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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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Entanglement properties of the harmonic chain

2002-10-30
Katrien Audenaert , Jens Eisert , Martin B. Plenio +1 more
We study the entanglement properties of a closed chain of harmonic oscillators that are coupled via a translationally invariant Hamiltonian, where the coupling acts only on the position operators. We … We study the entanglement properties of a closed chain of harmonic oscillators that are coupled via a translationally invariant Hamiltonian, where the coupling acts only on the position operators. We consider the ground state and thermal states of this system, which are Gaussian states. The entanglement properties of these states can be completely characterized analytically when one uses the logarithmic negativity as a measure of entanglement.
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Evenly distributed unitaries: On the structure of unitary designs

2007-05-01
David Groß , Katrien Audenaert , Jens Eisert
We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design … We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.
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Entanglement Cost under Positive-Partial-Transpose-Preserving Operations

2003-01-16
Katrien Audenaert , Martin B. Plenio , Jens Eisert
We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby … We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this entanglement measure. As examples we discuss general Werner states and arbitrary bipartite Gaussian states. Then we prove that for the antisymmetric Werner state PPT cost and PPT entanglement of distillation coincide. This is the first example of a truly mixed state for which entanglement manipulation is asymptotically reversible, which points towards a unique entanglement measure under PPT operations.
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Asymptotic Error Rates in Quantum Hypothesis Testing

2008-02-07
Katrien Audenaert , M. Nussbaum , Arleta Szkoła +1 more
We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for … We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the specified error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance. The proof relies on two new techniques that have been introduced in [quant-ph/0610027] and [quant-ph/0607216], respectively, and that are also well suited to prove the quantum generalisation of the Hoeffding bound, which is a modification of the Chernoff distance and specifies the optimal achievable asymptotic error rate in the context of asymmetric hypothesis testing. This has been done subsequently by Hayashi [quant-ph/0611013] and Nagaoka [quant-ph/0611289] for the special case where both hypotheses have full support. Moreover, quantum Stein's Lemma and quantum Sanov's theorem may be derived directly from the quantum Hoeffding bound combining it with a result obtained recently in [math/0703772]. The goal of this paper is to present the proofs of the above mentioned results in a unified way and in full generality (allowing hypothetic states with different supports). Additionally, we give an in-depth treatment of the properties of the quantum Chernoff distance. We argue that, although it is not a metric, it is a natural distance measure on the set of density operators, due to its clear operational meaning.
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Quantitative entanglement witnesses

2007-03-02
Jens Eisert , Fernando G. S. L. Brandão , Katrien Audenaert
Entanglement witnesses provide tools to detect entanglement in experimental situations without the need of having full tomographic knowledge about the state. If one estimates in an experiment an expectation value … Entanglement witnesses provide tools to detect entanglement in experimental situations without the need of having full tomographic knowledge about the state. If one estimates in an experiment an expectation value smaller than zero, one can directly infer that the state has been entangled, or specifically multi-partite entangled, in the first place. In this paper, we emphasize that all these tests—based on the very same data—give rise to quantitative estimates in terms of entanglement measures: 'If a test is strongly violated, one can also infer that the state was quantitatively very much entangled'. We consider various measures of entanglement, including the negativity, the entanglement of formation, and the robustness of entanglement, in the bipartite and multipartite setting. As examples, we discuss several experiments in the context of quantum state preparation that have recently been performed.
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When are correlations quantum?—verification and quantification of entanglement by simple measurements

2006-11-08
Katrien Audenaert , Martin B. Plenio
The verification and quantification of experimentally created entanglement by simple measurements, especially between distant particles, is an important basic task in quantum processing. When composite systems are subjected to local … The verification and quantification of experimentally created entanglement by simple measurements, especially between distant particles, is an important basic task in quantum processing. When composite systems are subjected to local measurements the measurement data will exhibit correlations, whether these systems are classical or quantum. Therefore, the observation of correlations in the classical measurement record does not automatically imply the presence of quantum correlations in the system under investigation. In this study, we explore the question of when correlations, or other measurement data, are sufficient to guarantee the existence of a certain amount of quantum correlations in the system and when additional information, such as the degree of purity of the system, is needed to do so. Various measurement settings are discussed, both numerically and analytically. Exact results and lower bounds on the least entanglement consistent with the observations are presented. The approach is suitable both for the bi-partite and the multi-partite setting.
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Asymptotic Relative Entropy of Entanglement

2001-11-06
Katrien Audenaert , Jens Eisert , E. Jané +3 more
We present an analytical formula for the asymptotic relative entropy of entanglement with respect to positive partial transpose states for Werner states of arbitrary dimension. We then demonstrate its validity … We present an analytical formula for the asymptotic relative entropy of entanglement with respect to positive partial transpose states for Werner states of arbitrary dimension. We then demonstrate its validity using methods from convex optimization. This is the first case in which the asymptotic value of a subadditive entanglement measure has been calculated.
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Asymptotic relative entropy of entanglement for orthogonally invariant states

2002-09-18
Katrien Audenaert , Bart De Moor , K. G. H. Vollbrecht +1 more
For a special class of bipartite states we calculate explicitly the asymptotic relative entropy of entanglement ${E}_{R}^{\ensuremath{\infty}}$ with respect to states having a positive partial transpose. This quantity is an … For a special class of bipartite states we calculate explicitly the asymptotic relative entropy of entanglement ${E}_{R}^{\ensuremath{\infty}}$ with respect to states having a positive partial transpose. This quantity is an upper bound to distillable entanglement. The states considered are invariant under rotations of the form $\mathrm{O}\ensuremath{\bigotimes}\mathrm{O},$ where O is any orthogonal matrix. We show that in this case ${E}_{R}^{\ensuremath{\infty}}$ is equal to another upper bound on distillable entanglement, constructed by Rains. To perform these calculations, we have introduced a number of results that are interesting in their own right: (i) the Rains bound is convex and continuous; (ii) under some weak assumption, the Rains bound is an upper bound to ${E}_{R}^{\ensuremath{\infty}};$ (iii) for states for which the relative entropy of entanglement ${E}_{R}$ is additive, the Rains bound is equal to ${E}_{R}.$
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Remarks on entanglement measures and non-local state distinguishability

2003-05-08
Jens Eisert , Katrien Audenaert , Martin B. Plenio
We investigate the properties of three entanglement measures that quantify the statistical distinguishability of a given state with the closest disentangled state that has the same reductions as the primary … We investigate the properties of three entanglement measures that quantify the statistical distinguishability of a given state with the closest disentangled state that has the same reductions as the primary state. In particular, we concentrate on the relative entropy of entanglement with reversed entries. We show that this quantity is an entanglement monotone which is strongly additive, thereby demonstrating that monotonicity under local quantum operations and strong additivity are compatible in principle. In accordance with the presented statistical interpretation which is provided, this entanglement monotone, however, has the property that it diverges on pure states, with the consequence that it cannot distinguish the degree of entanglement of different pure states. We also prove that the relative entropy of entanglement with respect to the set of disentangled states that have identical reductions to the primary state is an entanglement monotone. We finally investigate the trace-norm measure and demonstrate that it is also a proper entanglement monotone.
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Problems and Conjectures in Matrix and Operator Inequalities

2012-01-01
Katrien Audenaert , Fuad Kıttaneh
We present a treasure trove of open problems in matrix and operator inequalities, of a functional analytic nature, and with various degrees of hardness. We present a treasure trove of open problems in matrix and operator inequalities, of a functional analytic nature, and with various degrees of hardness.
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Strongly subadditive functions

2010-04-14
Katrien Audenaert , Fumio Hiai , D. Petz
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Comment on "Optimum Quantum Error Recovery using Semidefinite Programming"

2006-01-01
Michael Reimpell , Reinhard F. Werner , Katrien Audenaert
In a recent paper ([1]=quant-ph/0606035) it is shown how the optimal recovery operation in an error correction scheme can be considered as a semidefinite program. As a possible future improvement … In a recent paper ([1]=quant-ph/0606035) it is shown how the optimal recovery operation in an error correction scheme can be considered as a semidefinite program. As a possible future improvement it is noted that still better error correction might be obtained by optimizing the encoding as well. In this note we present the result of such an improvement, specifically for the four-bit correction of an amplitude damping channel considered in [1]. We get a strict improvement for almost all values of the damping parameter. The method (and the computer code) is taken from our earlier study of such correction schemes (quant-ph/0307138).

When are correlations quantum

2006-08-08
Katrien Audenaert , Martin B. Plenio

Anti Lie-Trotter formula

2014-01-01
Katrien Audenaert , Fumio Hiai
Let $A$ and $B$ be positive semidefinite matrices. The limit of the expression $Z_p:=(A^{p/2}B^pA^{p/2})^{1/p}$ as $p$ tends to $0$ is given by the well known Lie-Trotter-Kato formula. A similar formula … Let $A$ and $B$ be positive semidefinite matrices. The limit of the expression $Z_p:=(A^{p/2}B^pA^{p/2})^{1/p}$ as $p$ tends to $0$ is given by the well known Lie-Trotter-Kato formula. A similar formula holds for the limit of $G_p:=(A^p\,\#\,B^p)^{2/p}$ as $p$ tends to $0$, where $X\,\#\,Y$ is the geometric mean of $X$ and $Y$. In this paper we study the complementary limit of $Z_p$ and $G_p$ as $p$ tends to $\infty$, with the ultimate goal of finding an explicit formula, which we call the anti Lie-Trotter formula. We show that the limit of $Z_p$ exists and find an explicit formula in a special case. The limit of $G_p$ is shown for $2\times2$ matrices only.
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Notes on multiplicativity of maximal output purity for completely positive qubit maps

2008-03-01
Katrien Audenaert
A problem in quantum information theory that has received considerable attention in recent years is the question of multiplicativity of the so-called maximal output purity (MOP) of a quantum channel. … A problem in quantum information theory that has received considerable attention in recent years is the question of multiplicativity of the so-called maximal output purity (MOP) of a quantum channel. This quantity is defined as the maximum value of the purity one can get at the output of a channel by varying over all physical input states, when purity is measured by the Schatten $q$-norm, and is denoted by $\nu_q$. The multiplicativity problem is the question whether two channels used in parallel have a combined $\nu_q$ that is the product of the $\nu_q$ of the two channels. A positive answer would imply a number of other additivity results in QIT. Very recently, P. Hayden has found counterexamples for every value of $q>1$. Nevertheless, these counterexamples require that the dimension of these channels increases with $1-q$ and therefore do not rule out multiplicativity for $q$ in intervals $[1,q_0)$ with $q_0$ depending on the channel dimension. I argue that this would be enough to prove additivity of entanglement of formation and of the classical capacity of quantum channels. More importantly, no counterexamples have as yet been found in the important special case where one of the channels is a qubit-channel, i.e. its input states are 2-dimensional. In this paper I focus attention to this qubit case and I rephrase the multiplicativity conjecture in the language of block matrices and prove the conjecture in a number of special cases.
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Multiplicativity of Accessible Fidelity and Quantumness for Sets of Quantum States

2003-01-01
Katrien Audenaert , Christopher A. Fuchs , Christopher King +1 more
Two measures of sensitivity to eavesdropping for alphabets of quantum states were recently introduced by Fuchs and Sasaki in quant-ph/0302092. These are the accessible fidelity and quantumness. In this paper … Two measures of sensitivity to eavesdropping for alphabets of quantum states were recently introduced by Fuchs and Sasaki in quant-ph/0302092. These are the accessible fidelity and quantumness. In this paper we prove an important property of both measures: They are multiplicative under tensor products. The proof in the case of accessible fidelity shows a connection between the measure and characteristics of entanglement-breaking quantum channels.

Unlocking Heisenberg Sensitivity with Sequential Weak Measurement Preparation

2024-03-09
T. B. Lantaño , Dayou Yang , Katrien Audenaert +2 more
We propose a state preparation protocol based on sequential measurements of a central spin coupled with a spin ensemble, and investigate the usefulness of the generated multi-spin states for quantum … We propose a state preparation protocol based on sequential measurements of a central spin coupled with a spin ensemble, and investigate the usefulness of the generated multi-spin states for quantum enhanced metrology. Our protocol is shown to generate highly entangled spin states, devoid of the necessity for non-linear spin interactions. The metrological sensitivity of the resulting state surpasses the standard quantum limit, reaching the Heisenberg limit under symmetric coupling strength conditions. We also explore asymmetric coupling strengths, identifying specific preparation windows in time for optimal sensitivity. Our findings introduce a novel method for generating large-scale, non-classical, entangled states, enhancing quantum-enhanced metrology within current experimental capabilities.
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Unlocking Heisenberg Sensitivity with Sequential Weak Measurement Preparation

2025-01-14
T. B. Lantaño , Dayou Yang , Katrien Audenaert +2 more
We propose a state preparation protocol based on sequential measurements of a central spin coupled with a spin ensemble, and investigate the usefulness of the generated multi-spin states for quantum … We propose a state preparation protocol based on sequential measurements of a central spin coupled with a spin ensemble, and investigate the usefulness of the generated multi-spin states for quantum enhanced metrology. Our protocol is shown to generate highly entangled spin states, devoid of the necessity for non-linear spin interactions. The metrological sensitivity of the resulting state surpasses the standard quantum limit, reaching the Heisenberg limit under symmetric coupling strength conditions. We also explore asymmetric coupling strengths, identifying specific preparation windows in time for optimal sensitivity. Our findings introduce a novel method for generating large-scale, non-classical, entangled states, enabling quantum-enhanced metrology within current experimental capabilities.
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Unlocking Heisenberg Sensitivity with Sequential Weak Measurement Preparation

2025-01-14
T. B. Lantaño , Dayou Yang , Katrien Audenaert +2 more
We propose a state preparation protocol based on sequential measurements of a central spin coupled with a spin ensemble, and investigate the usefulness of the generated multi-spin states for quantum … We propose a state preparation protocol based on sequential measurements of a central spin coupled with a spin ensemble, and investigate the usefulness of the generated multi-spin states for quantum enhanced metrology. Our protocol is shown to generate highly entangled spin states, devoid of the necessity for non-linear spin interactions. The metrological sensitivity of the resulting state surpasses the standard quantum limit, reaching the Heisenberg limit under symmetric coupling strength conditions. We also explore asymmetric coupling strengths, identifying specific preparation windows in time for optimal sensitivity. Our findings introduce a novel method for generating large-scale, non-classical, entangled states, enabling quantum-enhanced metrology within current experimental capabilities.
View PDF Chat

Unlocking Heisenberg Sensitivity with Sequential Weak Measurement Preparation

2024-03-09
T. B. Lantaño , Dayou Yang , Katrien Audenaert +2 more
We propose a state preparation protocol based on sequential measurements of a central spin coupled with a spin ensemble, and investigate the usefulness of the generated multi-spin states for quantum … We propose a state preparation protocol based on sequential measurements of a central spin coupled with a spin ensemble, and investigate the usefulness of the generated multi-spin states for quantum enhanced metrology. Our protocol is shown to generate highly entangled spin states, devoid of the necessity for non-linear spin interactions. The metrological sensitivity of the resulting state surpasses the standard quantum limit, reaching the Heisenberg limit under symmetric coupling strength conditions. We also explore asymmetric coupling strengths, identifying specific preparation windows in time for optimal sensitivity. Our findings introduce a novel method for generating large-scale, non-classical, entangled states, enhancing quantum-enhanced metrology within current experimental capabilities.
View PDF Chat

Anti Lie-Trotter formula

2014-01-01
Katrien Audenaert , Fumio Hiai
Let $A$ and $B$ be positive semidefinite matrices. The limit of the expression $Z_p:=(A^{p/2}B^pA^{p/2})^{1/p}$ as $p$ tends to $0$ is given by the well known Lie-Trotter-Kato formula. A similar formula … Let $A$ and $B$ be positive semidefinite matrices. The limit of the expression $Z_p:=(A^{p/2}B^pA^{p/2})^{1/p}$ as $p$ tends to $0$ is given by the well known Lie-Trotter-Kato formula. A similar formula holds for the limit of $G_p:=(A^p\,\#\,B^p)^{2/p}$ as $p$ tends to $0$, where $X\,\#\,Y$ is the geometric mean of $X$ and $Y$. In this paper we study the complementary limit of $Z_p$ and $G_p$ as $p$ tends to $\infty$, with the ultimate goal of finding an explicit formula, which we call the anti Lie-Trotter formula. We show that the limit of $Z_p$ exists and find an explicit formula in a special case. The limit of $G_p$ is shown for $2\times2$ matrices only.
View PDF Chat

Problems and Conjectures in Matrix and Operator Inequalities

2012-01-01
Katrien Audenaert , Fuad Kıttaneh
We present a treasure trove of open problems in matrix and operator inequalities, of a functional analytic nature, and with various degrees of hardness. We present a treasure trove of open problems in matrix and operator inequalities, of a functional analytic nature, and with various degrees of hardness.
View PDF Chat

Strongly subadditive functions

2010-04-14
Katrien Audenaert , Fumio Hiai , D. Petz
View PDF Chat

Notes on multiplicativity of maximal output purity for completely positive qubit maps

2008-03-01
Katrien Audenaert
A problem in quantum information theory that has received considerable attention in recent years is the question of multiplicativity of the so-called maximal output purity (MOP) of a quantum channel. … A problem in quantum information theory that has received considerable attention in recent years is the question of multiplicativity of the so-called maximal output purity (MOP) of a quantum channel. This quantity is defined as the maximum value of the purity one can get at the output of a channel by varying over all physical input states, when purity is measured by the Schatten $q$-norm, and is denoted by $\nu_q$. The multiplicativity problem is the question whether two channels used in parallel have a combined $\nu_q$ that is the product of the $\nu_q$ of the two channels. A positive answer would imply a number of other additivity results in QIT. Very recently, P. Hayden has found counterexamples for every value of $q>1$. Nevertheless, these counterexamples require that the dimension of these channels increases with $1-q$ and therefore do not rule out multiplicativity for $q$ in intervals $[1,q_0)$ with $q_0$ depending on the channel dimension. I argue that this would be enough to prove additivity of entanglement of formation and of the classical capacity of quantum channels. More importantly, no counterexamples have as yet been found in the important special case where one of the channels is a qubit-channel, i.e. its input states are 2-dimensional. In this paper I focus attention to this qubit case and I rephrase the multiplicativity conjecture in the language of block matrices and prove the conjecture in a number of special cases.
View PDF Chat

Asymptotic Error Rates in Quantum Hypothesis Testing

2008-02-07
Katrien Audenaert , M. Nussbaum , Arleta Szkoła +1 more
We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for … We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the specified error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance. The proof relies on two new techniques that have been introduced in [quant-ph/0610027] and [quant-ph/0607216], respectively, and that are also well suited to prove the quantum generalisation of the Hoeffding bound, which is a modification of the Chernoff distance and specifies the optimal achievable asymptotic error rate in the context of asymmetric hypothesis testing. This has been done subsequently by Hayashi [quant-ph/0611013] and Nagaoka [quant-ph/0611289] for the special case where both hypotheses have full support. Moreover, quantum Stein's Lemma and quantum Sanov's theorem may be derived directly from the quantum Hoeffding bound combining it with a result obtained recently in [math/0703772]. The goal of this paper is to present the proofs of the above mentioned results in a unified way and in full generality (allowing hypothetic states with different supports). Additionally, we give an in-depth treatment of the properties of the quantum Chernoff distance. We argue that, although it is not a metric, it is a natural distance measure on the set of density operators, due to its clear operational meaning.
View PDF Chat

Evenly distributed unitaries: On the structure of unitary designs

2007-05-01
David Groß , Katrien Audenaert , Jens Eisert
We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design … We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.
View PDF Chat

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.
View PDF Chat

Quantitative entanglement witnesses

2007-03-02
Jens Eisert , Fernando G. S. L. Brandão , Katrien Audenaert
Entanglement witnesses provide tools to detect entanglement in experimental situations without the need of having full tomographic knowledge about the state. If one estimates in an experiment an expectation value … Entanglement witnesses provide tools to detect entanglement in experimental situations without the need of having full tomographic knowledge about the state. If one estimates in an experiment an expectation value smaller than zero, one can directly infer that the state has been entangled, or specifically multi-partite entangled, in the first place. In this paper, we emphasize that all these tests—based on the very same data—give rise to quantitative estimates in terms of entanglement measures: 'If a test is strongly violated, one can also infer that the state was quantitatively very much entangled'. We consider various measures of entanglement, including the negativity, the entanglement of formation, and the robustness of entanglement, in the bipartite and multipartite setting. As examples, we discuss several experiments in the context of quantum state preparation that have recently been performed.
View PDF Chat

When are correlations quantum?—verification and quantification of entanglement by simple measurements

2006-11-08
Katrien Audenaert , Martin B. Plenio
The verification and quantification of experimentally created entanglement by simple measurements, especially between distant particles, is an important basic task in quantum processing. When composite systems are subjected to local … The verification and quantification of experimentally created entanglement by simple measurements, especially between distant particles, is an important basic task in quantum processing. When composite systems are subjected to local measurements the measurement data will exhibit correlations, whether these systems are classical or quantum. Therefore, the observation of correlations in the classical measurement record does not automatically imply the presence of quantum correlations in the system under investigation. In this study, we explore the question of when correlations, or other measurement data, are sufficient to guarantee the existence of a certain amount of quantum correlations in the system and when additional information, such as the degree of purity of the system, is needed to do so. Various measurement settings are discussed, both numerically and analytically. Exact results and lower bounds on the least entanglement consistent with the observations are presented. The approach is suitable both for the bi-partite and the multi-partite setting.
View PDF Chat

When are correlations quantum

2006-08-08
Katrien Audenaert , Martin B. Plenio

Comment on "Optimum Quantum Error Recovery using Semidefinite Programming"

2006-01-01
Michael Reimpell , Reinhard F. Werner , Katrien Audenaert
In a recent paper ([1]=quant-ph/0606035) it is shown how the optimal recovery operation in an error correction scheme can be considered as a semidefinite program. As a possible future improvement … In a recent paper ([1]=quant-ph/0606035) it is shown how the optimal recovery operation in an error correction scheme can be considered as a semidefinite program. As a possible future improvement it is noted that still better error correction might be obtained by optimizing the encoding as well. In this note we present the result of such an improvement, specifically for the four-bit correction of an amplitude damping channel considered in [1]. We get a strict improvement for almost all values of the damping parameter. The method (and the computer code) is taken from our earlier study of such correction schemes (quant-ph/0307138).

Remarks on entanglement measures and non-local state distinguishability

2003-05-08
Jens Eisert , Katrien Audenaert , Martin B. Plenio
We investigate the properties of three entanglement measures that quantify the statistical distinguishability of a given state with the closest disentangled state that has the same reductions as the primary … We investigate the properties of three entanglement measures that quantify the statistical distinguishability of a given state with the closest disentangled state that has the same reductions as the primary state. In particular, we concentrate on the relative entropy of entanglement with reversed entries. We show that this quantity is an entanglement monotone which is strongly additive, thereby demonstrating that monotonicity under local quantum operations and strong additivity are compatible in principle. In accordance with the presented statistical interpretation which is provided, this entanglement monotone, however, has the property that it diverges on pure states, with the consequence that it cannot distinguish the degree of entanglement of different pure states. We also prove that the relative entropy of entanglement with respect to the set of disentangled states that have identical reductions to the primary state is an entanglement monotone. We finally investigate the trace-norm measure and demonstrate that it is also a proper entanglement monotone.
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Entanglement Cost under Positive-Partial-Transpose-Preserving Operations

2003-01-16
Katrien Audenaert , Martin B. Plenio , Jens Eisert
We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby … We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this entanglement measure. As examples we discuss general Werner states and arbitrary bipartite Gaussian states. Then we prove that for the antisymmetric Werner state PPT cost and PPT entanglement of distillation coincide. This is the first example of a truly mixed state for which entanglement manipulation is asymptotically reversible, which points towards a unique entanglement measure under PPT operations.
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Multiplicativity of Accessible Fidelity and Quantumness for Sets of Quantum States

2003-01-01
Katrien Audenaert , Christopher A. Fuchs , Christopher King +1 more
Two measures of sensitivity to eavesdropping for alphabets of quantum states were recently introduced by Fuchs and Sasaki in quant-ph/0302092. These are the accessible fidelity and quantumness. In this paper … Two measures of sensitivity to eavesdropping for alphabets of quantum states were recently introduced by Fuchs and Sasaki in quant-ph/0302092. These are the accessible fidelity and quantumness. In this paper we prove an important property of both measures: They are multiplicative under tensor products. The proof in the case of accessible fidelity shows a connection between the measure and characteristics of entanglement-breaking quantum channels.

Entanglement properties of the harmonic chain

2002-10-30
Katrien Audenaert , Jens Eisert , Martin B. Plenio +1 more
We study the entanglement properties of a closed chain of harmonic oscillators that are coupled via a translationally invariant Hamiltonian, where the coupling acts only on the position operators. We … We study the entanglement properties of a closed chain of harmonic oscillators that are coupled via a translationally invariant Hamiltonian, where the coupling acts only on the position operators. We consider the ground state and thermal states of this system, which are Gaussian states. The entanglement properties of these states can be completely characterized analytically when one uses the logarithmic negativity as a measure of entanglement.
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Asymptotic relative entropy of entanglement for orthogonally invariant states

2002-09-18
Katrien Audenaert , Bart De Moor , K. G. H. Vollbrecht +1 more
For a special class of bipartite states we calculate explicitly the asymptotic relative entropy of entanglement ${E}_{R}^{\ensuremath{\infty}}$ with respect to states having a positive partial transpose. This quantity is an … For a special class of bipartite states we calculate explicitly the asymptotic relative entropy of entanglement ${E}_{R}^{\ensuremath{\infty}}$ with respect to states having a positive partial transpose. This quantity is an upper bound to distillable entanglement. The states considered are invariant under rotations of the form $\mathrm{O}\ensuremath{\bigotimes}\mathrm{O},$ where O is any orthogonal matrix. We show that in this case ${E}_{R}^{\ensuremath{\infty}}$ is equal to another upper bound on distillable entanglement, constructed by Rains. To perform these calculations, we have introduced a number of results that are interesting in their own right: (i) the Rains bound is convex and continuous; (ii) under some weak assumption, the Rains bound is an upper bound to ${E}_{R}^{\ensuremath{\infty}};$ (iii) for states for which the relative entropy of entanglement ${E}_{R}$ is additive, the Rains bound is equal to ${E}_{R}.$
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Asymptotic Relative Entropy of Entanglement

2001-11-06
Katrien Audenaert , Jens Eisert , E. Jané +3 more
We present an analytical formula for the asymptotic relative entropy of entanglement with respect to positive partial transpose states for Werner states of arbitrary dimension. We then demonstrate its validity … We present an analytical formula for the asymptotic relative entropy of entanglement with respect to positive partial transpose states for Werner states of arbitrary dimension. We then demonstrate its validity using methods from convex optimization. This is the first case in which the asymptotic value of a subadditive entanglement measure has been calculated.
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Coauthor Papers Together
Martin B. Plenio 8
Jens Eisert 6
Reinhard F. Werner 3
Frank Verstraete 2
Fumio Hiai 2
Dayou Yang 2
Bart De Moor 2
Susana F. Huelga 2
T. B. Lantaño 2
Fuad Kıttaneh 1
Fernando G. S. L. Brandão 1
Lluís Masanes 1
Arleta Szkoła 1
Andreas Winter 1
D. Petz 1
R. Muñoz-Tapia 1
E. Bagán 1
K. G. H. Vollbrecht 1
Antonio Acín 1
Michael Reimpell 1
S. Virmani 1
E. Jané 1
M. Nussbaum 1
Christopher King 1
David Groß 1
Christopher A. Fuchs 1
John Calsamiglia 1

Entanglement measures and purification procedures

1998-03-01
Vlatko Vedral , Martin B. Plenio
We improve previously proposed conditions each measure of entanglement has to satisfy. We present a class of entanglement measures that satisfy these conditions and show that the quantum relative entropy … We improve previously proposed conditions each measure of entanglement has to satisfy. We present a class of entanglement measures that satisfy these conditions and show that the quantum relative entropy and Bures metric generate two measures of this class. We calculate the measures of entanglement for a number of mixed two spin-1/2 systems using the quantum relative entropy, and provide an efficient numerical method to obtain the measures of entanglement in this case. In addition, we prove a number of properties of our entanglement measure that have important physical implications. We briefly explain the statistical basis of our measure of entanglement in the case of the quantum relative entropy. We then argue that our entanglement measure determines an upper bound to the number of singlets that can be obtained by any purification procedure.
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Quantifying Entanglement

1997-03-24
Vlatko Vedral , Martin B. Plenio , M. A. Rippin +1 more
We present conditions every measure of entanglement has to satisfy, and construct a whole class of ``good'' entanglement measures. The generalization of our class of entanglement measures to more than … We present conditions every measure of entanglement has to satisfy, and construct a whole class of ``good'' entanglement measures. The generalization of our class of entanglement measures to more than two particles is straightforward. We present a measure which has a statistical operational basis that might enable experimental determination of the quantitative degree of entanglement.
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Statistical inference, distinguishability of quantum states, and quantum entanglement

1997-12-01
Vlatko Vedral , Martin B. Plenio , Kurt Jacobs +1 more
We argue from the point of view of statistical inference that the quantum relative entropy is a good measure for distinguishing between two quantum states (or two classes of quantum … We argue from the point of view of statistical inference that the quantum relative entropy is a good measure for distinguishing between two quantum states (or two classes of quantum states) described by density matrices. We extend this notion to describe the amount of entanglement between two quantum systems from a statistical point of view. Our measure is independent of the number of entangled systems and their dimensionality.
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Teleportation, entanglement and thermodynamics in the quantum world

1998-11-01
Martin B. Plenio , Vlatko Vedral
Quantum mechanics has many counter-intuitive consequences which contradict our intuition which is based on classical physics. Here we discuss a special aspect of quantum mechanics, namely the possibility of entanglement … Quantum mechanics has many counter-intuitive consequences which contradict our intuition which is based on classical physics. Here we discuss a special aspect of quantum mechanics, namely the possibility of entanglement between two or more particles. We will establish the basic properties of entanglement using quantum state teleportation. These principles will then allow us to formulate quantitative measures of entanglement. Finally we will show that the same general principles can also be used to prove seemingly difficult questions regarding entanglement dynamics very easily. This will be used to motivate the hope that we can construct a thermodynamics of entanglement.
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Computable measure of entanglement

2002-02-22
Guifré Vidal , Reinhard F. Werner
We present a measure of entanglement that can be computed effectively for any mixed state of an arbitrary bipartite system. We show that it does not increase under local manipulations … We present a measure of entanglement that can be computed effectively for any mixed state of an arbitrary bipartite system. We show that it does not increase under local manipulations of the system, and use it to obtain a bound on the teleportation capacity and on the distillable entanglement of mixed states.
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Bound on distillable entanglement

1999-07-01
Eric M. Rains
The best bound known on 2-locally distillable entanglement is that of Vedral and Plenio, involving a certain measure of entanglement based on relative entropy. It turns out that a related … The best bound known on 2-locally distillable entanglement is that of Vedral and Plenio, involving a certain measure of entanglement based on relative entropy. It turns out that a related argument can be used to give an even stronger bound; we give this bound, and examine some of its properties. In particular, and in contrast to the earlier bounds, the new bound is not additive in general. We give an example of a state for which the bound fails to be additive, as well as a number of states for which the bound is additive.
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Bounds on relative entropy of entanglement for multi-party systems

2001-08-24
Martin B. Plenio , Vlatko Vedral
We present upper and lower bounds to the relative entropy of entanglement of multi-party systems in terms of the bi-partite entanglements of formation and distillation and entropies of various subsystems. … We present upper and lower bounds to the relative entropy of entanglement of multi-party systems in terms of the bi-partite entanglements of formation and distillation and entropies of various subsystems. We point out implications of our results to the local reversible convertibility of multi-party pure states and discuss their physical basis in terms of deleting information.
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Asymptotic Relative Entropy of Entanglement

2001-11-06
Katrien Audenaert , Jens Eisert , E. Jané +3 more
We present an analytical formula for the asymptotic relative entropy of entanglement with respect to positive partial transpose states for Werner states of arbitrary dimension. We then demonstrate its validity … We present an analytical formula for the asymptotic relative entropy of entanglement with respect to positive partial transpose states for Werner states of arbitrary dimension. We then demonstrate its validity using methods from convex optimization. This is the first case in which the asymptotic value of a subadditive entanglement measure has been calculated.
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A semidefinite program for distillable entanglement

2001-01-01
Eric M. Rains
We show that the maximum fidelity obtained by a positive partial transpose (p.p.t.) distillation protocol is given by the solution to a certain semidefinite program. This gives a number of … We show that the maximum fidelity obtained by a positive partial transpose (p.p.t.) distillation protocol is given by the solution to a certain semidefinite program. This gives a number of new lower and upper bounds on p.p.t. distillable entanglement (and thus new upper bounds on 2-locally distillable entanglement). In the presence of symmetry, the semidefinite program simplifies considerably, becoming a linear program in the case of isotropic and Werner states. Using these techniques, we determine the p.p.t. distillable entanglement of asymmetric Werner states and "maximally correlated" states. We conclude with a discussion of possible applications of semidefinite programming to quantum codes and 1-local distillation.
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A comparison of entanglement measures

1999-01-01
Jens Eisert , Martin B. Plenio
Abstract We compare the entanglement of formation with a measure defined as the modulus of the negative eigenvalue of the partial transpose. In particular we investigate whether both measures give … Abstract We compare the entanglement of formation with a measure defined as the modulus of the negative eigenvalue of the partial transpose. In particular we investigate whether both measures give the same ordering of density operators with respect to the amount of entanglement
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Entanglement Cost under Positive-Partial-Transpose-Preserving Operations

2003-01-16
Katrien Audenaert , Martin B. Plenio , Jens Eisert
We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby … We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this entanglement measure. As examples we discuss general Werner states and arbitrary bipartite Gaussian states. Then we prove that for the antisymmetric Werner state PPT cost and PPT entanglement of distillation coincide. This is the first example of a truly mixed state for which entanglement manipulation is asymptotically reversible, which points towards a unique entanglement measure under PPT operations.
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Volume of the set of separable states

1998-08-01
Karol Życzkowski , Paweł Horodecki , Anna Sanpera +1 more
A natural measure in the space of density matrices describing N-dimensional quantum systems is proposed. We study the probability P that a quantum state chosen randomly with respect to the … A natural measure in the space of density matrices describing N-dimensional quantum systems is proposed. We study the probability P that a quantum state chosen randomly with respect to the natural measure is not entangled (is separable). We find analytical lower and upper bounds for this quantity. Numerical calculations give P = 0.632 for N=4 and P=0.384 for N=6, and indicate that P decreases exponentially with N. Analysis of a conditional measure of separability under the condition of fixed purity shows a clear dualism between purity and separability: entanglement is typical for pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable.
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A comparison of entanglement measures

1999-01-01
Jens Eisert , Martin B. Plenio
Abstract We compare the entanglement of formation with a measure defined as the modulus of the negative eigenvalue of the partial transpose. In particular we investigate whether both measures give … Abstract We compare the entanglement of formation with a measure defined as the modulus of the negative eigenvalue of the partial transpose. In particular we investigate whether both measures give the same ordering of density operators with respect to the amount of entanglement
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Mixed-state entanglement and quantum error correction

1996-11-01
Charles H. Bennett , David P. DiVincenzo , John A. Smolin +1 more
Entanglement purification protocols (EPPs) and quantum error-correcting codes (QECCs) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, … Entanglement purification protocols (EPPs) and quantum error-correcting codes (QECCs) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbitrary quantum state |\ensuremath{\xi}〉 can be transmitted at some rate Q through a noisy channel \ensuremath{\chi} without degradation. We prove that an EPP involving one-way classical communication and acting on mixed state M^(\ensuremath{\chi}) (obtained by sharing halves of Einstein-Podolsky-Rosen pairs through a channel \ensuremath{\chi}) yields a QECC on \ensuremath{\chi} with rate Q=D, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts ${\mathit{D}}_{1}$(M) and ${\mathit{D}}_{2}$(M) that can be locally distilled from it by EPPs using one- and two-way classical communication, respectively, and give an exact expression for E(M) when M is Bell diagonal. While EPPs require classical communication, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way communication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way communication is available. We exhibit a family of codes based on universal hashing able to achieve an asymptotic Q (or D) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single-error-correcting quantum block code. We prove that iff a QECC results in high fidelity for the case of no error then the QECC can be recast into a form where the encoder is the matrix inverse of the decoder. \textcopyright{} 1996 The American Physical Society.
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Strong converse and Stein's lemma in quantum hypothesis testing

2000-01-01
Hiroshi Nagaoka , Tetsuhiro Ogawa
The hypothesis testing problem for two quantum states is treated. We show a new inequality between the errors of the first kind and the second kind, which complements the result … The hypothesis testing problem for two quantum states is treated. We show a new inequality between the errors of the first kind and the second kind, which complements the result of Hiai and Petz (1991) to establish the quantum version of Stein's lemma. The inequality is also used to show a bound on the probability of errors of the first kind when the power exponent for the probability of errors of the second kind exceeds the quantum relative entropy, which yields the strong converse in quantum hypothesis testing. Finally, we discuss the relation between the bound and the power exponent derived by Han and Kobayashi (1989) in classical hypothesis testing.
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The proper formula for relative entropy and its asymptotics in quantum probability

1991-12-01
Fumio Hiai , Dėnes Petz
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Rigorous treatment of distillable entanglement

1999-07-01
Eric M. Rains
The notion of distillable entanglement is one of the fundamental concepts of quantum information theory. Unfortunately, there is an apparent mismatch between the intuitive and rigorous definitions of distillable entanglement. … The notion of distillable entanglement is one of the fundamental concepts of quantum information theory. Unfortunately, there is an apparent mismatch between the intuitive and rigorous definitions of distillable entanglement. To be precise, the existing rigorous definitions impose the constraint that the distillation protocol produce an output of constant dimension. It is therefore conceivable that this unnecessary constraint might have led to underestimation of the true distillable entanglement. We give a definition of distillable entanglement which removes this constraint, but could conceivably overestimate the true value. Since the definitions turn out to be equivalent, neither underestimation nor overestimation is possible, and both definitions are arguably correct.
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Entanglement measures under symmetry

2001-11-14
K. G. H. Vollbrecht , Reinhard F. Werner
We show how to simplify the computation of the entanglement of formation and the relative entropy of entanglement for states, which are invariant under a group of local symmetries. For … We show how to simplify the computation of the entanglement of formation and the relative entropy of entanglement for states, which are invariant under a group of local symmetries. For several examples of groups we characterize the state spaces, which are invariant under these groups. For specific examples we calculate the entanglement measures. In particular, we derive an explicit formula for the entanglement of formation for $(U\ensuremath{\bigotimes}U)$-invariant states, and we find a counterexample of the additivity conjecture for the relative entropy of entanglement.
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On Strong Superadditivity of the Entanglement of Formation

2004-04-01
Koenraad M. R. Audenaert , Samuel L. Braunstein
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Operator monotones, the reduction criterion and the relative entropy

2000-05-24
Martin B. Plenio , S. Virmani , Panayiotis Papadopoulos
We introduce the theory of operator monotone functions and employ it to derive a new inequality relating the quantum relative entropy and the quantum conditional entropy. We present applications of … We introduce the theory of operator monotone functions and employ it to derive a new inequality relating the quantum relative entropy and the quantum conditional entropy. We present applications of this new inequality and, in particular, we prove a new lower bound on the relative entropy of entanglement and other properties of entanglement measures.
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The “transition probability” in the state space of a ∗-algebra

1976-04-01
Armin Uhlmann

Separability of mixed states: necessary and sufficient conditions

1996-11-01
Michał Horodecki , Paweł Horodecki , Ryszard Horodecki
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Conditions for a Class of Entanglement Transformations

1999-07-12
Michael A. Nielsen
Suppose Alice and Bob jointly possess a pure state, $|\ensuremath{\psi}〉$. Using local operations on their respective systems and classical communication it may be possible for Alice and Bob to transform … Suppose Alice and Bob jointly possess a pure state, $|\ensuremath{\psi}〉$. Using local operations on their respective systems and classical communication it may be possible for Alice and Bob to transform $|\ensuremath{\psi}〉$ into another joint state $|\ensuremath{\varphi}〉$. This Letter gives necessary and sufficient conditions for this process of entanglement transformation to be possible. These conditions reveal a partial ordering on the entangled states and connect quantum entanglement to the algebraic theory of majorization. As a consequence, we find that there exist essentially different types of entanglement for bipartite quantum systems.
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On Error Exponents in Quantum Hypothesis Testing

2004-06-01
Tetsuhiro Ogawa , Masahito Hayashi
In the simple quantum hypothesis testing problem for two density operators, upper bounds on the error probabilities are shown based on a key operator inequality between a density operator and … In the simple quantum hypothesis testing problem for two density operators, upper bounds on the error probabilities are shown based on a key operator inequality between a density operator and a conditional expectation of it. Concerning the error exponents, the upper bounds lead to a noncommutative analog of the Hoeffding bound, which is identical with the classical counterpart if two density operators commute. The upper bounds also provide a simple proof of the direct part of the quantum Stein's lemma.
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Convex trace functions and the Wigner-Yanase-Dyson conjecture

1973-12-01
Élliott H. Lieb

Convex Optimization

2004-03-01
Stephen Boyd , Lieven Vandenberghe
Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. … Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.

An introduction to entanglement measures

2007-01-01
Martin B. Plenio , S. Virmani
We review the theory of entanglement measures, concentrating mostly on the finite dimensional two-party case. Topics covered include: single-copy and asymptotic entanglement manipulation; the entanglement of formation; the entanglement cost; … We review the theory of entanglement measures, concentrating mostly on the finite dimensional two-party case. Topics covered include: single-copy and asymptotic entanglement manipulation; the entanglement of formation; the entanglement cost; the distillable entanglement; the relative entropic measures; the squashed entanglement; log-negativity; the robustness monotones; the greatest cross-norm; uniqueness and extremality theorems. Infinite dimensional systems and multi-party settings will be discussed briefly.
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Entanglement in Quantum Information Theory

1998-04-30
Martin B. Plenio , Vlatko Vedral
Quantum mechanics has many counter-intuitive consequences which contradict our intuition which is based on classical physics. Here we discuss a special aspect of quantum mechanics, namely the possibility of entanglement … Quantum mechanics has many counter-intuitive consequences which contradict our intuition which is based on classical physics. Here we discuss a special aspect of quantum mechanics, namely the possibility of entanglement between two or more particles. We will establish the basic properties of entanglement using quantum state teleportation. These principles will then allow us to formulate quantitative measures of entanglement. Finally we will show that the same general principles can also be used to prove seemingly difficult questions regarding entanglement dynamics very easily. This will be used to motivate the hope that we can construct a thermodynamics of entanglement.
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On the Araki-Lieb-Thirring inequality

2007-01-04
Koenraad M. R. Audenaert
We prove an inequality that complements the famous Araki-Lieb-Thirring (ALT) inequality for positive matrices A and B, by giving a lower bound on the quantity Tr[A r B r A … We prove an inequality that complements the famous Araki-Lieb-Thirring (ALT) inequality for positive matrices A and B, by giving a lower bound on the quantity Tr[A r B r A r ] q in terms of Tr[ABA] rq for 0 � r � 1 and q � 0, whereas the ALT inequality gives an upper bound. The bound contains certain norms of A and B as additional ingredients and is therefore of a different nature than the Kantorovich type inequality obtained by Bourin (Math. Inequal. Appl. 8(2005) pp. 373–378) and others. Secondly, we also prove a generalisation of the ALT inequality to general matrices.

Conditions for the Local Manipulation of Gaussian States

2002-08-06
Jens Eisert , Martin B. Plenio
We present a general necessary and sufficient criterion for thepossibility of a state transformation fromone mixed Gaussian state to another of a bipartite continuous-variablesystem with two modes. Theclass of operations … We present a general necessary and sufficient criterion for thepossibility of a state transformation fromone mixed Gaussian state to another of a bipartite continuous-variablesystem with two modes. Theclass of operations that will be considered is the set of local Gaussiancompletely positive trace-preservingmaps.
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Logarithmic Negativity: A Full Entanglement Monotone That is not Convex

2005-08-26
Martin B. Plenio
It is proven that logarithmic negativity does not increase on average under a general positive partial transpose preserving operation (a set of operations that incorporate local operations and classical communication … It is proven that logarithmic negativity does not increase on average under a general positive partial transpose preserving operation (a set of operations that incorporate local operations and classical communication as a subset) and, in the process, a further proof is provided that the negativity does not increase on average under the same set of operations. Given that the logarithmic negativity is not a convex function this result is surprising, as it is generally considered that convexity describes the local physical process of losing information. The role of convexity and, in particular, its relation (or lack thereof) to physical processes is discussed and importance of continuity in this context is stressed.
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Bell inequalities and the separability criterion

2000-07-01
Barbara M. Terhal
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Means of positive linear operators

1980-10-01
Fumio Kubo , Tsuyoshi Andô

Distilling Gaussian States with Gaussian Operations is Impossible

2002-09-04
Jens Eisert , Stefan Scheel , Martin B. Plenio
We show that no distillation protocol for Gaussian quantum states exists that relies on (i) arbitrary local unitary operations that preserve the Gaussian character of the state and (ii) homodyne … We show that no distillation protocol for Gaussian quantum states exists that relies on (i) arbitrary local unitary operations that preserve the Gaussian character of the state and (ii) homodyne detection together with classical communication and postprocessing by means of local Gaussian unitary operations on two symmetric identically prepared copies. This is in contrast to the finite-dimensional case, where entanglement can be distilled in an iterative protocol using two copies at a time. The ramifications for the distribution of Gaussian states over large distances will be outlined. We also comment on the generality of the approach and sketch the most general form of a Gaussian local operation with classical communication in a bipartite setting.
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Concentrating partial entanglement by local operations

1996-04-01
Charles H. Bennett , H. J. Bernstein , Sandu Popescu +1 more
If two separated observers are supplied with entanglement, in the form of n pairs of particles in identical partly entangled pure states, one member of each pair being given to … If two separated observers are supplied with entanglement, in the form of n pairs of particles in identical partly entangled pure states, one member of each pair being given to each observer, they can, by local actions of each observer, concentrate this entanglement into a smaller number of maximally entangled pairs of particles, for example, Einstein-Podolsky-Rosen singlets, similarly shared between the two observers. The concentration process asymptotically conserves entropy of entanglement---the von Neumann entropy of the partial density matrix seen by either observer---with the yield of singlets approaching, for large n, the base-2 entropy of entanglement of the initial partly entangled pure state. Conversely, any pure or mixed entangled state of two systems can be produced by two classically communicating separated observers, drawing on a supply of singlets as their sole source of entanglement. \textcopyright{} 1996 The American Physical Society.
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Minimal Conditions for Local Pure-State Entanglement Manipulation

1999-08-16
Daniel Jonathan , Martin B. Plenio
We find a minimal set of necessary and sufficient conditions for the existence of a local procedure that converts a finite pure state into one of a set of possible … We find a minimal set of necessary and sufficient conditions for the existence of a local procedure that converts a finite pure state into one of a set of possible final states. This result provides a powerful method for obtaining optimal local entanglement manipulation protocols for pure initial states. As an example, we determine analytically the optimal distillable entanglement for arbitrary finite pure states. We also construct an explicit protocol achieving this bound.
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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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Entanglement of Formation of an Arbitrary State of Two Qubits

1998-03-09
William K. Wootters
The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed … The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state $\ensuremath{\rho}$ is the minimum average entanglement of an ensemble of pure states that represents \ensuremath{\rho}. An earlier paper conjectured an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix, and proved the formula for special states. The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing decompositions.
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Detecting Genuine Multipartite Entanglement with Two Local Measurements

2005-02-17
G. Tóth , Otfried Gühne
We present entanglement witness operators for detecting genuine multipartite entanglement. These witnesses are robust against noise and require only two local measurement settings when used in an experiment, independent of … We present entanglement witness operators for detecting genuine multipartite entanglement. These witnesses are robust against noise and require only two local measurement settings when used in an experiment, independent of the number of qubits. This allows detection of entanglement for an increasing number of parties without a corresponding increase in effort. The witnesses presented detect states close to Greenberger-Horne-Zeilinger, cluster, and graph states. Connections to Bell inequalities are also discussed.
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Entanglement of Pure States for a Single Copy

1999-08-02
Guifré Vidal
An optimal local conversion strategy between any two pure states of a bipartite system is presented. It is optimal in that the probability of success is the largest achievable if … An optimal local conversion strategy between any two pure states of a bipartite system is presented. It is optimal in that the probability of success is the largest achievable if the parties which share the system, and which can communicate classically, are only allowed to act locally on it. The study of optimal local conversions sheds some light on the entanglement of a single copy of a pure state. We propose a quantification of such an entanglement by means of a finite minimal set of new measures from which the optimal probability of conversion follows.
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On the quantification of entanglement in infinite-dimensional quantum systems

2002-04-19
Jens Eisert , Christoph Simon , Martin B. Plenio
We investigate entanglement measures in the infinite-dimensional regime. First, we discuss the peculiarities that may occur if the Hilbert space of a bi-partite system is infinite dimensional, most notably the … We investigate entanglement measures in the infinite-dimensional regime. First, we discuss the peculiarities that may occur if the Hilbert space of a bi-partite system is infinite dimensional, most notably the fact that the set of states with infinite entropy of entanglement is trace-norm dense in state space, implying that in any neighbourhood of every product state lies an arbitrarily strongly entangled state. The starting point for a clarification of this counterintuitive property is the observation that if one imposes the natural and physically reasonable constraint that the mean energy is bounded from above, then the entropy of entanglement becomes a trace-norm continuous functional. The considerations will then be extended to the asymptotic limit, and we will prove some asymptotic continuity properties. We proceed by investigating the entanglement of formation and the relative entropy of entanglement in the infinite-dimensional setting. Finally, we show that the set of entangled states is still trace-norm dense in state space, even under the constraint of a finite mean energy.
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A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations

1952-12-01
Herman Chernoff
In many cases an optimum or computationally convenient test of a simple hypothesis $H_0$ against a simple alternative $H_1$ may be given in the following form. Reject $H_0$ if $S_n … In many cases an optimum or computationally convenient test of a simple hypothesis $H_0$ against a simple alternative $H_1$ may be given in the following form. Reject $H_0$ if $S_n = \sum^n_{j=1} X_j \leqq k,$ where $X_1, X_2, \cdots, X_n$ are $n$ independent observations of a chance variable $X$ whose distribution depends on the true hypothesis and where $k$ is some appropriate number. In particular the likelihood ratio test for fixed sample size can be reduced to this form. It is shown that with each test of the above form there is associated an index $\rho$. If $\rho_1$ and $\rho_2$ are the indices corresponding to two alternative tests $e = \log \rho_1/\log \rho_2$ measures the relative efficiency of these tests in the following sense. For large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample of size $en$ with the second test. To obtain the above result, use is made of the fact that $P(S_n \leqq na)$ behaves roughly like $m^n$ where $m$ is the minimum value assumed by the moment generating function of $X - a$. It is shown that if $H_0$ and $H_1$ specify probability distributions of $X$ which are very close to each other, one may approximate $\rho$ by assuming that $X$ is normally distributed.

Robustness of entanglement

1999-01-01
Guifré Vidal , R. Tarrach
In the quest to completely describe entanglement in the general case of a finite number of parties sharing a physical system of finite-dimensional Hilbert space an entanglement magnitude is introduced … In the quest to completely describe entanglement in the general case of a finite number of parties sharing a physical system of finite-dimensional Hilbert space an entanglement magnitude is introduced for its pure and mixed states: robustness. It corresponds to the minimal amount of mixing with locally prepared states which washes out all entanglement. It quantifies in a sense the endurance of entanglement against noise and jamming. Its properties are studied comprehensively. Analytical expressions for the robustness are given for pure states of two-party systems, and analytical bounds for mixed states of two-party systems. Specific results are obtained mainly for the qubit-qubit system (qubit denotes quantum bit). As by-products local pseudomixtures are generalized, a lower bound for the relative volume of separable states is deduced, and arguments for considering convexity a necessary condition of any entanglement measure are put forward.
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Persistent Entanglement in Arrays of Interacting Particles

2001-01-29
Hans J. Briegel , Robert Raussendorf
We study the entanglement properties of a class of $N$ qubit quantum states that are generated in arrays of qubits with an Ising-type interaction. These states contain a large amount … We study the entanglement properties of a class of $N$ qubit quantum states that are generated in arrays of qubits with an Ising-type interaction. These states contain a large amount of entanglement as given by their Schmidt measure. They have also a high {\em persistency of entanglement} which means that $\sim N/2$ qubits have to be measured to disentangle the state. These states can be regarded as an entanglement resource since one can generate a family of other multi-particle entangled states such as the generalized GHZ states of $<N/2$ qubits by simple measurements and classical communication (LOCC).
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Norm inequalities for partitioned operators and an application

1990-03-01
Rajendra Bhatia , Fuad Kıttaneh

Continuity of relative entropy of entanglement

1999-12-01
Matthew J. Donald , Michał Horodecki
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Method of areas for manipulating the entanglement properties of one copy of a two-particle pure entangled state

1999-09-01
Lucién Hardy
We consider the problem of how to manipulate the entanglement properties of a general two-particle pure state, shared between Alice and Bob, by using only local operations at each end … We consider the problem of how to manipulate the entanglement properties of a general two-particle pure state, shared between Alice and Bob, by using only local operations at each end and classical communication between Alice and Bob. A method is developed in which this type of problem is found to be equivalent to a problem involving the cutting and pasting of certain shapes along with certain coloring problems. We consider two problems. First, we find the most general way of manipulating the state to obtain maximally entangled states. After such a manipulation, the entangled states $|11〉+|22〉+\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}+|\mathrm{mm}〉$ are obtained with probability ${p}_{m}.$ We obtain an expression for the optimal average entanglement obtainable. Also, some results of Lo and Popescu (e-print quant-ph/9707038) pertaining to this problem are given simple geometric proofs. Second, we consider how to manipulate one two-particle entangled state $|\ensuremath{\psi}〉$ to another $|{\ensuremath{\psi}}^{\ensuremath{'}}〉$ with certainty. We derive Nielsen's theorem (which states a necessary and sufficient condition for this to be possible) using the method of areas.
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Multiparty entanglement in graph states

2004-06-09
M. Hein , Jens Eisert , Hans J. Briegel
Graph states are multiparticle entangled states that correspond to mathematical graphs, where the vertices of the graph take the role of quantum spin systems and edges represent Ising interactions. They … Graph states are multiparticle entangled states that correspond to mathematical graphs, where the vertices of the graph take the role of quantum spin systems and edges represent Ising interactions. They are many-body spin states of distributed quantum systems that play a significant role in quantum error correction, multiparty quantum communication, and quantum computation within the framework of the one-way quantum computer. We characterize and quantify the genuine multiparticle entanglement of such graph states in terms of the Schmidt measure, to which we provide upper and lower bounds in graph theoretical terms. Several examples and classes of graphs will be discussed, where these bounds coincide. These examples include trees, cluster states of different dimensions, graphs that occur in quantum error correction, such as the concatenated [7,1,3]-CSS code, and a graph associated with the quantum Fourier transform in the one-way computer. We also present general transformation rules for graphs when local Pauli measurements are applied, and give criteria for the equivalence of two graphs up to local unitary transformations, employing the stabilizer formalism. For graphs of up to seven vertices we provide complete characterization modulo local unitary transformations and graph isomorphisms.
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Classical Information and Distillable Entanglement

2000-02-14
Jens Eisert , Timo Felbinger , Panayiotis Papadopoulos +2 more
We establish a quantitative connection between the amount of lost classical information about a quantum state and the concomitant loss of entanglement. Using methods that have been developed for the … We establish a quantitative connection between the amount of lost classical information about a quantum state and the concomitant loss of entanglement. Using methods that have been developed for the optimal purification of mixed states, we find a class of mixed states with known distillable entanglement. These results can be used to determine the quantum capacity of a quantum channel which randomizes the order of transmitted signals.
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The asymptotic entanglement cost of preparing a quantum state

2001-08-24
Patrick Hayden , Michał Horodecki , Barbara M. Terhal
We give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a bipartite state ρis equal to the regularized entanglement of formation of ρ. We give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a bipartite state ρis equal to the regularized entanglement of formation of ρ.
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