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arXiv:1801.06031 · Journal of Physics A Mathematical and Theoretical

Geometric coherence and quantum state discrimination

Chunhe Xiong, Junde Wu

Published Jun 1, 2018• 32 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/aac979

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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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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.
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 The link between quantum state discrimination and the no-signaling principle is applied to discriminating geometrically uniform states. Specifically, the original discrimination problem is converted to constructing the complementary matrix with predetermined off-diagonal entries. Two such constructions are given explicitly, and therefore three upper bounds on the success probability are derived. Those upper bounds are then employed to estimate the robustness of coherence, where we establish a larger class of states with equal robustness of coherence and l 1 -norm of coherence, and provide the exact answer of the robustness of coherence for any states that are diagonalized by Hadamard matrices.
Abstract Based on the concept of geometric coherence, we propose a wave-particle duality relation that establishes an equation between distinguishability and visibility for any pure state in the composite system of a general n-path interferometer and a which-path detector. This relation has the desired property that any increase of distinguishability will inevitably cause a decrease in visibility, thereby quantitatively capturing the essence of complementarity. Additionally, we reveal a connection between the fidelity function and discrimination of pure states. Specifically, we generalize the previous result in Xiong and Wu (2018 J. Phys. A 51 414005) by showing that for any quantum state ρ, the geometric coherence can be exactly expressed as the optimal guessing probability in the discrimination of the corresponding ensemble of pure states.