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

arXiv:0806.2699 · Physical Review A

Optimal positive-operator-valued measures for unambiguous state discrimination

B. F. Samsonov

Published Apr 8, 2009• 11 citations
arXivHTMLPDF

Abstract

Optimization of the mean efficiency for unambiguous (or error-free) discrimination among $N$ given linearly independent nonorthogonal states should be realized in a way to keep the probabilistic quantum-mechanical interpretation. This imposes a condition on a certain matrix to be positive semidefinite. We reformulated this condition in such a way that the conditioned optimization problem for the mean efficiency was reduced to finding an unconditioned maximum of a function defined on a unit $N$ sphere for equiprobable states and on an $N$ ellipsoid if the states are given with different probabilities. We established that for equiprobable states a point on the sphere with equal values of Cartesian coordinates, which we call symmetric point, plays a special role. Sufficient conditions for a vector set are formulated for which the mean efficiency for equiprobable states takes its maximal value at the symmetric point. This set, in particular, includes previously studied symmetric states. A subset of symmetric states, for which the optimal measurement corresponds to a positive-operator-valued measure (POVM) requiring a one-dimensional ancilla space is constructed. We presented our constructions of a POVM suitable for the ancilla space dimension varying from 1 to $N$ and the Neumark extension differing from the existing schemes by the property that it is straightforwardly applicable to the case when it is desirable to present the whole space $\text{system}+\text{ancilla}$ as the tensor product of a two-dimensional ancilla space and the $N$-dimensional system space.

DOI
10.1103/physreva.79.042312

Related papers

Optimized maximum-confidence discrimination of N mixed quantum states and application to symmetric states

Ulrike Herzog

We study an optimized measurement which discriminates $N$ mixed quantum states occurring with given prior probabilities. The measurement yields the maximum achievable confidence for each of the $N$ conclusive outcomes, thereby keeping the overall probability of inconclusive outcomes as small as possible. It corresponds to optimum unambiguous discrimination when for each outcome the confidence is equal to unity. Necessary and sufficient optimality conditions are derived and general properties of the optimum measurement are obtained. The results are applied to the optimized maximum-confidence discrimination of $N$ equiprobable symmetric mixed states. Analytical solutions are presented for a number of examples, including the discrimination of $N$ symmetric pure states spanning a $d$-dimensional Hilbert space ($d\ensuremath{\leqslant}N$) and the discrimination of $N$ symmetric mixed qubit states.

arXiv:1202.5435•Physical Review A•Mar 12, 2012•16 citations

Optimal unambiguous filtering of a quantum state: An instance in mixed state discrimination

János A. Bergou, Ulrike Herzog, Mark S. Hillery

Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e., allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the $N$ given nonorthogonal states ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ are divided into two sets and the first set consists of one state only while the second consists of all of the remaining states. We present the derivation of the optimal measurement strategy, in terms of a generalized measurement (positive-operator-valued measure), to distinguish $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [Phys. Rev. Lett. 90, 25901 (2003)] as the emphasis there was on appplication of the results to probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.

arXiv:quant-ph/0504138•Physical Review A•Apr 11, 2005•29 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

Ancilla dimensions needed to carry out positive-operator-valued measurement

Ping-Xing Chen, János A. Bergou, Shi-Yao Zhu +1 more

To implement a positive-operator-valued measurement (POVM), which is defined on the ${d}_{S}$-dimensional Hilbert space of a physical system, one has to extend the Hilbert space to include ${d}_{A}$ additional dimensions (called the ancilla). This is done via either the tensor product extension (TPE) or the direct sum extension (DSE). The implementation of a POVM utilizes the available resources more efficiently if it requires fewer additional dimensions. To determine how to implement a POVM with the least additional dimensions is, therefore, an important task in quantum information. We have determined the necessary and sufficient (hence minimal) number of the additional dimensions needed to implement the same POVM by the TPE and the DSE, respectively. If the POVM has $n$ elements and ${r}_{i}$ is the rank of the $i$th element, then the dimension of the minimal ancilla is ${d}_{A}={\ensuremath{\sum}}_{i=1}^{n}{r}_{i}\ensuremath{-}{d}_{S}$ for the DSE implementation, and this represents a lower bound for the added dimensions in the TPE implementation. In the proof, we explicitly construct the DSE implementation of a general POVM with elements of arbitrary rank. As an example, we determine ${d}_{A}$ for the unambiguous discrimination of $N$ linearly independent states and provide the full DSE implementation of a state-discriminating POVM for $N=2$.

Physical Review A•Dec 20, 2007•39 citations

Time-energy trade-off in unambiguous-state-discrimination positive operator-valued measures

Raam Uzdin, Omri Gat

Unambiguous-nonorthogonal-state discrimination has a fundamental importance in quantum information. Moreover, it can be used for entanglement distillation and secure communication. The discrimination is carried out by a positive operator-valued measure (POVM) generalized measurement, which is typically implemented by coupling the system to an ancilla. We find a trade-off between the needed energy resources and the evolution time needed to implement the POVM and express it in terms of an actionlike cost inequality. We find the realization that minimizes this actionlike cost and show that, in this case, the cost is determined by the maximal population transfer from the system to the ancilla. We demonstrate our findings in an example of a three-level system coupled to a laser.

arXiv:1309.2620•Physical Review A•Nov 21, 2013•4 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
Optimization of the mean efficiency for unambiguous (or error-free) discrimination among NNN given linearly independent nonorthogonal states should be realized in a way to keep the probabilistic quantum-mechanical interpretation. This imposes a condition on a certain matrix to be positive semidefinite. We reformulated this condition in such a way that the conditioned optimization problem for the mean efficiency was reduced to finding an unconditioned maximum of a function defined on a unit NNN sphere for equiprobable states and on an NNN ellipsoid if the states are given with different probabilities. We established that for equiprobable states a point on the sphere with equal values of Cartesian coordinates, which we call symmetric point, plays a special role. Sufficient conditions for a vector set are formulated for which the mean efficiency for equiprobable states takes its maximal value at the symmetric point. This set, in particular, includes previously studied symmetric states. A subset of symmetric states, for which the optimal measurement corresponds to a positive-operator-valued measure (POVM) requiring a one-dimensional ancilla space is constructed. We presented our constructions of a POVM suitable for the ancilla space dimension varying from 1 to NNN and the Neumark extension differing from the existing schemes by the property that it is straightforwardly applicable to the case when it is desirable to present the whole space system+ancilla\text{system}+\text{ancilla}system+ancilla as the tensor product of a two-dimensional ancilla space and the NNN-dimensional system space.
We study an optimized measurement which discriminates NNN mixed quantum states occurring with given prior probabilities. The measurement yields the maximum achievable confidence for each of the NNN conclusive outcomes, thereby keeping the overall probability of inconclusive outcomes as small as possible. It corresponds to optimum unambiguous discrimination when for each outcome the confidence is equal to unity. Necessary and sufficient optimality conditions are derived and general properties of the optimum measurement are obtained. The results are applied to the optimized maximum-confidence discrimination of NNN equiprobable symmetric mixed states. Analytical solutions are presented for a number of examples, including the discrimination of NNN symmetric pure states spanning a ddd-dimensional Hilbert space ($d\ensuremath{\leqslant}N$) and the discrimination of NNN symmetric mixed qubit states.
Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e., allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the NNN given nonorthogonal states ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ are divided into two sets and the first set consists of one state only while the second consists of all of the remaining states. We present the derivation of the optimal measurement strategy, in terms of a generalized measurement (positive-operator-valued measure), to distinguish $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [Phys. Rev. Lett. 90, 25901 (2003)] as the emphasis there was on appplication of the results to probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.
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.
To implement a positive-operator-valued measurement (POVM), which is defined on the dS{d}_{S}dS​-dimensional Hilbert space of a physical system, one has to extend the Hilbert space to include dA{d}_{A}dA​ additional dimensions (called the ancilla). This is done via either the tensor product extension (TPE) or the direct sum extension (DSE). The implementation of a POVM utilizes the available resources more efficiently if it requires fewer additional dimensions. To determine how to implement a POVM with the least additional dimensions is, therefore, an important task in quantum information. We have determined the necessary and sufficient (hence minimal) number of the additional dimensions needed to implement the same POVM by the TPE and the DSE, respectively. If the POVM has nnn elements and ri{r}_{i}ri​ is the rank of the iiith element, then the dimension of the minimal ancilla is ${d}_{A}={\ensuremath{\sum}}_{i=1}^{n}{r}_{i}\ensuremath{-}{d}_{S}$ for the DSE implementation, and this represents a lower bound for the added dimensions in the TPE implementation. In the proof, we explicitly construct the DSE implementation of a general POVM with elements of arbitrary rank. As an example, we determine dA{d}_{A}dA​ for the unambiguous discrimination of NNN linearly independent states and provide the full DSE implementation of a state-discriminating POVM for N=2N=2N=2.
Unambiguous-nonorthogonal-state discrimination has a fundamental importance in quantum information. Moreover, it can be used for entanglement distillation and secure communication. The discrimination is carried out by a positive operator-valued measure (POVM) generalized measurement, which is typically implemented by coupling the system to an ancilla. We find a trade-off between the needed energy resources and the evolution time needed to implement the POVM and express it in terms of an actionlike cost inequality. We find the realization that minimizes this actionlike cost and show that, in this case, the cost is determined by the maximal population transfer from the system to the ancilla. We demonstrate our findings in an example of a three-level system coupled to a laser.
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.