Quantum average correlation based on average coherence

Abstract: 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.

• The paper introduces a unified, basis-independent formalism for average coherence and quantum correlation using Wigner-Yanase skew information.
• It demonstrates the equivalence of MUB and Haar unitary averaging methods to establish a novel average correlation monotone with clear operational meaning.
• It establishes a complementarity relation linking wave-particle duality with system-environment correlations, enhancing resource theory insights.

Quantum Average Correlation and Coherence: A Rigorous Construct via Skew Information

Introduction

The quantification and structural analysis of quantum coherence and quantum correlations have been central to quantum information theory, with broad applications ranging from quantum metrology to quantum thermodynamics. This paper, "Quantum average correlation based on average coherence" (2604.23501), addresses intrinsic, basis-independent measures of both quantum coherence and correlation, operationalizing them through the Wigner-Yanase skew information. The work provides a unified, invariant formalism for average coherence and introduces a novel average correlation monotone, anchored in the difference between global and local skew information, with robust theoretical properties and concrete links to wave-particle duality.

Formulation of Average Coherence

Average coherence is constructed in two mathematically equivalent forms:

• MUB Averaging: Coherence is averaged over all complete sets of mutually unbiased bases (MUBs). For a $d$-dimensional Hilbert space (with $d$ a prime power), a complete set of $d+1$ MUBs allows full uniform sampling over complementary observables. The average coherence is given as:

$$C_{mub}(\rho) = \frac{1}{d+1} \sum_{t=1}^{d+1} C(\rho|\Pi_t),$$

where each $C(\rho|\Pi_t)$ uses Wigner-Yanase skew information.

• Haar Unitary Averaging: Coherence is averaged over all possible orthonormal bases, equivalently integrating over the unitary group endowed with the Haar measure:

$$C_{\mathcal{U}}(\rho) = \int_{\mathcal{U}} C(\rho|U\Pi U^\dagger)dU.$$

The key finding is the equivalence of these two approaches:

$$C_{mub}(\rho) = C_{\mathcal{U}}(\rho) = \frac{d-(\mathrm{tr} \sqrt{\rho})^2}{d+1}.$$

This provides a canonical, basis-independent coherence measure tightly linked to intrinsic state properties.

Quantum Average Correlation: Definition and Properties

Extending the average coherence framework, the authors define the quantum average correlation for a bipartite system $\rho^{AB}$:

$$Q(\rho^{AB}|\Pi) = C(\rho^{AB}|\Pi \otimes \mathbb{I}_B) - C(\rho^A|\Pi).$$

Average correlation is then defined via:

• MUB Averaging:

$$Q_{mub}(\rho^{AB}) = \frac{1}{d_A + 1} \sum_{t=1}^{d_A + 1} Q(\rho^{AB}|\Pi_t)$$

• Unitary Averaging:

$d$0

• Orthonormal Operator Basis Averaging:

Employs an orthonormal operator basis $d$1 for the observables on subsystem $d$2.

For all three definitions, the resulting analytic expression is:

$d$3

Key properties established:

• Non-negativity: $d$4, with equality if and only if the state is a product state.
• Contractivity under Local Quantum Channels: $d$5 does not increase under local CPTP maps on subsystem $d$6.
• Local Unitary Invariance: $d$7 remains invariant under local unitary transformations.
• Basis Independence: All three averaging procedures (MUB, Haar, operator basis) yield identical results, confirming the measure’s canonical status.

Complementarity Relation: Wave-Particle Duality and System-Environment Correlations

A significant contribution is the establishment of a complementarity relation for bipartite pure states $d$8, correlating wave-particle duality measures with system-environment quantum correlation:

• The wave property $d$9 and particle property $d+1$0 for a path basis $d+1$1 satisfy

$d+1$2

This exact complementarity quantitatively describes the trade-off: higher system-environment correlation strictly reduces the sum of local wave and particle characteristics. When $d+1$3 vanishes (product state), the wave-particle terms retain the standard normalization, but correlation nontrivially redistributes the informational content.

This result generalizes previous entropic duality frameworks and underscores the fundamental role of averaged quantum correlations in the physics of interference and decoherence.

Implications for Quantum Information Theory

The rigorous, basis-invariant formulation of average coherence and average correlation based on skew information has several implications:

• Resource Theory Foundations: The equivalence of MUB, Haar, and operator-basis averages supports a robust canonical resource quantifier, directly linked to distinguishability and symmetry invariance.
• Operational Scenarios: The measure $d+1$4 is tightly connected to quantification of quantum correlations, extending beyond entanglement to more general, possibly discord-like, correlations.
• Wave-Particle-Entanglement Triality: The complementarity relation situates average correlation explicitly as the “missing” informational component in multi-path interference, suggesting new ways to assess environment-induced decoherence and its effect on observable quantum features.
• Algorithmic and Experimental Relevance: The analytic form of $d+1$5 allows direct computation (given the spectral decomposition of reduced and global states), which is advantageous for high-dimensional and multipartite settings relevant in quantum simulation and quantum technologies.

Outlook and Future Directions

This framework invites several extensions:

• Multipartite Systems: Generalization to multipartite or networked quantum systems, potentially yielding new insights into the distribution and scaling of correlations.
• Dynamical Scenarios: Investigation of the behavior of $d+1$6 under quantum channels, open-system evolution, and in non-Markovian regimes may provide deeper understanding of decoherence and information flow.
• Operational Interpretation: Exploring concrete operational tasks (e.g., state discrimination, metrology, thermodynamic work extraction) where $d+1$7 precisely quantifies resource utility compared to entanglement or other correlation measures.
• Connections to Quantum Thermodynamics: Given the close connection between skew information, coherence, and extractable work, further integration with frameworks of work, entropy production, and quantum heat engines may be anticipated.

Conclusion

The paper establishes a mathematically rigorous and physically motivated formalism for quantifying basis-independent average coherence and average correlation via the Wigner-Yanase skew information. The analytic equivalence of distinct averaging methods, the satisfaction of essential correlation measure axioms, and the derived complementarity relation linking system-environment correlation to wave-particle duality form a substantial theoretical advance. These results anchor skew information-based quantifiers at the forefront of the study of quantum resources, with broad relevance for quantum information, foundations, and the emerging interface with quantum thermodynamics and quantum technology (2604.23501).