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Physical Review A

Minimum-error quantum state discrimination: Extremely tight upper and lower bounds with application in digital quantum communication

Amir Mohammad Yaghoobianzadeh, Jawad A. Salehi

Published Oct 1, 2024• 6 citations
Publisher

Abstract

This paper proposes a class of tight upper and lower bounds for the optimal probability of correctly discriminating $K$ 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.

DOI
10.1103/physreva.110.042401

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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.
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 prove a tight and close-to-optimal lower bound on the effectiveness of local quantum measurements (without classical communication) at discriminating any two bipartite quantum states. Our result implies, for example, that any two orthogonal quantum states of a nA×nBn_{A}\times n_{B}nA​×nB​ bipartite quantum system can be discriminated via local measurements with an error probability no larger than 12(1−1cmin⁡{nA,nB})\frac {1}2 \left ({1 - \frac {1}{c \min \{n_{A}, n_{B}\}} }\right)21​(1−cmin{nA​,nB​}1​) , where 1≤c≤221\leq c\leq 2\sqrt {2}1≤c≤22​ is a universal constant, and our bound scales provably optimally with the local dimensions nA,nBn_{A},n_{B}nA​,nB​ . Mathematically, this is achieved by showing that the distinguishability norm ∥⋅∥LO\|\cdot \|_{ \mathrm {LO}}∥⋅∥LO​ associated with local measurements satisfies that ∥⋅∥1≤22min⁡{nA,nB}∥⋅∥LO\|\cdot \|_{1}\leq 2\sqrt {2} \min \{n_{A},n_{B}\} \|\cdot \|_{ \mathrm {LO}}∥⋅∥1​≤22​min{nA​,nB​}∥⋅∥LO​ , where ∥⋅∥1\|\cdot \|_{1}∥⋅∥1​ is the trace norm.