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Physica Scripta

Average coherence relative to symmetric measurements and generalized symmetric measurements

Yajing FAN, Xueying Sha

Published May 15, 2026• 0 citations
Publisher

Abstract

Abstract Quantum coherence is one of the characteristic features of quantum mechanics and underpins many quantum mysteries. In this work, we investigate the average coherence of a quantum state with respect to symmetric positive operator-valued measures using metric-adjusted skew information. Specifically, we derive explicit expressions for the average coherence relative to mutually unbiased measurements and to general symmetric informationally complete measurements, and establish a complementary relation between average coherence and maximal coherence. Furthermore, we introduce a measure of average coherence based on metric-adjusted skew information for states with respect to a generalized symmetric measurements (GSMs). When these GSMs are required to be equinumerous, the corresponding average coherence reduces to that for symmetric measurements. Finally, we evaluate and compare various average coherence measures with different operator monotone functions for qubit states, Werner states, and isotropic states.

DOI
10.1088/1402-4896/ae6e9f

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Abstract Quantum coherence is one of the characteristic features of quantum mechanics and underpins many quantum mysteries. In this work, we investigate the average coherence of a quantum state with respect to symmetric positive operator-valued measures using metric-adjusted skew information. Specifically, we derive explicit expressions for the average coherence relative to mutually unbiased measurements and to general symmetric informationally complete measurements, and establish a complementary relation between average coherence and maximal coherence. Furthermore, we introduce a measure of average coherence based on metric-adjusted skew information for states with respect to a generalized symmetric measurements (GSMs). When these GSMs are required to be equinumerous, the corresponding average coherence reduces to that for symmetric measurements. Finally, we evaluate and compare various average coherence measures with different operator monotone functions for qubit states, Werner states, and isotropic states.
Quantum coherence is one of the characteristic features of quantum mechanics and underpins many quantum mysteries. To eliminate the influence of the reference basis on the coherence of a quantum state and uncover its intrinsic properties, it is common to study coherence by averaging over different reference bases. Using the metric-adjusted skew information, we explore three natural approaches to average coherence of a state: average over all orthonormal bases, average over all elements of an operator orthonormal base, and average over a complete family of mutually unbiased bases. We establish the equivalence among these three types of average coherence and interpret the unified average coherence as the coherence of a quantum state relative to a depolarizing channel. Additionally, we employ the unified average coherence to introduce a notion of quantum fff entropy (where fff is an operator monotone function associated with the metric-adjusted skew information) and demonstrate that quantum fff entropy possesses properties analogous to the ubiquitous von Neumann entropy. Furthermore, we illuminate some connections between quantum fff entropy and quasientropy, and compare fff entropy with von Neumann entropy, R\'enyi entropy, and Tsallis entropy for some typical states.
Quantum coherence is an important quantum resource which plays a pivotal role in the field of quantum information. Based on metric adjusted skew information, we define a measure of quantum uncertainty to study average coherence under conical 2-designs generalized equiangular measurements, and prove the equivalence of this measure to the scaled average coherence based on metric adjusted skew information under a set of unitary groups, operator orthonormal bases, and mutually unbiased bases. We also derive two trade-off relations by this measure and solve a conjecture. Furthermore, we give two entanglement criteria by this measure and conical 2-designs generalized equiangular measurement, respectively, and illustrate the effectiveness of them by explicit examples.
ABSTRACT Quantum entanglement and coherence, serving as cornerstone resources for quantum information science, fundamentally depend on quantum measurements for their detection and quantification, with quantum uncertainty serving as the essential link that connects these components. In this work, by integrating metric‐adjusted skew information with informationally complete symmetric measurements, we establish a universal connection between measurement‐induced quantum uncertainty and the fundamental quantum uncertainty measure. Then, we demonstrate that the total uncertainty associated with symmetric measurements is universally proportional to the intrinsic quantum uncertainty of the state. Subsequently, we derived a series of entanglement criteria by employing the proposed framework and demonstrated their validity and comprehensiveness through illustrative example. In addition, we employ symmetric measurements with metric‐adjusted skew information to formulate generalized measures of average coherence, and derive several measurement‐based complementary trade‐off relations for average coherence associated with a given quantum state. This framework offers a robust quantitative approach to understanding quantum resources, with significant practical implications for the analysis of quantum entanglement and coherence.
In this paper, we introduce a family of correlation measures relative to a local channel in terms of the metric-adjusted skew information for a global state and reveal their fundamental properties. We quantify the correlation of quantum states with respect to several typical channels, including unitary channels, twirling channels, and weak measurements, and illustrate that the correlation with respect to the twirling channel induced by the fully unitary group is consistent with that relative to the completely depolarizing channel. In particular, we further evaluate the correlations of the Bell diagonal states, Werner states, and isotropic states and make a comparative study for this family of correlation measures with different operator monotone functions of the two-qubit Werner states and isotropic states, respectively.
We investigate quantum average correlations and complementarity relations based on metric-adjusted skew information. Several natural averaging procedures are considered, including complete families of mutually unbiased bases, all orthonormal bases, operator orthonormal bases, and twirling channels induced by the unitary group. All these approaches lead to the same closed expression, which identifies the resulting average correlation as an intrinsic quantity independent of the averaging scheme. By defining measures of wave and particle features via metric-adjusted skew information, we establish complementarity relations among wave and particle features, quantum entropy, and average correlation. These results provide a unified framework for investigating quantum average correlations and complementarity relations in terms of metric-adjusted skew information.