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Journal of Physics A Mathematical and Theoretical

Geometric block-coherence and quantum state discrimination

Danting Tang, Ping Li, Mingfei Ye, Yongming Li

Published Dec 7, 2021• 0 citations
Publisher

Abstract

Abstract Quantum coherence with respect to orthonormal bases has been studied extensively in the past few years. From the perspective of operational meaning, geometric coherence can be equal to the minimum error probability to discriminate a set of pure states (2018 J. Phys. A: Math. Theor. 51 414005). By regarding coherence as a physical resource, Baumgratz et al (2014 Phys. Rev. Lett. 113 140401) presented a comprehensive framework for coherence. Recently, geometric block-coherence as an effective block-coherence measure has been proposed. In this paper, we reveal an equivalence relationship between mixed quantum state discrimination (QSD) task and geometric block-coherence, which provides an operational interpretation for geometric block-coherence and generalizes the main result in coherence resource theory. Meanwhile, we show that partial coherence is a special case of block-coherence. By linking the relationship between geometric partial coherence and QSD tasks, we show that the value range of the two measures is the same. Finally, we reveal the relationship between geometric positive operator-valued measure-based coherence and QSD task.

DOI
10.1088/1751-8121/ac3f87

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Abstract The operational meaning of coherence measure lies at the very heart of the coherence theory. In this paper, we provide an operational interpretation for geometric coherence, by proving that the geometric coherence of a quantum state is equal to the minimum error probability to discriminate a set of pure states with von Neumann measurement. On the other hand, we also show that a task to ambiguously discriminate a set of linearly independent pure states can be regarded as a problem of calculating geometric coherence. That is, we reveal an equivalence relation between ambiguous quantum state discrimination and geometric coherence. Based on this equivalence, moreover, we improve the upper bound for geometric coherence and give the explicit expression of geometric coherence for a class of states. Besides, we establish a complementarity relation of geometric coherence and path distinguishability, with which the relationship between l 1 -norm of coherence and geometric coherence is obtained. Finally, with geometric coherence, we study multiple copies of quantum state discrimination and give an example to show how to discriminate two pure states.
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A central problem in quantum resource theory is to give operational meaning to quantum resources that can provide clear advantages in certain physical tasks compared to the convex set of resource-free states. We propose to extend this basic principle by defining the relative superiority of resources over a specific convex set of resource states, also provide a relative advantage in physical tasks based on this extended principle. This allows the generalized robustness measure to quantify the relative maximal advantage due to a given resource state over a specific convex set of resource states in the subchannel discrimination, thereby showing that the operational interpretation of resource measures also holds in a relative perspective. In addition, we offer a new framework for defining the deficiency of a given state in physical tasks compared to the set of maximum resource states. The geometric measure we provide satisfies the conditions of the framework for quantum coherence and entanglement, and it accurately quantifies the minimal disadvantage due to a given state compared to maximumresourcestates inthe subchannel discrimination in certain situations. These two extensions and new interpretations expand the scope of quantum resource theories and provide a more comprehensive operational interpretation.
Abstract Quantum coherence is an important quantum resource and it is intimately related to various research fields. The geometric coherence is a coherence measure both operationally and geometrically. We study the trade-off relation of geometric coherence in qubit systems. We first derive an upper bound for the geometric coherence by the purity of quantum states. Based on this, a complementarity relation between the quantum coherence and the mixedness is established. We then derive the quantum uncertainty relations of the geometric coherence on two and three general measurement bases in terms of the incompatibility respectively, which turn out to be state-independent for pure states. These trade-off relations provide the limit to the amount of quantum coherence. As a byproduct, the complementarity relation between the minimum error probability for discriminating a pure-states ensemble and the mixedness of quantum states is established.
Coherence measures and their operational interpretations lay the cornerstone of coherence theory. In this paper, we introduce a class of coherence measures with $\ensuremath{\alpha}$ affinity, say $\ensuremath{\alpha}$ affinity of coherence for $\ensuremath{\alpha}\ensuremath{\in}(0,1)$. Furthermore, we obtain the analytic formulas for these coherence measures and study their corresponding convex roof extension. We provide an operational interpretation for 1/21/21/2 affinity of coherence by showing that it is equal to the error probability to discrimination a set of pure states with the least-square measurement. By employing this relationship we regain the optimal measurement for equiprobable quantum state discrimination. Moreover, we compare these coherence quantifiers and establish a complementarity relation between the 1/21/21/2 affinity of coherence and path distinguishability for some special cases.
Abstract We investigate minimum-error (ME) discrimination for mixed qubit states using a geometric approach. By analyzing positive operator-valued measure (POVM) solutions and introducing Lagrange operator Γ\Gamma Γ Γ , we develop a four-step structured instruction to find Γ\Gamma Γ Γ for N mixed qubit states. Our method covers optimal solutions for two, three, and four mixed qubit states, including a novel result for four qubit states. We introduce geometric-based POVM classes and non-decomposable subsets for constructing optimal solutions, enabling us to find all possible answers for the general problem of minimum-error discrimination for N mixed qubit states with arbitrary a priori probabilities.