposed. Existing techniques for state reconstruction based on the inversion of measured data are shown to be overestimated since they do not guarantee the positive definiteness of the reconstructed density matrix.
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Elliptic Rydberg states as direction indicators

2003-10-09
Netanel H. Lindner , Asher Peres , Daniel R. Terno
The orientation in space of a Cartesian coordinate system can be indicated by the two vectorial constants of motion of a classical Keplerian orbit: the angular momentum and the Laplace-Runge-Lenz … The orientation in space of a Cartesian coordinate system can be indicated by the two vectorial constants of motion of a classical Keplerian orbit: the angular momentum and the Laplace-Runge-Lenz vector. In quantum mechanics, the states of a hydrogen atom that mimic classical elliptic orbits are the coherent states of the SO(4) rotation group. It is known how to produce these states experimentally. They have minimal dispersions of the two conserved vectors and can be used as direction indicators. We compare the fidelity of this transmission method with that of the idealized optimal method.
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Beating noise with abstention in state estimation

2012-10-16
B. Gendra , E. Ronco-Bonvehi , John Calsamiglia +2 more
We address the problem of estimating pure qubit states with non-ideal (noisy) measurements in the multiple-copy scenario, where the data consists of a number N of identically prepared qubits. We … We address the problem of estimating pure qubit states with non-ideal (noisy) measurements in the multiple-copy scenario, where the data consists of a number N of identically prepared qubits. We show that the average fidelity of the estimates can increase significantly if the estimation protocol allows for inconclusive answers, or abstentions. We present the optimal such protocol and compute its fidelity for a given probability of abstention. The improvement over standard estimation, without abstention, can be viewed as an effective noise reduction. These and other results are exemplified for small values of N. For asymptotically large N, we derive analytical expressions of the fidelity and the probability of abstention, and show that for a fixed fidelity gain the latter decreases with N at an exponential rate given by a Kulback-Leibler (relative) entropy. As a byproduct, we obtain an asymptotic expression in terms of this very entropy of the probability that a system of N qubits, all prepared in the same state, has a given total angular momentum. We also discuss an extreme situation where noise increases with N and where estimation with abstention provides a most significant improvement as compared to the standard approach.
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Covariant quantum measurements may not be optimal

2002-07-01
Asher Peres , Petra F. Scudo
Abstract Quantum particles, such as spins, can be used for communicating spatial directions to observers who share no common coordinate frame. We show that if the emitter's signals are the … Abstract Quantum particles, such as spins, can be used for communicating spatial directions to observers who share no common coordinate frame. We show that if the emitter's signals are the orbit of a group, then the optimal detection method may not be a covariant measurement (contrary to widespread belief). It may be advantageous for the receiver to use a different group and an indirect estimation method: first, an ordinary measurement supplies redundant numerical parameters; the latter are then used for a nonlinear optimal identification of the signal.
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PROBABILISTIC AND STATISTICAL ASPECTS OF QUANTUM THEORY (North-Holland Series in Statistics and Probability, 1)

1984-01-01
E. B. Davies
By A. S. Holevo: pp. 316. US$85.00. Dfl.225.00. (North-Holland Publishing Company, 1982.) By A. S. Holevo: pp. 316. US$85.00. Dfl.225.00. (North-Holland Publishing Company, 1982.)

Fidelity Balance in Quantum Operations

2001-02-12
Konrad Banaszek
I derive a tight bound between the quality of estimating the state of a single copy of a d-level system, and the degree the initial state has to be altered … I derive a tight bound between the quality of estimating the state of a single copy of a d-level system, and the degree the initial state has to be altered in the course of this procedure. This result provides a complete analytical description of the quantum mechanical trade-off between the information gain and the quantum state disturbance expressed in terms of mean fidelities. I also discuss consequences of this bound for quantum teleportation using nonmaximally entangled states.
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Optimal estimation of quantum dynamics

2001-10-18
Antonio Acín , E. Jané , Guifré Vidal
We construct the optimal strategy for the estimation of an unknown unitary transformation $U\in SU(d)$. This includes, in addition to a convenient measurement on a probe system, finding which is … We construct the optimal strategy for the estimation of an unknown unitary transformation $U\in SU(d)$. This includes, in addition to a convenient measurement on a probe system, finding which is the best initial state on which $U$ is to act. When $U\in SU(2)$, such an optimal strategy can be applied to estimate simultaneously both the direction and the strength of a magnetic field, and shows how to use a spin 1/2 particle to transmit information about a whole coordinate system instead of only a direction in space.
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Implementation of a Nondeterministic Optical Noiseless Amplifier

2010-03-24
Franck Ferreyrol , Marco Barbieri , Rémi Blandino +3 more
Quantum mechanics imposes that any amplifier that works independently on the phase of the input signal has to introduce some excess noise. The impossibility of such a noiseless amplifier is … Quantum mechanics imposes that any amplifier that works independently on the phase of the input signal has to introduce some excess noise. The impossibility of such a noiseless amplifier is rooted in the unitarity and linearity of quantum evolution. A possible way to circumvent this limitation is to interrupt such evolution via a measurement, providing a random outcome able to herald a successful-and noiseless-amplification event. Here we show a successful realization of such an approach; we perform a full characterization of an amplified coherent state using quantum homodyne tomography, and observe a strong heralded amplification, with about a 6 dB gain and a noise level significantly smaller than the minimal allowed for any ordinary phase-independent device.
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Infrared finite electron propagator

1997-09-15
E. Bagán , Martin Lavelle , David McMullan
We investigate the properties of a dressed electron which reduces, in a particular class of gauges, to the usual fermion. A one loop calculation of the propagator is presented. We … We investigate the properties of a dressed electron which reduces, in a particular class of gauges, to the usual fermion. A one loop calculation of the propagator is presented. We show explicitly that an infra-red finite, multiplicative, mass shell renormalisation is possible for this dressed electron, or, equivalently, for the usual fermion in the abovementioned gauges. The results are in complete accord with previous conjectures.
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Advances in quantum metrology

2011-03-31
Vittorio Giovannetti , Seth Lloyd , Lorenzo Maccone
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Collective versus local measurements in a qubit mixed-state estimation

2004-01-23
E. Bagán , M. Baig , R. Muñoz-Tapia +1 more
We discuss the problem of estimating a qubit mixed state. We give the optimal estimation that can be inferred from any given set of measurements. For collective measurements and for … We discuss the problem of estimating a qubit mixed state. We give the optimal estimation that can be inferred from any given set of measurements. For collective measurements and for a large number N of copies, we show that the error in the estimation varies as $1/N.$ For local measurements, we focus on the simpler case of states lying on the equatorial plane of the Bloch sphere. We show that the error using plain tomography varies as ${1/N}^{1/4},$ while our approach leads to an error proportional to ${1/N}^{3/4}.$
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Estimating the spectrum of a density operator

2001-10-15
Michael Keyl , Reinhard F. Werner
Given N quantum systems prepared according to the same density operator \rho, we propose a measurement on the N-fold system which approximately yields the spectrum of \rho. The projections of … Given N quantum systems prepared according to the same density operator \rho, we propose a measurement on the N-fold system which approximately yields the spectrum of \rho. The projections of the proposed observable decompose the Hilbert space according to the irreducible representations of the permutations on N points, and are labeled by Young frames, whose relative row lengths estimate the eigenvalues of \rho in decreasing order. We show convergence of these estimates in the limit N\to\infty, and that the probability for errors decreases exponentially with a rate we compute explicitly.
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Entanglement-assisted alignment of reference frames using a dense covariant coding

2004-05-14
E. Bagán , M. Baig , R. Muñoz-Tapia
We present a procedure inspired by dense coding, which enables a highly efficient transmission of information of a continuous nature. The procedure requires the sender and the recipient to share … We present a procedure inspired by dense coding, which enables a highly efficient transmission of information of a continuous nature. The procedure requires the sender and the recipient to share a maximally entangled state. We deal with the concrete problem of aligning reference frames or trihedra by means of a quantum system. We find the optimal covariant measurement and compute the corresponding average error, which has a remarkably simple close form. The connection of this procedure with that of estimating unitary transformations on qubits is briefly discussed.
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