Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Quantum measurement for a group-covariant state set

Kenji Nakahira, Tsuyoshi Sasaki Usuda

Published Jan 10, 2013• 10 citations
Publisher

Abstract

We consider a group covariant quantum state set with a group in which each element corresponds to a unitary or antiunitary operator. This type of quantum state set can express a broad class of quantum state sets, including geometrically uniform state sets and self-symmetric state sets. We derive that for any quantum measurement for a group-covariant state set (both with and without a certain fraction of inconclusive results), a group covariant quantum measurement exists with respect to the same group with the same performance, that is, the same probability of correct detection and the same rate of inconclusive results. We then show that a group covariant optimal measurement exists for any group-covariant state set. In some cases, we derive an optimal measurement for a group-covariant state set.

DOI
10.1103/physreva.87.012308

Related papers

Mixed-quantum-state detection with inconclusive results

Yonina C. Eldar

We consider the problem of designing an optimal quantum detector with a fixed rate of inconclusive results that maximizes the probability of correct detection, when distinguishing between a collection of mixed quantum states. We show that the design of the optimal detector can be formulated as a semidefinite programming problem, and derive a set of necessary and sufficient conditions for an optimal measurement. We then develop a sufficient condition for the scaled inverse measurement to maximize the probability of correct detection for the case in which the rate of inconclusive results exceeds a certain threshold. Using this condition we derive the optimal measurement for linearly independent pure-state sets, and for mixed-state sets with a broad class of symmetries. Specifically, we consider geometrically uniform (GU) state sets and compound geometrically uniform (CGU) state sets with generators that satisfy a certain constraint. We then show that the optimal measurements corresponding to GU and CGU state sets with arbitrary generators are also GU and CGU, respectively, with generators that can be computed very efficiently in polynomial time within any desired accuracy by solving a reduced size semidefinite programming problem.

arXiv:quant-ph/0211121•Physical Review A•Apr 15, 2003•76 citations

A Semidefinite Programming Approach to Optimal Unambiguous Discrimination of Quantum States

Yonina C. Eldar

Abstract—In this paper, we consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum. Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal. Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We first consider geometrically uniform (GU) state sets that are defined over a group of unitary matrices and are generated by a single generating vector. We then consider compound GU state sets which are generated by a group of unitary matrices using multiple generating vectors, where the generating vectors satisfy a certain (weighted) norm constraint. Index Terms—Compound geometrically uniform (CGU) quantum states, equal-probability measurement (EPM), geometrically uniform (GU) quantum states, quantum detection, semidefinite programming, unambiguous discrimination. I.

•Jun 14, 2002•83 citations

A semidefinite programming approach to optimal unambiguous discrimination of quantum states

Yonina C. Eldar

We consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum. Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal. Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We first consider geometrically uniform (GU) state sets that are defined over a group of unitary matrices and are generated by a single generating vector. We then consider compound GU state sets which are generated by a group of unitary matrices using multiple generating vectors, where the generating vectors satisfy a certain (weighted) norm constraint.

arXiv:quant-ph/0206093•IEEE Transactions on Information Theory•Feb 1, 2003•120 citations

Generalized quantum state discrimination problems

Kenji Nakahira, Kentaro Kato, Tsuyoshi Sasaki Usuda

We address a broad class of optimization problems of finding quantum measurements, which includes the problems of finding an optimal measurement in the Bayes criterion and a measurement maximizing the average correct probability with a fixed rate of inconclusive results. Our approach can deal with any problem in which each of the objective and constraint functions is formulated by the sum of the traces of the multiplication of a Hermitian operator and a detection operator. We first derive dual problems and necessary and sufficient conditions for an optimal measurement. We also consider the minimax version of these problems and provide necessary and sufficient conditions for a minimax solution. Finally, for optimization problem having a certain symmetry, there exists an optimal solution with the same symmetry. Examples are shown to illustrate how our results can be used.

arXiv:1501.05744•Physical Review A•May 6, 2015•31 citations

Optimal covariant quantum measurements

Erkka Haapasalo, Juha-Pekka Pellonpää

Abstract We discuss symmetric quantum measurements and the associated covariant observables modelled, respectively, as instruments and positive-operator-valued measures. The emphasis of this work are the optimality properties of the measurements, namely, extremality, informational completeness, and the rank-1 property which contrast the complementary class of (rank-1) projection-valued measures. The first half of this work concentrates solely on finite-outcome measurements symmetric w.r.t. finite groups where we derive exhaustive characterizations for the pointwise Kraus-operators of covariant instruments and necessary and sufficient extremality conditions using these Kraus-operators. We motivate the use of covariance methods by showing that observables covariant with respect to symmetric groups contain a family of representatives from both of the complementary optimality classes of observables and show that even a slight deviation from a rank-1 projection-valued measure can yield an extreme informationally complete rank-1 observable. The latter half of this work derives similar results for continuous measurements in (possibly) infinite dimensions. As an example we study covariant phase space instruments, their structure, and extremality properties.

arXiv:2009.14080•Journal of Physics A Mathematical and Theoretical•Feb 17, 2021•5 citations

Minimum-Bayes-cost discrimination for symmetric quantum states

Kenji Nakahira, Tsuyoshi Sasaki Usuda

We derive the measurement strategy that optimizes the Bayes criterion, not necessarily the strategy minimizing the error probability, for distinguishing between collections of quantum states. We present a sufficient condition concerning the rank of each detection operator of an optimal measurement for the Bayes criterion. We also derive an analytical expression for an optimal measurement for the Bayes criterion for an Abelian geometrically uniform state set or for a cyclic state set under a certain constraint.

Physical Review A•Dec 6, 2012•5 citations
We consider a group covariant quantum state set with a group in which each element corresponds to a unitary or antiunitary operator. This type of quantum state set can express a broad class of quantum state sets, including geometrically uniform state sets and self-symmetric state sets. We derive that for any quantum measurement for a group-covariant state set (both with and without a certain fraction of inconclusive results), a group covariant quantum measurement exists with respect to the same group with the same performance, that is, the same probability of correct detection and the same rate of inconclusive results. We then show that a group covariant optimal measurement exists for any group-covariant state set. In some cases, we derive an optimal measurement for a group-covariant state set.
We consider the problem of designing an optimal quantum detector with a fixed rate of inconclusive results that maximizes the probability of correct detection, when distinguishing between a collection of mixed quantum states. We show that the design of the optimal detector can be formulated as a semidefinite programming problem, and derive a set of necessary and sufficient conditions for an optimal measurement. We then develop a sufficient condition for the scaled inverse measurement to maximize the probability of correct detection for the case in which the rate of inconclusive results exceeds a certain threshold. Using this condition we derive the optimal measurement for linearly independent pure-state sets, and for mixed-state sets with a broad class of symmetries. Specifically, we consider geometrically uniform (GU) state sets and compound geometrically uniform (CGU) state sets with generators that satisfy a certain constraint. We then show that the optimal measurements corresponding to GU and CGU state sets with arbitrary generators are also GU and CGU, respectively, with generators that can be computed very efficiently in polynomial time within any desired accuracy by solving a reduced size semidefinite programming problem.
Abstract—In this paper, we consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum. Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal. Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We first consider geometrically uniform (GU) state sets that are defined over a group of unitary matrices and are generated by a single generating vector. We then consider compound GU state sets which are generated by a group of unitary matrices using multiple generating vectors, where the generating vectors satisfy a certain (weighted) norm constraint. Index Terms—Compound geometrically uniform (CGU) quantum states, equal-probability measurement (EPM), geometrically uniform (GU) quantum states, quantum detection, semidefinite programming, unambiguous discrimination. I.
We consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum. Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal. Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We first consider geometrically uniform (GU) state sets that are defined over a group of unitary matrices and are generated by a single generating vector. We then consider compound GU state sets which are generated by a group of unitary matrices using multiple generating vectors, where the generating vectors satisfy a certain (weighted) norm constraint.
We address a broad class of optimization problems of finding quantum measurements, which includes the problems of finding an optimal measurement in the Bayes criterion and a measurement maximizing the average correct probability with a fixed rate of inconclusive results. Our approach can deal with any problem in which each of the objective and constraint functions is formulated by the sum of the traces of the multiplication of a Hermitian operator and a detection operator. We first derive dual problems and necessary and sufficient conditions for an optimal measurement. We also consider the minimax version of these problems and provide necessary and sufficient conditions for a minimax solution. Finally, for optimization problem having a certain symmetry, there exists an optimal solution with the same symmetry. Examples are shown to illustrate how our results can be used.
Abstract We discuss symmetric quantum measurements and the associated covariant observables modelled, respectively, as instruments and positive-operator-valued measures. The emphasis of this work are the optimality properties of the measurements, namely, extremality, informational completeness, and the rank-1 property which contrast the complementary class of (rank-1) projection-valued measures. The first half of this work concentrates solely on finite-outcome measurements symmetric w.r.t. finite groups where we derive exhaustive characterizations for the pointwise Kraus-operators of covariant instruments and necessary and sufficient extremality conditions using these Kraus-operators. We motivate the use of covariance methods by showing that observables covariant with respect to symmetric groups contain a family of representatives from both of the complementary optimality classes of observables and show that even a slight deviation from a rank-1 projection-valued measure can yield an extreme informationally complete rank-1 observable. The latter half of this work derives similar results for continuous measurements in (possibly) infinite dimensions. As an example we study covariant phase space instruments, their structure, and extremality properties.
We derive the measurement strategy that optimizes the Bayes criterion, not necessarily the strategy minimizing the error probability, for distinguishing between collections of quantum states. We present a sufficient condition concerning the rank of each detection operator of an optimal measurement for the Bayes criterion. We also derive an analytical expression for an optimal measurement for the Bayes criterion for an Abelian geometrically uniform state set or for a cyclic state set under a certain constraint.