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Chinese Control Conference (CCC)

Minimum-error quantum state discrimination based on semidefinite programming

Wang Wengang, Guohua Dong, Mingshan Li

Published Jul 1, 2008• 2 citations
Publisher

Abstract

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.

DOI
10.1109/chicc.2008.4605546

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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.
For the general problem of optimum unambiguous discrimination of quantum states, it is very hard to get the explicit analytical form for optimum detection operators. Therefore, it is necessary to find an efficient numerical method for designing optimum detection operators. According to the theory of semidefinite programming, it is demonstrated that the design of the optimal quantum measurement can be formulated as a standard semidefinite programming problem, and the optimality conditions for semidefinite programming can be used to perspicuously derive a set of optimality conditions for optimum detection operators.By means of the standard semidefinite programming, the maximum value of the success probability of detection and the optimum detection operators can be obtained directly in polynomial time. It is exemplified that this method is simple to be implemented and can be used to design optimum detection operators for quantum-state discrimination efficiently.
Quantum state discrimination is a fundamental primitive in quantum information processing, underpinning tasks in quantum communication, sensing, and learning. We consider the general Bayes framework, as introduced by Helstrom, for state discrimination when, instead of a classical description of the candidate states, one has access to their \emph{source code}: the quantum circuit that prepares them. We show that the semidefinite program (SDP) for the discrimination problem can be reformulated in terms of the Gram matrix of these states, reducing the SDP variable dimensions from dLdLdL to NLNLNL, where ddd is the Hilbert space dimension, NNN is the number of candidate states, and LLL is the number of possible guesses. Importantly, we further introduce a quantum pre-processing procedure which efficiently constructs the reduced semidefinite program from the source code, enabling our method to operate directly on quantum data. We consider two applications. First, we characterize the optimal identifications for quantum changepoint problems under several reward structures, including multiple-changepoint settings that were previously computationally inaccessible. Second, we consider a quantum error classification problem and show how our reduction makes it tractable for systems of hundreds of qubits.
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.
Abstract Designing optimal measurement operators for quantum state discrimination (QSD) is an important problem in quantum communications and cryptography applications. Prior works have demonstrated that optimal quantum measurement operators can be obtained by solving a convex semidefinite program (SDP). However, solving the SDP can represent a high computational burden for many real‐time quantum communication systems. To address this issue, a majorisation‐minimisation (MM)‐based algorithm, called Quantum Majorisation‐Minimisation (QMM) is proposed for solving the QSD problem. In QMM, the authors reparametrise the original objective, then tightly upper‐bound it at any given iterate, and obtain the next iterate as a closed‐form solution to the upper‐bound minimisation problem. Our numerical simulations demonstrate that the proposed QMM algorithm significantly outperforms the state‐of‐the‐art SDP algorithm in terms of speed, while maintaining comparable performance for solving QSD problems in quantum communication applications.
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.
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.