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arXiv:1102.0361 · Physical Review Letters

No-Signaling Principle Can Determine Optimal Quantum State Discrimination

Joonwoo Bae, Won-Young Hwang, Yeong Deok Han

Published Oct 20, 2011• 38 citations
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Abstract

We provide a general framework of utilizing the no-signaling principle in derivation of the guessing probability in the minimum-error quantum state discrimination. We show that, remarkably, the guessing probability can be determined by the no-signaling principle. This is shown by proving that, in the semidefinite programing for the discrimination, the optimality condition corresponds to the constraint that quantum theory cannot be used for a superluminal communication. Finally, a general bound to the guessing probability is presented in a closed form.

DOI
10.1103/physrevlett.107.170403

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We provide a general framework of utilizing the no-signaling principle in derivation of the guessing probability in the minimum-error quantum state discrimination. We show that, remarkably, the guessing probability can be determined by the no-signaling principle. This is shown by proving that, in the semidefinite programing for the discrimination, the optimality condition corresponds to the constraint that quantum theory cannot be used for a superluminal communication. Finally, a general bound to the guessing probability is presented in a closed form.
In this paper we consider the general problem of designing the optimal quantum detection for minimum-error quantum state discrimination. Since it is hard to get an explicit solution and we must resort to numerical methods for the general case. We show the problem can be cast into a semidefinite programming(SDP) problem, we use the theory of SDP to solve it. Based on the idea, we propose a method for minimum-error discrimination by formulating the design of optimal quantum detection as a standard SDP problem; furthermore, based on the optimality conditions for SDP, we derive a set of optimality conditions for quantum detection. It is exemplified that this method is simple to implement and can be used to design optimum detection operators for minimum-error discrimination efficiently.
Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.
We show a geometric formulation for minimum-error discrimination of qubit states that can be applied to arbitrary sets of qubit states given with arbitrary a priori probabilities. In particular, when qubit states are given with equal a priori probabilities, we provide a systematic way of finding optimal discrimination and the complete solution in a closed form. This generally gives a bound to cases when prior probabilities are unequal. Then it is shown that the guessing probability does not depend on detailed relations among the given states, such as the angles between them, but on a property that can be assigned by the set of given states itself. This also shows how a set of quantum states can be modified such that the guessing probability remains the same. Optimal measurements are also characterized accordingly, and a general method of finding them is provided.