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Physical Review A

Quantum state discrimination via generalized bounds on Löwner partial order sets

Amir Mohammad Yaghoobianzadeh, Jawad A. Salehi

Published Feb 5, 2025• 3 citations
Publisher

Abstract

This paper introduces a novel class of upper and lower bounds for a set of Hermitian operators with $K$ elements. These bounds, derived from the L\"owner partial ordering, represent the maximal lower and minimal upper bounds for a pair of Hermitian operators. Additionally, the paper redefines the minimum Bayes cost and minimum error quantum state discrimination problems by identifying the maximal lower and minimal upper bounds for quantum density operators. These proposed bounds are then utilized to establish a tight bound for the error probability of quantum state discrimination problem. Furthermore, the paper provides a geometric interpretation of these bounds, offering valuable insights into the quantum state discrimination problem and aiding in the understanding of the optimal measurement strategy, especially for qubits.

DOI
10.1103/physreva.111.022404

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This paper introduces a novel class of upper and lower bounds for a set of Hermitian operators with KKK elements. These bounds, derived from the L"owner partial ordering, represent the maximal lower and minimal upper bounds for a pair of Hermitian operators. Additionally, the paper redefines the minimum Bayes cost and minimum error quantum state discrimination problems by identifying the maximal lower and minimal upper bounds for quantum density operators. These proposed bounds are then utilized to establish a tight bound for the error probability of quantum state discrimination problem. Furthermore, the paper provides a geometric interpretation of these bounds, offering valuable insights into the quantum state discrimination problem and aiding in the understanding of the optimal measurement strategy, especially for qubits.
This paper proposes a class of tight upper and lower bounds for the optimal probability of correctly discriminating KKK quantum signals. The tightness of these bounds is shown analytically and numerically. These bounds are optimal for a binary quantum signal. The proposed upper bounds are based on a generalized trace distance measure between more than two quantum states. Also, we propose projector measurements that are good candidates for the near-optimal projector measurement. Also, an equivalent expression for the bounds is given based on the Gram matrix, which is more appropriate for numerical calculations, especially for digital quantum communication purposes, such as investigating the error probability of classical communication by using high-dimension quantum pulse position modulation signals.
We derive the measurement strategy that optimizes the Bayes criterion, not necessarily the strategy minimizing the error probability, for distinguishing between collections of quantum states. We present a sufficient condition concerning the rank of each detection operator of an optimal measurement for the Bayes criterion. We also derive an analytical expression for an optimal measurement for the Bayes criterion for an Abelian geometrically uniform state set or for a cyclic state set under a certain constraint.
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.
For the optimal success probability under minimum-error discrimination between $r\ensuremath{\ge}2$ arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds and specify the relations between these new general bounds and the known general bounds, lower and upper. We also present the example where the values of the new general lower and upper bounds on the optimal success probability are tighter than the values of most of the general analytical bounds known in the literature. The new upper bound on the optimal success probability explicitly generalizes to r>2r>2r>2 the form of the Helstrom bound. For r=2r=2r=2, each of our new bounds, lower and upper, reduces to the Helstrom bound.
For the optimal success probability under minimum-error discrimination between r≥2r\geq2r≥2 arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds and specify the relations between these new general bounds and the general bounds known in the literature. We also present the example where the new general analytical bounds, lower and upper, on the optimal success probability are tighter than most of the general analytical bounds known in the literature. The new upper bound on the optimal success probability explicitly generalizes to r>2r>2r>2 the form of the Helstrom bound. For r=2r=2r=2, each of our new bounds, lower and upper, reduces to the Helstrom bound.
Two types of errors can occur when discriminating pairs of quantum states. Asymmetric state discrimination involves minimizing the probability of one type of error, subject to a constraint on the other. We give explicit expressions bounding the set of achievable errors, using the trace norm, the fidelity, and the quantum Chernoff bound. The upper bound is asymptotically tight and the lower bound is exact for pure states. Unlike asymptotic bounds, our bounds give error values instead of exponents, so can give more precise results when applied to finite-copy state discrimination problems.