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:1401.3701

Minimum Decision Cost for Operators

Bernhard K. Meister

Published Jan 15, 2014• 0 citations
arXivHTMLPDF

Abstract

State discrimination with the aim to minimize the error probability is a well studied problem. Instead, here the binary decision problem for operators with a given prior is investigated. A black box containing the unknown operator is probed by selected wave functions. The output is analyzed with conventional methods developed for state discrimination. An error probability bound for all binary operator choices is provided, and it is shown how probe entanglement enhances the result.

DOI
10.48550/arxiv.1401.3701

Related papers

Another look at the optimal Bayes cost in the binary decision problem

Bernhard K. Meister

The problem of quantum state discrimination between two wave functions of a particle in a square well potential is considered. The optimal minimum-error probability for the state discrimination is known to be given by the Helstrom bound. A new strategy is introduced here whereby the square well is compressed isoenergetically, modifying the wave-functions. The new contracted chamber is then probed using the conventional optimal strategy, and the error probability is calculated. It is shown that in some cases the Helstrom bound can be violated, i.e. the state discrimination can be realized with a smaller error probability.

arXiv:1001.3583•Jan 20, 2010•0 citations

Decision problems with quantum black boxes

Mark S. Hillery, Erika Andersson, Stephen Mark Barnett +1 more

We examine how to distinguish between unitary operators, when the exact form of the possible operators is not known. Instead we are supplied with ‘programs’ in the form of unitary transforms, which can be used as references for identifying the unknown unitary transform. All unitary transforms should be used as few times as possible. This situation is analogous to programmable state discrimination. One difference, however, is that the quantum state to which we apply the unitary transforms may be entangled, leading to a richer variety of possible strategies. By suitable selection of an input state and generalized measurement of the output state, both unambiguous and minimum-error discrimination can be achieved. Pairwise comparison of operators, comparing each transform to be identified with a program transform, is often a useful strategy. There are, however, situations in which more complicated strategies perform better. This is the case especially when the number of allowed applications of program operations is different from the number of the transforms to be identified.

arXiv:1109.4823•Journal of Modern Optics•Aug 26, 2009•4 citations

Reexamination of the optimal Bayes cost in the binary decision problem

Bernhard K. Meister

The problem of quantum state discrimination between two wave functions of a particle in a square well potential is considered. The optimal minimum-error probability is known to be given by the Helstrom bound. A new strategy is introduced by inserting an impenetrable barrier in the middle of the square well, which is either a nodal or non-nodal point of the wave function. The energy required to insert the barrier is dependent on the initial state. This enables the experimenter to gain additional information beyond the standard probing of the state envisaged by Helstrom and to improve the distinguishability of the states. It is shown that under some conditions the Helstrom bound can be violated, i.e. the state discrimination can be realized with a smaller error probability.

arXiv:1106.5196•Jun 26, 2011•0 citations

The Bayes cost in the binary decision problem

Bernhard K. Meister

The problem of quantum state discrimination between two wave functions on a ring is considered. The optimal minimum-error probability is known to be given by the Helstrom bound. A new strategy is introduced by inserting either adiabatically or instantaneously an impenetrable barrier. The insertion point, independent of the shape of the initial wave function, becomes a node. The resulting modified wave functions can be, if the initial functions are judiciously chosen, distinguished with a smaller error probability, and as a consequence the Helstrom bound can be violated under idealised conditions.

arXiv:1110.5284•Oct 24, 2011•0 citations

RESEARCH ARTICLE Decision problems with quantum black boxes

Mark S. Hillery, Erika Andersson, Stephen Mark Barnett +1 more

(September 23, 2011)We examine how to distinguish between unitary operators, when the exact form of thepossible operators is not known. Instead we are supplied with “programs” in the form ofunitary transforms, which can be used as references for identifying the unknown unitarytransform. All unitary transforms should be used as few times as possible. This situationis analoguous to programmable state discrimination. One difference, however, is that thequantum state to which we apply the unitary transforms may be entangled, leading to aricher variety of possible strategies. By suitable selection of an input state and generalizedmeasurement of the output state, both unambiguous and minimum-error discrimination canbe achieved. Pairwise comparison of operators, comparing each transform to be identifiedwith a program transform, is often a useful strategy. There are, however, situations in whichmore complicated strategies perform better. This is the case especially when the number ofallowed applications of program operations is different from the number of the transforms tobe identified.Keywords:unambiguous discrimination; optimum discrimination; operator comparison;generalized measurements

•Jan 1, 2011•0 citations

Structure of minimum-error quantum state discrimination

Joonwoo Bae

Distinguishing different quantum states is a fundamental task having practical applications in information processing. Despite the effort devoted so far, however, strategies for optimal discrimination are known only for specific examples. In this paper we consider the problem of minimum-error quantum state discrimination where one attempts to minimize the average error. We show the general structure of minimum-error state discrimination as well as useful properties to derive analytic solutions. Based on the general structure, we present a geometric formulation of the problem, which can be applied to cases where quantum state geometry is clear. We also introduce equivalent classes of sets of quantum states in terms of minimum-error discrimination: sets of quantum states in an equivalent class that share the same guessing probability. In particular, for qubit states where the state geometry is found with the Bloch sphere, we illustrate that for an arbitrary set of qubit states, the minimum-error state discrimination with equal prior probabilities can be analytically solved, that is, optimal measurement and the guessing probability are explicitly obtained.

arXiv:1210.2845•New Journal of Physics•Jul 18, 2013•81 citations
State discrimination with the aim to minimize the error probability is a well studied problem. Instead, here the binary decision problem for operators with a given prior is investigated. A black box containing the unknown operator is probed by selected wave functions. The output is analyzed with conventional methods developed for state discrimination. An error probability bound for all binary operator choices is provided, and it is shown how probe entanglement enhances the result.
The problem of quantum state discrimination between two wave functions of a particle in a square well potential is considered. The optimal minimum-error probability for the state discrimination is known to be given by the Helstrom bound. A new strategy is introduced here whereby the square well is compressed isoenergetically, modifying the wave-functions. The new contracted chamber is then probed using the conventional optimal strategy, and the error probability is calculated. It is shown that in some cases the Helstrom bound can be violated, i.e. the state discrimination can be realized with a smaller error probability.
We examine how to distinguish between unitary operators, when the exact form of the possible operators is not known. Instead we are supplied with ‘programs’ in the form of unitary transforms, which can be used as references for identifying the unknown unitary transform. All unitary transforms should be used as few times as possible. This situation is analogous to programmable state discrimination. One difference, however, is that the quantum state to which we apply the unitary transforms may be entangled, leading to a richer variety of possible strategies. By suitable selection of an input state and generalized measurement of the output state, both unambiguous and minimum-error discrimination can be achieved. Pairwise comparison of operators, comparing each transform to be identified with a program transform, is often a useful strategy. There are, however, situations in which more complicated strategies perform better. This is the case especially when the number of allowed applications of program operations is different from the number of the transforms to be identified.
The problem of quantum state discrimination between two wave functions of a particle in a square well potential is considered. The optimal minimum-error probability is known to be given by the Helstrom bound. A new strategy is introduced by inserting an impenetrable barrier in the middle of the square well, which is either a nodal or non-nodal point of the wave function. The energy required to insert the barrier is dependent on the initial state. This enables the experimenter to gain additional information beyond the standard probing of the state envisaged by Helstrom and to improve the distinguishability of the states. It is shown that under some conditions the Helstrom bound can be violated, i.e. the state discrimination can be realized with a smaller error probability.
The problem of quantum state discrimination between two wave functions on a ring is considered. The optimal minimum-error probability is known to be given by the Helstrom bound. A new strategy is introduced by inserting either adiabatically or instantaneously an impenetrable barrier. The insertion point, independent of the shape of the initial wave function, becomes a node. The resulting modified wave functions can be, if the initial functions are judiciously chosen, distinguished with a smaller error probability, and as a consequence the Helstrom bound can be violated under idealised conditions.
(September 23, 2011)We examine how to distinguish between unitary operators, when the exact form of thepossible operators is not known. Instead we are supplied with “programs” in the form ofunitary transforms, which can be used as references for identifying the unknown unitarytransform. All unitary transforms should be used as few times as possible. This situationis analoguous to programmable state discrimination. One difference, however, is that thequantum state to which we apply the unitary transforms may be entangled, leading to aricher variety of possible strategies. By suitable selection of an input state and generalizedmeasurement of the output state, both unambiguous and minimum-error discrimination canbe achieved. Pairwise comparison of operators, comparing each transform to be identifiedwith a program transform, is often a useful strategy. There are, however, situations in whichmore complicated strategies perform better. This is the case especially when the number ofallowed applications of program operations is different from the number of the transforms tobe identified.Keywords:unambiguous discrimination; optimum discrimination; operator comparison;generalized measurements
Distinguishing different quantum states is a fundamental task having practical applications in information processing. Despite the effort devoted so far, however, strategies for optimal discrimination are known only for specific examples. In this paper we consider the problem of minimum-error quantum state discrimination where one attempts to minimize the average error. We show the general structure of minimum-error state discrimination as well as useful properties to derive analytic solutions. Based on the general structure, we present a geometric formulation of the problem, which can be applied to cases where quantum state geometry is clear. We also introduce equivalent classes of sets of quantum states in terms of minimum-error discrimination: sets of quantum states in an equivalent class that share the same guessing probability. In particular, for qubit states where the state geometry is found with the Bloch sphere, we illustrate that for an arbitrary set of qubit states, the minimum-error state discrimination with equal prior probabilities can be analytically solved, that is, optimal measurement and the guessing probability are explicitly obtained.