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arXiv:2308.08600 · Journal of High Energy Physics

Rényi mutual information in quantum field theory, tensor networks, and gravity

Jonah Kudler-Flam, Laimei Nie, Akash Vijay

Published Jun 27, 2024• 6 citations
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Abstract

A bstract We explore a large class of correlation measures called the α − z Rényi mutual informations (RMIs). Unlike the commonly used notion of RMI involving linear combinations of Rényi entropies, the α − z RMIs are positive semi-definite and monotonically decreasing under local quantum operations, making them sensible measures of total (quantum and classical) correlations. This follows from their descendance from Rényi relative entropies. In addition to upper bounding connected correlation functions between subsystems, we prove the much stronger statement that for certain values of α and z , the α − z RMIs also lower bound certain connected correlation functions. We develop an easily implementable replica trick which enables us to compute the α − z RMIs in a variety of many-body systems including conformal field theories, free fermions, random tensor networks, and holography.

DOI
10.1007/jhep06(2024)195

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A bstract We explore a large class of correlation measures called the α − z Rényi mutual informations (RMIs). Unlike the commonly used notion of RMI involving linear combinations of Rényi entropies, the α − z RMIs are positive semi-definite and monotonically decreasing under local quantum operations, making them sensible measures of total (quantum and classical) correlations. This follows from their descendance from Rényi relative entropies. In addition to upper bounding connected correlation functions between subsystems, we prove the much stronger statement that for certain values of α and z , the α − z RMIs also lower bound certain connected correlation functions. We develop an easily implementable replica trick which enables us to compute the α − z RMIs in a variety of many-body systems including conformal field theories, free fermions, random tensor networks, and holography.
We study a proper definition of Rényi mutual information (RMI) in quantum field theory as defined via the Petz Rényi relative entropy. Unlike the standard definition, the RMI we compute is a genuine measure of correlations between subsystems, as evidenced by its non-negativity and monotonicity under local operations. Furthermore, the RMI is UV finite and well defined in the continuum limit. We develop a replica path integral approach for the RMI in quantum field theories and evaluate it explicitly in 1+1D conformal field theory using twist fields. We prove that it bounds connected correlation functions and check our results against exact numerics in the massless free fermion theory.
Quantum mutual information is an important tool for characterizing correlations in quantum many-body systems, but its numerical evaluation is often prohibitively expensive. While some variants of Rényi Mutual Information (RMI) are computationally more tractable, it is not clear whether they correctly capture the long-wavelength physics or are dominated by UV effects, which is of key importance in lattice simulations. We analyze the relevance of lattice effects on the family of ααα-zzz Rényi mutual informations for ground states of models with conformal field theory descriptions. On the example of massless free fermions we identify distinct regions in the ααα-zzz plane, where RMI corrections due to the lattice are relevant or irrelevant. We further support these findings with MPS calculations on the transverse field Ising model (TFIM). Our results, accompanied by the open-source Julia package QMICalc...jl, provide guidance to using RMI in quantum-many body physics numerical computations.
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.
By developing a method to represent the Renyi entropies via a replica trick on classical statistical mechanical systems, we introduce a procedure to calculate the Renyi mutual information (RMI) in any Monte Carlo simulation. Through simulations on several classical models, we demonstrate that the RMI can detect finite-temperature critical points, and even identify their universality class, without knowledge of an order parameter or other thermodynamic estimators. Remarkably, in addition to critical points mediated by symmetry breaking, the RMI is able to detect topological vortex-unbinding transitions, as we explicitly demonstrate on simulations of the XYXYXY model.
Relative entropy is a measure of distinguishability for quantum states, and it plays a central role in quantum information theory. The family of Renyi entropies generalizes to Renyi relative entropies that include, as special cases, most entropy measures used in quantum information theory. We construct a Euclidean path-integral approach to Renyi relative entropies in conformal field theory, then compute the fidelity and the relative entropy of states in one spatial dimension at zero and finite temperature using a replica trick. In contrast to the entanglement entropy, the relative entropy is free of ultraviolet divergences, and is obtained as a limit of certain correlation functions. The relative entropy of two states provides an upper bound on their trace distance.
The mutual information is a measure of classical and quantum correlations of great interest in quantum information. It is also relevant in quantum many-body physics, by virtue of satisfying an area law for thermal states and bounding all correlation functions. However, calculating it exactly or approximately is often challenging in practice. Here, we consider alternative definitions based on Rényi divergences. Their main advantage over their von Neumann counterpart is that they can be expressed as a variational problem whose cost function can be efficiently evaluated for families of states like matrix product operators while preserving all desirable properties of a measure of correlations. In particular, we show that they obey a thermal area law in great generality, and that they upper bound all correlation functions. We also investigate their behavior on certain tensor network states and on classical thermal distributions.