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arXiv:2506.07344 · Journal of Chemical Theory and Computation

Mutual Correlation

Francesco A. Evangelista

Published Jul 29, 2025• 1 citation
arXivHTMLPDF

Abstract

High Resolution Image Download MS PowerPoint Slide Quantifying correlation and complexity in quantum many-body states is central to advancing theoretical and computational chemistry, physics, and quantum information science. This work introduces a novel framework, mutual correlation, based on the Frobenius norm squared of the two-body reduced density matrix cumulant. Through systematic partitioning of the cumulant norm, mutual correlation quantifies nonadditive correlations among interacting subsystems. To assess the ability of mutual correlation to identify orbital interactions, we performed benchmark studies on model systems, including H 10, N 2, and p -benzyne, and performed a formal and numerical comparison with orbital mutual information. Maximally correlated orbitals, obtained by maximizing a nonlinear cost function of the mutual correlation, are also considered to identify a basis-independent partitioning of correlation. This study suggests that mutual correlation is a broadly applicable metric, useful in active space selection and the interpretation of electronic states.

DOI
10.1021/acs.jctc.5c00766

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High Resolution Image Download MS PowerPoint Slide Quantifying correlation and complexity in quantum many-body states is central to advancing theoretical and computational chemistry, physics, and quantum information science. This work introduces a novel framework, mutual correlation, based on the Frobenius norm squared of the two-body reduced density matrix cumulant. Through systematic partitioning of the cumulant norm, mutual correlation quantifies nonadditive correlations among interacting subsystems. To assess the ability of mutual correlation to identify orbital interactions, we performed benchmark studies on model systems, including H 10, N 2, and p -benzyne, and performed a formal and numerical comparison with orbital mutual information. Maximally correlated orbitals, obtained by maximizing a nonlinear cost function of the mutual correlation, are also considered to identify a basis-independent partitioning of correlation. This study suggests that mutual correlation is a broadly applicable metric, useful in active space selection and the interpretation of electronic states.
Orbital entropies, pair entropies, and mutual information have become popular tools for analysis of strongly correlated wave functions. They can quantitatively measure how strongly an orbital participates in the electron correlation and reveal the correlation pattern between different orbitals. However, this pattern can become rather complicated and sometimes difficult to interpret for large active spaces and is not invariant with respect to the spin projection (MsM_sMs​) component of the spin multiplet state. We introduce a modified spin-free orbital entropy, pair entropy, and mutual information, which simplify the correlation analysis and are invariant with respect to MsM_sMs​. By comparison of these quantities with their "original" spin-including counterparts one can distinguish static correlation due to spin couplings from the "genuine" strong correlation due to a multiconfigurational character of the wave function. Recently, Evangelista introduced mutual correlation computed from the two-body cumulant as an alternative correlation measure, which is easier to compute for some methods. We present here a spin-free analogue of this quantity, which is MsM_sMs​ invariant, and perform a comparison of the entropy-based and cumulant-based correlation measures in both spin-free and original variants. We illustrate the approaches on iron-sulfur bound complexes with one and two iron atoms.
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.
Natural orbitals, defined in electronic structure and quantum chemistry as the molecular orbitals diagonalizing the one-particle reduced density matrix of the ground state, have been conjectured for decades to be the perfect reference orbitals to describe electron correlation. In the present work we applied the Wave function-Adapted Hamiltonian Through Orbital Rotation (WAHTOR) method to study correlated empirical ansätze for quantum computing. In all representative molecules considered, we show that the converged orbitals are coinciding with natural orbitals. Interestingly, the resulting quantum mutual information matrix built on such orbitals is also maximally sparse, providing a clear picture that such orbital choice is indeed able to provide the optimal basis to describe electron correlation. The correlation is therefore encoded in a smaller number of qubit pairs contributing to the quantum mutual information matrix.
Orbital rotations are used in quantum many-body physics and chemistry to reduce the complexity of quantum states by compressing their correlation into a small set of relevant orbitals. In striking contrast, quantum information processing strives for the opposite: orbitals that maximize entanglement as a resource. Motivated by this, we prove that Slater determinants of NNN fermions on a lattice of dA+dB=dd_A + d_B = ddA​+dB​=d sites, although free of particle correlation, can carry up to min⁡{dA,dB,N,d−N}\min\{d_A, d_B, N, d-N\}min{dA​,dB​,N,d−N} ebits of orbital entanglement. Given enough particles and holes, this equals the maximum over all fermionic states. For arbitrary free states, we derive the maximal orbital total correlation in closed form for all bipartitions, the finest partition, and all partitions of up to five sites, using new majorization relations for Hermitian matrices. Taken together, these results turn mean-field states, the starting point of most many-body methods in physics and chemistry, into a viable resource for quantum information processing.
Orbital entropies, pair entropies and mutual information have become popular tools for the analysis of strongly correlated wave functions. They can quantitatively measure how strongly an orbital (e.g. from the DMRG active space) participates in the strong correlation and reveals the correlation pattern between different orbitals. However, this pattern can become rather complicated and sometimes difficult to interpret for large active spaces and is not invariant with respect to the spin projection (Ms) component of the spin multiplet state. We introduce a modified spin-free orbital entropy, pair entropy and mutual information, which simplify the correlation analysis and are invariant with respect to Ms. By comparing these quantities with their ‘original’ spin-including counterparts, one can distinguish static correlation due to spin couplings from the ‘genuine’ strong correlation due to a multiconfigurational character of the wave function. We illustrate the approach on a model consisting of a non-interacting dimer of triplet diradicals and on a more realistic example of iron-sulphur bound complexes with one and two iron atoms.