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

Quantum Information and Computation

On impact of a priori classical knowledge of discriminated states on the optimal unambiguous discrimination

Ming Zhang, Z.-T. Zhou, Hui Dai, D.-W. Hu

Published Nov 1, 2008• 2 citations
Publisher

Abstract

Due to the fundamental limitations related to the Heisenberg uncertainty principle and the non-cloning theorem, it is impossible, even in principle, to determine the quantum state of a single system without a priori knowledge of it. To discriminate nonorthogonal quantum states in some optimal way, a priori knowledge of the discriminated states has to be relied upon. In this paper, we thoroughly investigate some impact of a priori classical knowledge of two quantum states on the optimal unambiguous discrimination. It is exemplified that a priori classical knowledge of the discriminated states, incomplete or complete, can be utilized to improve the optimal success probabilities, whereas the lack of a prior classical knowledge can not be compensated even by more resources.

DOI
10.26421/qic8.10-5

Related papers

Unambiguous pure-state identification without classical knowledge

A. Hayashi, M. Horibe, T. Hashimoto

We study how to unambiguously identify a given quantum pure state with one of the two reference pure states when no classical knowledge on the reference states is given but a certain number of copies of each reference quantum state are presented. By unambiguous identification, we mean that we are not allowed to make a mistake but our measurement can produce an inconclusive result. Assuming the two reference states are independently distributed over the whole pure state space in a unitary invariant way, we determine the optimal mean success probability for an arbitrary number of copies of the reference states and a general dimension of the state space. It is explicitly shown that the obtained optimal mean success probability asymptotically approaches that of the unambiguous discrimination as the number of the copies of the reference states increases.

arXiv:quant-ph/0510015•Physical Review A•Jan 20, 2006•49 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

Distinguishing mixed quantum states: Minimum-error discrimination versus optimum unambiguous discrimination

Ulrike Herzog, János A. Bergou

We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.

arXiv:quant-ph/0403124•Physical Review A•Aug 5, 2004•64 citations

Quantum State Seperation, Unambiguous Discrimination and Exact Cloning

Anthony Chefles, Stephen Mark Barnett

Unambiguous discrimination and exact cloning reduce the square-overlap between quantum states, exemplifying the more general type of procedure we term state separation. We obtain the maximum probability with which two equiprobable quantum states can be separated by an arbitrary degree, and find that the established bounds on the success probabilities for discrimination and cloning are special cases of this general bound. The latter also gives the maximum probability of successfully producing N exact copies of a quantum system whose state is chosen secretly from a known pair, given M initial realizations of the state, where N > M. We also discuss the relationship between this bound and that on unambiguous state discrimination.

European Quantum Electronics Conference (EQEC)•Aug 24, 2005•0 citations

Minimax Strategy of Optimal Unambiguous State Discrimination

Wen‐Hai Zhang, Long-Bao Yu, Zhuo‐Liang Cao +1 more

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.

Communications in Theoretical Physics•Aug 1, 2012•0 citations

Optimum Discrimination of Non-orthogonal Quantum States: An Optical Realization

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

Since the state of a quantum system is not an observable, it cannot be determined completely, in principle, if no a priori information is available. In this case, one needs to resort to various estimation strategies. The situation is completely different if a priori knowledge is available. Even unambigous discrimination is possible if the system is always prepared in a state that is selected from a known set of quantum states that are not orthogonal but linearly independent. This type of discrimination strategy has a higher probability of success by using generalised measurements than by using standard von Neumann measurements. We discuss here possible optical realizations, via generalized interferometers and single photon states, for the implementation of strategies that maximize the success rate. The implementations are based on exact analytical results. We also discuss possible applications for entanglement enhancement and quantum communication protocols.

Fortschritte der Physik•Oct 1, 2001•7 citations
Due to the fundamental limitations related to the Heisenberg uncertainty principle and the non-cloning theorem, it is impossible, even in principle, to determine the quantum state of a single system without a priori knowledge of it. To discriminate nonorthogonal quantum states in some optimal way, a priori knowledge of the discriminated states has to be relied upon. In this paper, we thoroughly investigate some impact of a priori classical knowledge of two quantum states on the optimal unambiguous discrimination. It is exemplified that a priori classical knowledge of the discriminated states, incomplete or complete, can be utilized to improve the optimal success probabilities, whereas the lack of a prior classical knowledge can not be compensated even by more resources.
We study how to unambiguously identify a given quantum pure state with one of the two reference pure states when no classical knowledge on the reference states is given but a certain number of copies of each reference quantum state are presented. By unambiguous identification, we mean that we are not allowed to make a mistake but our measurement can produce an inconclusive result. Assuming the two reference states are independently distributed over the whole pure state space in a unitary invariant way, we determine the optimal mean success probability for an arbitrary number of copies of the reference states and a general dimension of the state space. It is explicitly shown that the obtained optimal mean success probability asymptotically approaches that of the unambiguous discrimination as the number of the copies of the reference states increases.
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 two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.
Unambiguous discrimination and exact cloning reduce the square-overlap between quantum states, exemplifying the more general type of procedure we term state separation. We obtain the maximum probability with which two equiprobable quantum states can be separated by an arbitrary degree, and find that the established bounds on the success probabilities for discrimination and cloning are special cases of this general bound. The latter also gives the maximum probability of successfully producing N exact copies of a quantum system whose state is chosen secretly from a known pair, given M initial realizations of the state, where N > M. We also discuss the relationship between this bound and that on unambiguous state discrimination.
Since the state of a quantum system is not an observable, it cannot be determined completely, in principle, if no a priori information is available. In this case, one needs to resort to various estimation strategies. The situation is completely different if a priori knowledge is available. Even unambigous discrimination is possible if the system is always prepared in a state that is selected from a known set of quantum states that are not orthogonal but linearly independent. This type of discrimination strategy has a higher probability of success by using generalised measurements than by using standard von Neumann measurements. We discuss here possible optical realizations, via generalized interferometers and single photon states, for the implementation of strategies that maximize the success rate. The implementations are based on exact analytical results. We also discuss possible applications for entanglement enhancement and quantum communication protocols.