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arXiv:2604.23504

Quantum average correlations and complementarity relations via metric-adjusted skew information

Xiaoyu Ma, Qing-Hua Zhang, Cong Xu

quant-phSubmitted Apr 26, 2026• Announced Apr 28, 2026• 0 citations
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Abstract

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.

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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.
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This paper studies the quantification and structural properties of quantum average correlation based on average coherence. Motivated by two mathematically equivalent approaches to define average coherence: one by averaging over complete sets of mutually unbiased bases, and the other by integrating over all orthogonal bases under the Haar measure, we define an average correlation for bipartite systems as the difference between global and local skew information. This correlation measure is shown to satisfy essential properties including non negativity, contractivity under local quantum channels, and local unitary invariance. We further prove the equivalence between the average correlation defined via mutually unbiased bases and that defined via unitary groups. Finally, we derive a complementarity relation that connects wave-particle duality with the average correlation between a system and its environment.
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.
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 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.
In terms of the metric-adjusted skew information (an important and versatile class of quantum Fisher information), which generalizes the seminal Wigner-Yanase skew information arising naturally from the study of quantum measurement, we propose a family of coherence measures of states relative to quantum channels, and reveal their basic properties such as unitary covariance, convexity, and monotonicity. Furthermore, we evaluate these coherence measures of states relative to several prototypical quantum channels, and make a comparative study for this family of coherence measures with relative entropy of coherence. This provides a general approach to coherence of states relative to quantum channels, which also captures decoherence on the states caused by quantum channels and asymmetry of states relative to quantum channels.