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arXiv:2412.08000 · Entropy

Axiomatic Approach to Measures of Total Correlations

Gabriel L. Moraes, R. M. Angelo, Ana C. S. Costa

Published Dec 15, 2024• 1 citation
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Abstract

Correlations play a pivotal role in various fields of science, particularly in quantum mechanics, yet their proper quantification remains a subject of debate. In this work, we aimed to discuss the challenge of defining a reliable measure of total correlations. We first outlined the essential properties that an effective correlation measure should satisfy and reviewed existing measures, including quantum mutual information, the p-norm of the correlation matrix, and the recently defined quantum Pearson correlation coefficient. Additionally, we introduced new measures based on Rényi and Tsallis relative entropies, as well as the Kullback–Leibler divergence. Our analysis revealed that while quantum mutual information, the p-norm, and the Pearson measure exhibit equivalence for two-qubit systems, they all suffer from an ordering problem. Despite criticisms regarding its reliability, we argued that QMI remains a valid measure of total correlations.

DOI
10.3390/e26121098

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Correlations play a pivotal role in various fields of science, particularly in quantum mechanics, yet their proper quantification remains a subject of debate. In this work, we aimed to discuss the challenge of defining a reliable measure of total correlations. We first outlined the essential properties that an effective correlation measure should satisfy and reviewed existing measures, including quantum mutual information, the p-norm of the correlation matrix, and the recently defined quantum Pearson correlation coefficient. Additionally, we introduced new measures based on Rényi and Tsallis relative entropies, as well as the Kullback–Leibler divergence. Our analysis revealed that while quantum mutual information, the p-norm, and the Pearson measure exhibit equivalence for two-qubit systems, they all suffer from an ordering problem. Despite criticisms regarding its reliability, we argued that QMI remains a valid measure of total correlations.
Correlations disguised in various forms underlie a host of important phenomena in classical and quantum systems, such as information and energy exchanges. The quantum mutual information and the norm of the correlation matrix are both considered as proper measures of total correlations. We demonstrate that, when applied to the same system, these two measures can actually show significantly different behavior except at least in two limiting cases: when there are no correlations and when there is maximal quantum entanglement. We further quantify the discrepancy by providing analytic formulas for time derivatives of the measures for an interacting bipartite system evolving unitarily. We argue that to properly account for correlations one should consider the full information provided by the correlation matrix (and reduced states of the subsystems). Scalar quantities such as the norm of the correlation matrix or the quantum mutual information can only capture a part of the complex features of correlations. As a concrete example, we show that in describing heat exchange associated with correlations neither of these quantities can fully capture the underlying physics. As a byproduct, we also prove an area law for quantum mutual information in systems with local and short-range interactions, without need to assume Markovianity or final thermal equilibrium.
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We built a new set of suitable measures of correlations for bipartite quantum states based upon a recently introduced theoretical framework [Bussandri et al. in Quantum Inf. Proc. 18:57, 2019]. We applied these measures to examine the behavior of correlations in two-qubit states with maximally mixed marginals independently interacting with non-dissipative decohering environments in different dynamical scenarios of physical relevance. In order to get further insight about the physical meaning of the behavior of these correlation measures we compared our results with those obtained by means of well-known correlation measures such as quantum mutual information and quantum discord. On one hand, we found that the behaviors of total and classical correlations, as assessed by means of the measures introduced in this work, are qualitatively in agreement with the behavior displayed by quantum mutual information and the measure of classical correlations typically used to calculate quantum discord. We also found that the optimization of all the measures of classical correlations depends upon a single parameter and the optimal value of this parameter turns out to be the same in all cases. On the other hand, regarding the measures of quantum correlations used in our studies, we found that in general their behavior does not follow the standard quantum discord D . As the quantification by means of standard quantum discord and the measures of quantum correlations introduced in this work depends upon the assumption that total correlations are additive, our results indicate that this property needs a deeper and systematic study in order to gain a further understanding regarding the possibility to obtain reliable quantifiers of quantum correlations within this additive scheme.
Quantum conditional mutual information (QCMI) is a fundamental measure of conditional independence, with central applications throughout quantum information theory and many-body physics. Despite its importance, finding a fully quantum Rényi generalization that satisfies all desirable structural properties has remained an open question. In this work, we introduce a quantum Rényi conditional mutual information possessing these desirable properties: it is monotone under local quantum channels, additive under tensor products, monotone in the Rényi parameter, and converges to the QCMI in the limit α→1α\to 1α→1. We establish bounds on this quantity in terms of relative entropies to the Petz-recovered state, showing that it provides a distinct measure of approximate conditional independence. As a consequence, we derive recovery-based upper and lower bounds on the QCMI in terms of quantum generalizations of the Kullback-Leibler divergence, improving upon the best known results in certain regimes. Finally, we establish bounds in terms of relative entropies to the sets of Markov and separable states and derive Rényi chain rules, which appear to be novel even in the classical setting.
The concept of coherence is one of cornerstones in physics. The development of quantum information science has lead to renewed interest in properly approaching the coherence at the quantum world. Various measures could be proposed for quantifying the coherence of a quantum state with respect to the prescribed orthonormal basis. To be a proper measure of coherence, each candidate should enjoy certain properties. It seems that the monotonicity property plays a crucial role here. Indeed, there is known an intuitive measure of coherence that does not share this condition. We study coherence measures induced by quantum divergences of the Tsallis type. It is proved that considered measures of coherence satisfy all the required properties. The only point is that monotonicity under incoherent selective measurements has to be reformulated. The proposed formulation can naturally be treated as a parametric extension of its standard form. This extension is reduced to the standard form when the standard relative entropy is dealt. Finally, two coherence measures quadratic in moduli of matrix elements are compared from the monotonicity viewpoint.