Abstract
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.
Highlights
One of the most important features of quantum systems is the nature of their correlations, which differ from their classical counterparts, and lie behind the complexity of many-body quantum states
We show that they satisfy desirable properties of the mutual information, including an area law for thermal states, which constitutes our main technical result
As a measure of bipartite correlations, they satisfy a number of desirable properties, including area laws for thermal states
Summary
One of the most important features of quantum systems is the nature of their correlations, which differ from their classical counterparts, and lie behind the complexity of many-body quantum states. The resulting quantity can be computed via a variety of numerical and analytical means and has been shown to characterize phenomena such as quantum [11, 12] and thermal [13] phase transitions, or the correlations in many-body localization [14] It lacks a number of important properties, which prevent it from being a sensible measure of correlations. We focus on two particular cases and explain how to compute them in practice, at least when the input state is represented via tensor networks We show that they satisfy desirable properties of the mutual information, including an area law for thermal states, which constitutes our main technical result. The technical proofs, as well as further details, can be found in the Appendix
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