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:1202.5435 · Physical Review A

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

Ulrike Herzog

Published Mar 12, 2012• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.85.032312

Related papers

Discrimination of two mixed quantum states with maximum confidence and minimum probability of inconclusive results

Ulrike Herzog

We study an optimized measurement that discriminates two mixed quantum states with maximum confidence for each conclusive result, thereby keeping the overall probability of inconclusive results as small as possible. When the rank of the detection operators associated with the two different conclusive outcomes does not exceed unity, we obtain a general solution. As an application, we consider the discrimination of two mixed qubit states. Moreover, for the case of higher-rank detection operators we give a solution for particular states. The relation of the optimized measurement to other discrimination schemes is also discussed.

arXiv:0812.3174•Physical Review A•Mar 19, 2009•25 citations

Optimal state discrimination with a fixed rate of inconclusive results: Analytical solutions and relation to state discrimination with a fixed error rate

Ulrike Herzog

We study an optimum measurement for quantum state discrimination, which maximizes the probability of correct results when the probability of inconclusive results is fixed at a given value. The measurement describes minimum-error discrimination if this value is zero, while under certain conditions it corresponds to optimized maximum-confidence discrimination, or to optimum unambiguous discrimination, respectively, when the fixed value reaches a definite minimum. Using operator conditions that determine the optimum measurement, we derive analytical solutions for the discrimination of two mixed qubit states, including the case of two pure states occurring with arbitrary prior probabilities, and for the discrimination of $N$ symmetric states, both pure and mixed. We also consider a case where the given density operators resolve the identity operator, and we specify the optimality conditions for partially symmetric states. Moreover, we show that from the complete solution for arbitrary values of the fixed rate of inconclusive results one can always obtain the optimum measurement in another strategy where the error rate is fixed, and vice versa.

arXiv:1206.4412•Physical Review A•Sep 14, 2012•45 citations

Optimal measurements for the discrimination of quantum states with a fixed rate of inconclusive results

Ulrike Herzog

We study the discrimination of $N$ mixed quantum states in an optimal measurement that maximizes the probability of correct results while the probability of inconclusive results is fixed at a given value. After considering the discrimination of $N$ states in a $d$-dimensional Hilbert space, we focus on the discrimination of qubit states. We develop a method to determine an optimal measurement for discriminating arbitrary qubit states, taking into account that often the optimal measurement is not unique and the maximum probability of correct results can be achieved by several different measurements. Analytical results are derived for a number of examples, mostly for the discrimination between qubit states which possess a partial symmetry, but also for discriminating $N$ equiprobable qubit states and for the dicrimination between a pure and a uniformly mixed state in $d$ dimensions. In the special case where the fixed rate of inconclusive results is equal to zero, our method provides a treatment for the minimum-error discrimination of arbitrary qubit states which differs from previous approaches.

arXiv:1504.03917•Physical Review A•Apr 29, 2015•24 citations

Optimum measurement for unambiguously discriminating two mixed states: General considerations and special cases

Ulrike Herzog, János A Bergou

Based on our previous publication [U. Herzog and J. A. Bergou, Phys. Rev. A 71 , 050301(R)(2005)] we investigate the optimum measurement for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. Unambiguous discrimination of nonorthogonal states is possible in a probabilistic way, at the expense of a nonzero probability of inconclusive results, where the measurement fails. Along with a discussion of the general problem, we give an example illustrating our method of solution. We also provide general inequalities for the minimum achievable failure probability and discuss in more detail the necessary conditions that must be fulfilled when its absolute lower bound, proportional to the fidelity of the states, can be reached.

arXiv:quant-ph/0510170•Journal of Physics Conference Series•Apr 1, 2006•2 citations

Experimental Realization of Maximum Confidence Quantum State Discrimination for the Extraction of Quantum Information

Peter J. Mosley, Sarah Croke, Ian A. Walmsley +1 more

We present the first experimental demonstration of the maximum confidence measurement strategy for quantum state discrimination. Applying this strategy to an arbitrary set of states assigns to each input state a measurement outcome which, when realized, gives the highest possible confidence that the state was indeed present. The theoretically optimal measurement for discriminating between three equiprobable symmetric qubit states is implemented in a polarization-based free-space interferometer. The maximum confidence in the measurement result is 2/3. This is the first explicit demonstration that an improvement in the confidence over the optimal minimum error measurement is possible for linearly dependent states.

arXiv:quant-ph/0610237•Physical Review Letters•Nov 9, 2006•55 citations

Discrimination between geometrically uniform quantum states with inconclusive results

Kenji Nakahira, Tsuyoshi Sasaki Usuda, Kentaro Kato

We consider the problem of discrimination between a set of quantum states with the maximum probability of correct detection for a fixed rate of inconclusive results. We show that this type of measurement can be decomposed into two steps. The first step is the discrimination of whether the measurement result is inconclusive. The second step, which is performed in the case in which a conclusive result was obtained, is minimum error discrimination of output states of the first step. Using this result, we derive a closed-form analytical expression for an optimal measurement for both a geometrically uniform pure-state set and a set of binary doubly symmetric mixed states for any probability of an inconclusive result.

Physical Review A•Sep 14, 2012•19 citations
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⩽Nd\ensuremath{\leqslant}Nd\ensuremath⩽N) and the discrimination of NNN symmetric mixed qubit states.
We study an optimized measurement that discriminates two mixed quantum states with maximum confidence for each conclusive result, thereby keeping the overall probability of inconclusive results as small as possible. When the rank of the detection operators associated with the two different conclusive outcomes does not exceed unity, we obtain a general solution. As an application, we consider the discrimination of two mixed qubit states. Moreover, for the case of higher-rank detection operators we give a solution for particular states. The relation of the optimized measurement to other discrimination schemes is also discussed.
We study an optimum measurement for quantum state discrimination, which maximizes the probability of correct results when the probability of inconclusive results is fixed at a given value. The measurement describes minimum-error discrimination if this value is zero, while under certain conditions it corresponds to optimized maximum-confidence discrimination, or to optimum unambiguous discrimination, respectively, when the fixed value reaches a definite minimum. Using operator conditions that determine the optimum measurement, we derive analytical solutions for the discrimination of two mixed qubit states, including the case of two pure states occurring with arbitrary prior probabilities, and for the discrimination of NNN symmetric states, both pure and mixed. We also consider a case where the given density operators resolve the identity operator, and we specify the optimality conditions for partially symmetric states. Moreover, we show that from the complete solution for arbitrary values of the fixed rate of inconclusive results one can always obtain the optimum measurement in another strategy where the error rate is fixed, and vice versa.
We study the discrimination of NNN mixed quantum states in an optimal measurement that maximizes the probability of correct results while the probability of inconclusive results is fixed at a given value. After considering the discrimination of NNN states in a ddd-dimensional Hilbert space, we focus on the discrimination of qubit states. We develop a method to determine an optimal measurement for discriminating arbitrary qubit states, taking into account that often the optimal measurement is not unique and the maximum probability of correct results can be achieved by several different measurements. Analytical results are derived for a number of examples, mostly for the discrimination between qubit states which possess a partial symmetry, but also for discriminating NNN equiprobable qubit states and for the dicrimination between a pure and a uniformly mixed state in ddd dimensions. In the special case where the fixed rate of inconclusive results is equal to zero, our method provides a treatment for the minimum-error discrimination of arbitrary qubit states which differs from previous approaches.
Based on our previous publication [U. Herzog and J. A. Bergou, Phys. Rev. A 71 , 050301(R)(2005)] we investigate the optimum measurement for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. Unambiguous discrimination of nonorthogonal states is possible in a probabilistic way, at the expense of a nonzero probability of inconclusive results, where the measurement fails. Along with a discussion of the general problem, we give an example illustrating our method of solution. We also provide general inequalities for the minimum achievable failure probability and discuss in more detail the necessary conditions that must be fulfilled when its absolute lower bound, proportional to the fidelity of the states, can be reached.
We present the first experimental demonstration of the maximum confidence measurement strategy for quantum state discrimination. Applying this strategy to an arbitrary set of states assigns to each input state a measurement outcome which, when realized, gives the highest possible confidence that the state was indeed present. The theoretically optimal measurement for discriminating between three equiprobable symmetric qubit states is implemented in a polarization-based free-space interferometer. The maximum confidence in the measurement result is 2/3. This is the first explicit demonstration that an improvement in the confidence over the optimal minimum error measurement is possible for linearly dependent states.
We consider the problem of discrimination between a set of quantum states with the maximum probability of correct detection for a fixed rate of inconclusive results. We show that this type of measurement can be decomposed into two steps. The first step is the discrimination of whether the measurement result is inconclusive. The second step, which is performed in the case in which a conclusive result was obtained, is minimum error discrimination of output states of the first step. Using this result, we derive a closed-form analytical expression for an optimal measurement for both a geometrically uniform pure-state set and a set of binary doubly symmetric mixed states for any probability of an inconclusive result.