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

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.

• The paper presents a unifying MASI framework that quantifies quantum coherence and correlations in a basis-independent manner.
• It rigorously shows that averaging over MUBs, operator bases, or using the Haar measure produces identical, closed-form expressions for coherence.
• The study establishes precise complementarity relations combining wave-particle duality, quantum entropy, and system-environment correlations, with significant implications for resource theories.

Quantum Average Correlations and Complementarity via Metric-Adjusted Skew Information

Introduction

This paper provides a comprehensive and unifying treatment of quantum average correlations and complementarity relations based on the metric-adjusted skew information (MASI) formalism. The authors systematically connect averaging procedures over mutually unbiased bases (MUBs), operator orthonormal bases, all orthonormal bases (via Haar measure), and unitary twirling channels, demonstrating their equivalence in the context of quantum correlations. Through this framework, novel complementarity relations are established that relate wave and particle features, quantum entropy, and average quantum correlations. These developments yield basis-independent, intrinsic quantifiers for the quantum correlations in finite-dimensional systems, with direct connections to resource-theoretic and information-geometric paradigms.

Metric-Adjusted Skew Information and Coherence Quantification

MASI, introduced by Hansen, generalizes standard skew information by associating each operator monotone function $f$ with a Morozova–Chentsov function $c_f$, providing a family of monotone Riemannian metrics on quantum state space [Hansen, 2008]. For a state $\rho$ and observable $A$, the MASI is defined as

$$I^c_\rho(A) = \frac{f(0)}{2} \mathrm{Tr}\left(\mathbf{i}[\rho,A] c_f(L_\rho, R_\rho) \mathbf{i}[\rho,A]\right),$$

with $L_\rho, R_\rho$ denoting left and right multiplication superoperators, respectively. MASI is non-negative, convex in $\rho$, and vanishes iff $[\rho,A]=0$, making it a robust quantifier of quantum uncertainty and coherence.

To characterize coherence independently of the measurement basis, three averaging schemes are considered:

• Average over MUBs: For $d$-dimensional Hilbert spaces, the average is taken over $(d+1)$ MUBs (when they exist), each generating a set of projectors.
• Operator orthonormal bases: An average is taken with respect to an orthonormal basis $c_f$0 in operator space, independent of basis choice.
• Haar measure over all orthonormal bases: The average is taken over unitary conjugations of a fixed basis using the Haar measure over the unitary group.

The main structural result is that these averages yield identical, basis-independent expressions for average coherence, directly involving spectral data of $c_f$1:

$c_f$2

This basis independence is fundamental for attributing intrinsic status to the resulting quantifiers.

Average Quantum Correlations and Their Equivalence

The methodology is extended to bipartite states $c_f$3. The basis-dependent local coherence on subsystem $c_f$4 is quantified, then the average correlation is defined as the difference between the local coherence of the full system and that of the marginal:

$c_f$5

Averaging over MUBs, all orthonormal bases, or operator orthonormal bases, or equivalently, over the unitary twirling channel associated with the Haar measure, leads to

$c_f$6

where $c_f$7 are the eigenvalues of $c_f$8 and $c_f$9. This equivalence of average quantum correlations under various averaging procedures validates the intrinsic, basis-independent character of the MASI-based correlation quantifier.

The specializations to Wigner–Yanase skew information and quantum Fisher information yield closed-form, basis-independent expressions:

• Wigner–Yanase: $\rho$0,
• Fisher Information: $\rho$1.

Wave-Particle Duality and MASI-Based Quantification

By leveraging MASI, the paper defines precise measures for the wave and particle features in finite-dimensional quantum interference setups:

• Wave feature $\rho$2: Sum of MASI over the projectors associated with path basis elements, quantifying coherence (superposition).
• Particle feature $\rho$3: Sum of MASI over the off-diagonal transition operators, quantifying path distinguishability.

The paper establishes analytical properties: non-negativity, convexity, invariance under basis relabeling, and provides saturation conditions for extremal states (pure and maximally mixed). Notably, the maximum attainable sum of wave and particle features is bounded by $\rho$4, with equality iff $\rho$5 is pure.

Complementarity and Conservation Relations

A distinctive set of complementarity relations arise naturally:

• The sum $\rho$6, with equality for pure states.
• Incorporating quantum $\rho$7-entropy $\rho$8 (as defined by MASI), the exact conservation relation is achieved:

$\rho$9

This framework includes and generalizes previous complementarity formulations, extending them to the full MASI family.

In the bipartite context, for a pure state $A$0, a weighted sum of local wave and particle features and average correlation, corrected by the purity of the environment, saturates to a dimension-dependent constant:

$A$1

This result encapsulates an exact complementarity involving system, environment, and system-environment correlations, recovering the single-system relation when the environment is pure.

Implications and Perspectives

This work establishes MASI-based average quantum correlations as fundamental, intrinsic quantities, robust under different operational perspectives (measurement, observable, channel, and symmetry averaging). These measures naturally quantify nonclassical correlations and fit into information-geometric frameworks, offering significant advantages for resource-theoretic analysis, coherence theory, and quantification of multipartite and non-Markovian quantum correlations.

The complementarity relations unify wave-particle duality, entropy, and quantum correlations, illuminating structural constraints in the information content of quantum states and providing guidance for investigation of quantum resources under decoherence and operational transformations.

Potential avenues for future research include:

• Operational Significance: Evaluating the role of MASI-based average quantum correlations in state merging, remote state preparation, quantum thermodynamics, and communication tasks.
• Dynamics: Characterizing evolution under quantum channels, including non-Markovianity, and quantifying resilience against noise.
• Multisystem and CV Extension: Generalization to multipartite scenarios and continuous-variable systems.
• Experimental Accessibility: Developing schemes for direct experimental estimation of MASI-based quantifiers.

Conclusion

The paper delivers a rigorous, unified formulation of quantum average correlations and complementarity, anchored in metric-adjusted skew information. The demonstration of the equivalence of several averaging procedures produces basis-independent, intrinsic measures for quantum correlations. Quantification of wave and particle features and detailed complementarity relations further integrate coherence, entropy, and correlations under a single, mathematically coherent framework. This has significant implications for quantum information theory, quantum foundations, and future quantum technologies.

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Reference:

X. Ma, Q.-H. Zhang, and C. Xu, "Quantum average correlations and complementarity relations via metric-adjusted skew information" (2604.23504).