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

Communications in Theoretical Physics

Minimax Strategy of Optimal Unambiguous State Discrimination

Wen‐Hai Zhang, Long-Bao Yu, Zhuo‐Liang Cao, Liu Ye

Published Aug 1, 2012• 0 citations
Publisher

Abstract

In this paper, we consider the minimax strategy to unambiguously discriminate two pure nonorthogonal quantum states without knowing a priori probability. By exploiting the positive-operator valued measure, we derive the upper bound of the minimax measurement of the optimal unambiguous state discrimination. Based on the linear optical devices, we propose an experimentally feasible scheme to implement a minimax measure of a general pair of two nonorthogonal quantum states.

DOI
10.1088/0253-6102/58/2/07

Related papers

Minimax quantum-state discrimination

Giacomo Mauro D’Ariano, Massimiliano Federico Sacchi, Jonas Kahn

We derive the optimal measurement for quantum-state discrimination without a priori probabilities, i.e., in a minimax strategy instead of the usually considered Bayesian one. We consider both minimal-error and unambiguous discrimination problems, and provide the relation between the optimal measurements according to the two schemes. We show that there are instances in which the minimum risk cannot be achieved by an orthogonal measurement, and this is a common feature of the minimax estimation strategy.

arXiv:quant-ph/0504048•Physical Review A•Sep 8, 2005•50 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

Optical Realization of Optimal Unambiguous Discrimination for Pure and Mixed Quantum States

Masoud Mohseni, Aephraim M. Steinberg, János A. Bergou

Quantum mechanics forbids deterministic discrimination among nonorthogonal states. Nonetheless, the capability to distinguish nonorthogonal states unambiguously is an important primitive in quantum information processing. In this work, we experimentally implement generalized measurements in an optical system and demonstrate the first optimal unambiguous discrimination between three nonorthogonal states, with a success rate of 55%, to be compared with the 25% maximum achievable using projective measurements. Furthermore, we present the first realization of unambiguous discrimination between a pure state and a nonorthogonal mixed state.

arXiv:quant-ph/0401002•Physical Review Letters•Nov 9, 2004•119 citations

Optimum unambiguous discrimination between subsets of nonorthogonal quantum states

Yuqing Sun, János A. Bergou, Mark S. Hillery

It is known that unambiguous discrimination among nonorthogonal but linearly independent quantum states is possible with a certain probability of success. Here, we consider a variant of that problem. Instead of discriminating among all of the different states, we shall only discriminate between two subsets of them. In particular, for the case of three nonorthogonal states, ${|{\ensuremath{\psi}}_{1}〉,|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉},$ we show that the optimal strategy to distinguish $|{\ensuremath{\psi}}_{1}〉$ from the set ${|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉}$ has a higher success rate than if we wish to discriminate among all three states. Somewhat surprisingly, for unambiguous discrimination the subsets need not be linearly independent. A fully analytical solution is presented, and we also show how to construct generalized interferometers (multiport) which provide an optical implementation of the optimal strategy.

arXiv:quant-ph/0112051•Physical Review A•Sep 23, 2002•72 citations

Experimental scheme for unambiguous discrimination of linearly independent symmetric states

O. Jiménez, X. Sánchez-Lozano, Enrique B. Inostroza +2 more

We propose an experimental setup for discriminating four linearly independent nonorthogonal symmetric quantum states. The setup is based on linear optics only and can be configured to implement both optimal unambiguous state discrimination [Chefles and Barnett, Phys. Lett. A 250, 223 (1998)] and minimum error discrimination. In both cases, the setup is characterized by an optimal success probability. The experimental setup can be generalized to the case of discrimination among $N$ linearly nonorthogonal symmetric quantum states. We also study the discrimination between two incoherent superpositions of symmetric states. In this case, the setup also achieves an optimal success probability in the case of unambiguous discrimination as well as minimum error discrimination.

arXiv:0705.3813•Physical Review A•Dec 11, 2007•19 citations

On the feasibility conditions of quantum state discrimination

Chung-Chin Lu, Shiuan-Hao Kuo

In this paper, we consider the quantum unambiguous discrimination problem of a set of linearly independent pure states. With a constructive procedure, we establish a necessary and sufficient condition for feasible POVM measurements for unambiguous quantum state discrimination. We develop methods to calculate the optimal discrimination efficiencies for minimizing the inconclusive probability and the optimal discrimination efficiencies for the minimax criterion. Finally, we prove a necessary and sufficient condition to have a feasible POVM measurement which achieves the minimum inconclusive probability criterion and the minimax criterion simultaneously and show that geometrically uniform states with equal a priori probabilities meet this condition.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2017•0 citations
In this paper, we consider the minimax strategy to unambiguously discriminate two pure nonorthogonal quantum states without knowing a priori probability. By exploiting the positive-operator valued measure, we derive the upper bound of the minimax measurement of the optimal unambiguous state discrimination. Based on the linear optical devices, we propose an experimentally feasible scheme to implement a minimax measure of a general pair of two nonorthogonal quantum states.
We derive the optimal measurement for quantum-state discrimination without a priori probabilities, i.e., in a minimax strategy instead of the usually considered Bayesian one. We consider both minimal-error and unambiguous discrimination problems, and provide the relation between the optimal measurements according to the two schemes. We show that there are instances in which the minimum risk cannot be achieved by an orthogonal measurement, and this is a common feature of the minimax estimation strategy.
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.
Quantum mechanics forbids deterministic discrimination among nonorthogonal states. Nonetheless, the capability to distinguish nonorthogonal states unambiguously is an important primitive in quantum information processing. In this work, we experimentally implement generalized measurements in an optical system and demonstrate the first optimal unambiguous discrimination between three nonorthogonal states, with a success rate of 55%, to be compared with the 25% maximum achievable using projective measurements. Furthermore, we present the first realization of unambiguous discrimination between a pure state and a nonorthogonal mixed state.
It is known that unambiguous discrimination among nonorthogonal but linearly independent quantum states is possible with a certain probability of success. Here, we consider a variant of that problem. Instead of discriminating among all of the different states, we shall only discriminate between two subsets of them. In particular, for the case of three nonorthogonal states, ${|{\ensuremath{\psi}}_{1}〉,|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉},$ we show that the optimal strategy to distinguish $|{\ensuremath{\psi}}_{1}〉$ from the set ${|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉}$ has a higher success rate than if we wish to discriminate among all three states. Somewhat surprisingly, for unambiguous discrimination the subsets need not be linearly independent. A fully analytical solution is presented, and we also show how to construct generalized interferometers (multiport) which provide an optical implementation of the optimal strategy.
We propose an experimental setup for discriminating four linearly independent nonorthogonal symmetric quantum states. The setup is based on linear optics only and can be configured to implement both optimal unambiguous state discrimination [Chefles and Barnett, Phys. Lett. A 250, 223 (1998)] and minimum error discrimination. In both cases, the setup is characterized by an optimal success probability. The experimental setup can be generalized to the case of discrimination among NNN linearly nonorthogonal symmetric quantum states. We also study the discrimination between two incoherent superpositions of symmetric states. In this case, the setup also achieves an optimal success probability in the case of unambiguous discrimination as well as minimum error discrimination.
In this paper, we consider the quantum unambiguous discrimination problem of a set of linearly independent pure states. With a constructive procedure, we establish a necessary and sufficient condition for feasible POVM measurements for unambiguous quantum state discrimination. We develop methods to calculate the optimal discrimination efficiencies for minimizing the inconclusive probability and the optimal discrimination efficiencies for the minimax criterion. Finally, we prove a necessary and sufficient condition to have a feasible POVM measurement which achieves the minimum inconclusive probability criterion and the minimax criterion simultaneously and show that geometrically uniform states with equal a priori probabilities meet this condition.