scholarly journals Uniqueness of Minimax Strategy in View of Minimum Error Discrimination of Two Quantum States

Entropy ◽  
2019 ◽  
Vol 21 (7) ◽  
pp. 671 ◽  
Author(s):  
Jihwan Kim ◽  
Donghoon Ha ◽  
Younghun Kwon
Keyword(s):  
Optimal Strategy ◽  
Quantum States ◽  
Minimax Strategy ◽  
Minimum Error ◽  
Optimal Measurement ◽  
Single Side ◽  
Quantum Ensemble ◽  
Zero Sum ◽  
Necessary And Sufficient ◽  
Specific Quantum

This study considers the minimum error discrimination of two quantum states in terms of a two-party zero-sum game, whose optimal strategy is a minimax strategy. A minimax strategy is one in which a sender chooses a strategy for a receiver so that the receiver may obtain the minimum information about quantum states, but the receiver performs an optimal measurement to obtain guessing probability for the quantum ensemble prepared by the sender. Therefore, knowing whether the optimal strategy of the game is unique is essential. This is because there is no alternative if the optimal strategy is unique. This paper proposes the necessary and sufficient condition for an optimal strategy of the sender to be unique. Also, we investigate the quantum states that exhibit the minimum guessing probability when a sender’s minimax strategy is unique. Furthermore, we show that a sender’s minimax strategy and a receiver’s minimum error strategy cannot be unique if one can simultaneously diagonalize two quantum states, with the optimal measurement of the minimax strategy. This implies that a sender can confirm that the optimal strategy of only a single side (a sender or a receiver but not both of them) is unique by preparing specific quantum states.

Minimum guesswork discrimination between quantum states

2015 ◽  
Vol 15 (9&10) ◽  
pp. 737-758
Author(s):  
Weien Chen ◽  
Yongzhi Cao ◽  
Hanpin Wang ◽  
Yuan Feng
Keyword(s):  
Error Probability ◽  
Sufficient Conditions ◽  
Optimization Criterion ◽  
Quantum States ◽  
Actual State ◽  
Information Theoretic ◽  
Minimum Error ◽  
State Discrimination ◽  
Semidefinite Programming Problem ◽  
Necessary And Sufficient

Error probability is a popular and well-studied optimization criterion in discriminating non-orthogonal quantum states. It captures the threat from an adversary who can only query the actual state once. However, when the adversary is able to use a brute-force strategy to query the state, discrimination measurement with minimum error probability does not necessarily minimize the number of queries to get the actual state. In light of this, we take Massey's guesswork as the underlying optimization criterion and study the problem of minimum guesswork discrimination. We show that this problem can be reduced to a semidefinite programming problem. Necessary and sufficient conditions when a measurement achieves minimum guesswork are presented. We also reveal the relation between minimum guesswork and minimum error probability. We show that the two criteria generally disagree with each other, except for the special case with two states. Both upper and lower information-theoretic bounds on minimum guesswork are given. For geometrically uniform quantum states, we provide sufficient conditions when a measurement achieves minimum guesswork. Moreover, we give the necessary and sufficient condition under which making no measurement at all would be the optimal strategy.

scholarly journals Quantum nonlocality without entanglement: explicit dependence on prior probabilities of nonorthogonal mirror-symmetric states

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Donghoon Ha ◽  
Younghun Kwon
Keyword(s):  
Party System ◽  
Functional Dependence ◽  
Quantum States ◽  
Quantum Nonlocality ◽  
Explicit Dependence ◽  
Classical Communication ◽  
Minimum Error ◽  
Prior Probabilities ◽  
Quantum Ensemble ◽  
Perfect Discrimination

AbstractIn the case of a multi-party system, through local operations and classical communication (LOCC), each party may not perform perfect discrimination of quantum states that are separable and orthogonal. This property of quantum ensemble is called “nonlocality without entanglement” since each local party has a limit to full information of given quantum states. When this property is extended to the case of minimum-error discrimination, one can see that it is revealed when a nonlocal measurement provides more information about the unentangled states than LOCC does. One may infer the fact that the property depends on quantum states composing the quantum ensemble. However, an essential but unsettled question about the property is whether an explicit dependence on prior probabilities in terms of minimum-error discrimination could be shown in nonlocality without entanglement. In a simple term, one can ask whether different quantum ensembles made of the same separable quantum states could exhibit explicitly different behavior of the nonlocality. We answer this question in the positive, and we furthermore provide the explicit functional dependence of guessing probability on prior probabilities for the mirror-symmetric ensemble.

A note on the degree conjecture for separability of multipartite quantum states

2021 ◽  
pp. 2050048
Author(s):  
Zhen Wang ◽  
Ming-Jing Zhao ◽  
Zhi-Xi Wang
Keyword(s):  
Quantum States ◽  
Point Graph ◽  
Mixed States ◽  
Degree Condition ◽  
Graph Laplacians ◽  
Nearest Point ◽  
Necessary And Sufficient ◽  
Ppt Criterion ◽  
Math Phys ◽  
Degree Criterion

The degree conjecture for bipartite quantum states which are normalized graph Laplacians was first put forward by Braunstein et al. [Phys. Rev. A 73 (2006) 012320]. The degree criterion, which is equivalent to PPT criterion, is simpler and more efficient to detect the separability of quantum states associated with graphs. Hassan et al. settled the degree conjecture for the separability of multipartite quantum states in [J. Math. Phys. 49 (2008) 0121105]. It is proved that the conjecture is true for pure multipartite quantum states. However, the degree condition is only necessary for separability of a class of quantum mixed states. It does not apply to all mixed states. In this paper, we show that the degree conjecture holds for the mixed quantum states of nearest point graph. As a byproduct, the degree criterion is necessary and sufficient for multipartite separability of [Formula: see text]-qubit quantum states associated with graphs.

scholarly journals Efficient optimal minimum error discrimination of symmetric quantum states

2010 ◽  
Vol 81 (1) ◽  
Author(s):  
Antonio Assalini ◽  
Gianfranco Cariolaro ◽  
Gianfranco Pierobon
Keyword(s):  
Quantum States ◽  
Minimum Error ◽  
Symmetric Quantum

scholarly journals Optimal doubling strategy against a suboptimal opponent

2005 ◽  
Vol 42 (3) ◽  
pp. 867-872
Author(s):  
Konstantinos V. Katsikopoulos ◽  
Özgür Şimşek
Keyword(s):  
Continuous Function ◽  
Optimal Strategy ◽  
Numerical Results ◽  
Repeated Application ◽  
Zero Sum Games ◽  
Game Value ◽  
Zero Sum

For two-person zero-sum games, where the probability of each player winning is a continuous function of time and is known to both players, the mutually optimal strategy for proposing and accepting a doubling of the game value is known. We present an algorithm for deriving the optimal doubling strategy of a player who is aware of the suboptimal strategy followed by the opponent. We also present numerical results about the magnitude of the benefits; the results support the claim that repeated application of the algorithm by both players leads to the mutually optimal strategy.

scholarly journals Adaptive estimation and discrimination of Holevo-Werner channels

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
Thomas P. W. Cope ◽  
Stefano Pirandola
Keyword(s):  
Quantum Information ◽  
Error Probability ◽  
Adaptive Estimation ◽  
Analytical Formula ◽  
Quantum States ◽  
Minimum Error ◽  
Chernoff Bound ◽  
Werner States

AbstractThe class of quantum states known as Werner states have several interesting properties, which often serve to illuminate unusual properties of quantum information. Closely related to these states are the Holevo- Werner channels whose Choi matrices are Werner states. Exploiting the fact that these channels are teleportation covariant, and therefore simulable by teleportation, we compute the ultimate precision in the adaptive estimation of their channel-defining parameter. Similarly, we bound the minimum error probability affecting the adaptive discrimination of any two of these channels. In this case, we prove an analytical formula for the quantum Chernoff bound which also has a direct counterpart for the class of depolarizing channels. Our work exploits previous methods established in [Pirandola and Lupo, PRL

Payoff Sensitivity of Linear Quadratic Differential Games to Parameter Change

1981 ◽  
Vol 103 (1) ◽  
pp. 36-38
Author(s):  
C. T. Leondes ◽  
T. K. Sui
Keyword(s):  
Differential Games ◽  
Quadratic Differential ◽  
Optimal Strategy ◽  
Small Variation ◽  
Matrix Equations ◽  
Parameter Change ◽  
Linear Quadratic ◽  
System Parameters ◽  
Linear Quadratic Differential Games ◽  
Zero Sum

Both maximizing and minimizing players are concerned with the change in payoff due to small variation of system parameters. A technique is developed to derive linear algebraic matrix equations which can be used to determine the payoff sensitivity of all the parameters in linear zero- sum differential games with constant feedback. Above all, this technique is applicable for determining both the optimal strategy and payoff.

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