We perform generalized measurements of a qubit by realizing the qubit as a coin in a photonic quantum walk and subjecting the walker to projective measurements. Our experimental technique can ā¦
We perform generalized measurements of a qubit by realizing the qubit as a coin in a photonic quantum walk and subjecting the walker to projective measurements. Our experimental technique can be used to realize, photonically, any rank-1 single-qubit positive-operator-valued measure via constructing an appropriate interferometric quantum-walk network and then projectively measuring the walker's position at the final step.
We describe the structure of the extended Clifford Group (defined to be the group consisting of all operators, unitary and anti-unitary, which normalize the generalized Pauli group (or Weyl-Heisenberg group ā¦
We describe the structure of the extended Clifford Group (defined to be the group consisting of all operators, unitary and anti-unitary, which normalize the generalized Pauli group (or Weyl-Heisenberg group as it is often called)). We also obtain a number of results concerning the structure of the Clifford Group proper (i.e. the group consisting just of the unitary operators which normalize the generalized Pauli group). We then investigate the action of the extended Clifford group operators on symmetric informationally complete POVMs (or SIC-POVMs) covariant relative to the action of the generalized Pauli group. We show that each of the fiducial vectors which has been constructed so far (including all the vectors constructed numerically by Renes et al) is an eigenvector of one of a special class of order 3 Clifford unitaries. This suggests a strengthening of a conjuecture of Zauner's. We give a complete characterization of the orbits and stability groups in dimensions 2-7. Finally, we show that the problem of constructing fiducial vectors may be expected to simplify in the infinite sequence of dimensions 7, 13, 19, 21, 31,... . We illustrate this point by constructing exact expressions for fiducial vectors in dimensions 7 and 19.
We show that in prime dimensions not equal to 3, each group covariant symmetric informationally complete positive operator valued measure (SIC POVM) is covariant with respect to a unique HeisenbergāWeyl ā¦
We show that in prime dimensions not equal to 3, each group covariant symmetric informationally complete positive operator valued measure (SIC POVM) is covariant with respect to a unique HeisenbergāWeyl (HW) group. Moreover, the symmetry group of the SIC POVM is a subgroup of the Clifford group. Hence, two SIC POVMs covariant with respect to the HW group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended Clifford group. In dimension 3, each group covariant SIC POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC POVM is a subgroup of at least one of the Clifford groups of these HW groups, respectively. There may exist two or three orbits of equivalent SIC POVMs for each group covariant SIC POVM, depending on the order of its symmetry group. We then establish a complete equivalence relation among group covariant SIC POVMs in dimension 3, and classify inequivalent ones according to the geometric phases associated with fiducial vectors. Finally, we uncover additional SIC POVMs by regrouping of the fiducial vectors from different SIC POVMs which may or may not be on the same orbit of the extended Clifford group.
We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform. This ā¦
We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and, despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in Cn which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.
Following the work of Chevalley and Warning, Ax obtained a bound on the p-divisibility of exponential sums involving multi-variable polynomials of fixed degree d over a finite field of characteristic ā¦
Following the work of Chevalley and Warning, Ax obtained a bound on the p-divisibility of exponential sums involving multi-variable polynomials of fixed degree d over a finite field of characteristic p. This bound was subsequently improved by Katz. More recently, Moreno and Moreno, and Adolphson and Sperber, derived bounds that in many instances improved upon the AxāKatz result. Here we derive a tight bound on the p-divisibility of the exponential sums. While exact computation of this bound requires the solution of a system of modular equations, approximations are provided which in several classes of examples, improve on the results of Chevalley and Warning, Ax and Katz, Adolphson and Sperber, and Moreno and Moreno. All of the above results readily translate into bounds on the p-divisibility of the number of zeros of multi-variable polynomials. An important consequence of one of our main results is a method to find classes of examples for which bounds on divisibility of the number of solutions of a system of polynomial equations over finite fields are tight. In particular, we give classes of examples for which the MorenoāMoreno bound is tight. It is important to note that we have also found applications of our results to coding theory (computation of the covering radius of certain codes) and to Waring's problem over finite fields. These will be described elsewhere. 2000 Mathematics Subject Classification 11L07 (primary), 11G25 (secondary).
We define a sequence of polynomials Pd ā Z[x, y]. such that Pd is absolutely irreducible, of degree d, has low height, and has at least d2 + 2d + ā¦
We define a sequence of polynomials Pd ā Z[x, y]. such that Pd is absolutely irreducible, of degree d, has low height, and has at least d2 + 2d + 3 integral solutions to Pd(X, y) = 0. We know of no other nontrivial family of polynomials of increasing degree with as many integral solutions in terms of their degree.
We present an elementary proof of the quantum de Finetti representation theorem, a quantum analog of de Finettiās classical theorem on exchangeable probability assignments. This contrasts with the original proof ā¦
We present an elementary proof of the quantum de Finetti representation theorem, a quantum analog of de Finettiās classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potential for generalization. The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. The quantum de Finetti theorem, in a closely analogous fashion, deals with exchangeable density-operator assignments and provides an operational definition of the concept of an āunknown quantum stateā in quantum-state tomography. This result is especially important for information-based interpretations of quantum mechanics, where quantum states, like probabilities, are taken to be states of knowledge rather than states of nature. We further demonstrate that the theorem fails for real Hilbert spaces and discuss the significance of this point.
Quantum key distribution (QKD) protocols are cryptographic techniques with security based only on the laws of quantum mechanics. Two prominent QKD schemes are the BB84 and B92 protocols that use ā¦
Quantum key distribution (QKD) protocols are cryptographic techniques with security based only on the laws of quantum mechanics. Two prominent QKD schemes are the BB84 and B92 protocols that use four and two quantum states, respectively. In 2000, Phoenix et al. proposed a new family of three state protocols that offers advantages over the previous schemes. Until now, an error rate threshold for security of the symmetric trine spherical code QKD protocol has only been shown for the trivial intercept/resend eavesdropping strategy. In this paper, we prove the unconditional security of the trine spherical code QKD protocol, demonstrating its security up to a bit error rate of 9.81%. We also discuss on how this proof applies to a version of the trine spherical code QKD protocol where the error rate is evaluated from the number of inconclusive events.
We consider the existence in arbitrary finite dimensions d of a POVM comprised of d^2 rank-one operators all of whose operator inner products are equal. Such a set is called ā¦
We consider the existence in arbitrary finite dimensions d of a POVM comprised of d^2 rank-one operators all of whose operator inner products are equal. Such a set is called a ``symmetric, informationally complete'' POVM (SIC-POVM) and is equivalent to a set of d^2 equiangular lines in C^d. SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.
Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained ā¦
Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scottās code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.
Symmetric informationally complete positive operator-valued measures (SIC-POVMs) have many applications in quantum information. However, it is not easy to construct SIC-POVMs and there are only a few known classes of ā¦
Symmetric informationally complete positive operator-valued measures (SIC-POVMs) have many applications in quantum information. However, it is not easy to construct SIC-POVMs and there are only a few known classes of them, and we do not even know whether there exists an infinite class of them, thus constructing approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs) has its own meaning. In this paper, we use character sums over finite fields to present two constructions of ASIC-POVMs. We show that there are some classes of infinite families of ASIC-POVMs by using some special functions over finite fields.
A quantum measurement is Fisher symmetric if it provides uniform and maximal information on all parameters that characterize the quantum state of interest. Using (complex projective) 2-designs, we construct measurements ā¦
A quantum measurement is Fisher symmetric if it provides uniform and maximal information on all parameters that characterize the quantum state of interest. Using (complex projective) 2-designs, we construct measurements on a pair of identically prepared quantum states that are Fisher symmetric for all pure states. Such measurements are optimal in achieving the minimal statistical error without adaptive measurements. We then determine all collective measurements on a pair that are Fisher symmetric for the completely mixed state and for all pure states simultaneously. For a qubit, these measurements are Fisher symmetric for all states. The minimal optimal measurements are tied to the elusive symmetric informationally complete measurements, which reflects a deep connection between local symmetry and global symmetry. In the study, we derive a fundamental constraint on the Fisher information matrix of any collective measurement on a pair, which offers a useful tool for characterizing the tomographic efficiency of collective measurements.
We show that the Clifford group---the normaliser of the Weyl-Heisenberg group---can be represented by monomial phase-permutation matrices if and only if the dimension is a square number. This simplifies expressions ā¦
We show that the Clifford group---the normaliser of the Weyl-Heisenberg group---can be represented by monomial phase-permutation matrices if and only if the dimension is a square number. This simplifies expressions for SIC vectors, and has other applications to SICs and to Mutually Unbiased Bases. Exact solutions for SICs in dimension 16 are presented for the first time.
It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that ā¦
It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of $k$-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).
The author advocates two specific mathematical notations from his popular course and joint textbook, Concrete Mathematics. The first of these, extending an idea of Iverson, is the notation [P] for ā¦
The author advocates two specific mathematical notations from his popular course and joint textbook, Concrete Mathematics. The first of these, extending an idea of Iverson, is the notation [P] for the function which is 1 when the Boolean condition P is true and 0 otherwise. This notation can encourage and clarify the use of characteristic functions and Kronecker deltas in sums and integrals.
The second notation puts Stirling numbers on the same footing as binomial coefficients. Since binomial coefficients are written on two lines in parentheses and read n choose k, Stirling numbers of the first kind should be written on two lines in brackets and read n cycle k, while Stirling numbers of the second kind should be written in braces and read n subset k. (I might say n partition k.) The written form was first suggested by Imanuel Marx. The virtues of this notation are that Stirling partition numbers frequently appear in combinatorics, and that it more clearly presents functional relations similar to those satisfied by binomial coefficients.
We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of ā¦
We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC POVMs contains weak SIC POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC POVM.
Two-to-one (2-to-1) mappings over finite fields play an important role in symmetric cryptography. In particular they allow to design APN functions, bent functions and semi-bent functions. In this paper we ā¦
Two-to-one (2-to-1) mappings over finite fields play an important role in symmetric cryptography. In particular they allow to design APN functions, bent functions and semi-bent functions. In this paper we provide a systematic study of two-to-one mappings that are defined over finite fields. We characterize such mappings by means of the Walsh transforms. We also present several constructions, including an AGW-like criterion, constructions with the form of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x^{r}h(x^{(q-1)/d})$ </tex-math></inline-formula> , those from permutation polynomials, from linear translators and from APN functions. Then we present 2-to-1 polynomial mappings in classical classes of polynomials: linearized polynomials and monomials, low degree polynomials, Dickson polynomials and Muller-Cohen-Matthews polynomials, etc. Lastly, we show applications of 2-to-1 mappings over finite fields for constructions of bent Boolean and vectorial bent functions, semi-bent functions, planar functions and permutation polynomials. In all those respects, we shall review what is known and provide several new results.
Abstract The existence of $d^2$ pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in $d$-dimensional Hilbert space is known only for finitely many dimensions $d$. ā¦
Abstract The existence of $d^2$ pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in $d$-dimensional Hilbert space is known only for finitely many dimensions $d$. We prove that, if there exists a set of real units in a certain ray class field (depending on $d$) satisfying certain algebraic properties, a SIC-POVM exists, when $d$ is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at $s=0$ and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.
Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1) every POVM element is ā¦
Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1) every POVM element is rank one, (2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and (3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two, we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.
Symmetric informationally complete measurements (SICs) are elegant, celebrated, and broadly useful discrete structures in Hilbert space. We introduce a more sophisticated discrete structure compounded by several SICs. A SIC compound ā¦
Symmetric informationally complete measurements (SICs) are elegant, celebrated, and broadly useful discrete structures in Hilbert space. We introduce a more sophisticated discrete structure compounded by several SICs. A SIC compound is defined to be a collection of d3 vectors in d-dimensional Hilbert space that can be partitioned in two different ways: into d SICs and into d2 orthonormal bases. While a priori their existence may appear unlikely when d>2, we surprisingly find an explicit construction for d=4. Remarkably this SIC compound admits a close relation to mutually unbiased bases, as is revealed through quantum state discrimination. Going beyond fundamental considerations, we leverage these exotic properties to construct a protocol for quantum key distribution and analyze its security under general eavesdropping attacks. We show that SIC compounds enable secure key generation in the presence of errors that are large enough to prevent the success of the generalization of the six-state protocol.Received 8 July 2020Accepted 14 August 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.043122Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum communicationQuantum cryptographyQuantum foundationsQuantum measurementsGeneral PhysicsQuantum Information
We report on a new computer study of the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite ā¦
We report on a new computer study of the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and are the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory. We provide numerical solutions in all dimensions dā¤67 and, moreover, a putatively complete list of WeylāHeisenberg covariant solutions for dā¤50. A symmetry analysis of this list leads to new algebraic solutions in dimensions d=24, 35, and 48, which are given together with algebraic solutions for d=4,ā¦,15, and 19.
Examples of symmetric informationally complete positive operator valued measures (SIC-POVMs) have been constructed in every dimension less than or equal to 67. However, it remains an open question whether they ā¦
Examples of symmetric informationally complete positive operator valued measures (SIC-POVMs) have been constructed in every dimension less than or equal to 67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra gl(d,C). In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl(d,C). We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable.
An equiangular tight frame (ETF) yields a type of optimal packing of lines in a Euclidean space. ETFs seem to be rare, and all known infinite families of them arise ā¦
An equiangular tight frame (ETF) yields a type of optimal packing of lines in a Euclidean space. ETFs seem to be rare, and all known infinite families of them arise from some type of combinatorial design. In this paper, we introduce a new method for constructing ETFs. We begin by showing that it is sometimes possible to construct multiple ETFs for the same space that are "mutually unbiased" in a way that is analogous to the quantum-information-theoretic concept of mutually unbiased bases. We then show that taking certain tensor products of these mutually unbiased ETFs with other ETFs sometimes yields infinite families of new complex ETFs.
Mutually unbiased bases (MUB) and symmetric informationally complete positive operator-valued measure (SIC-POVM) are both important objects in quantum information theory.While people do not know if there exists a complete MUB ā¦
Mutually unbiased bases (MUB) and symmetric informationally complete positive operator-valued measure (SIC-POVM) are both important objects in quantum information theory.While people do not know if there exists a complete MUB for non-prime-power dimension,several versions of approximately MUB have been considered by relaxed the inner product condition.So far there are only finite number of K such that SICPOVMs in C k have been found.As in the MUB case,several versions of approximately SIC-POVM have been considered by relaxed the inner product condition.In this paper,we use the definitions of approximate MUB and SIC-POVM given by Klappenecker et al.For prime power q,we present simple constructions of q approximately MUB (AMUB) for dimension q-1,q+1 AMUB for dimension q-1,which shows the number of orthonormal bases of an AMUB in C k can be more than K+1,and q AMUB for dimension q+1 by Gauss and Jacobi sums.We also present a construction of approximately SIC-POVM (ASIC-POVM) in dimension q-1 by Gauss sum.