Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München, Germany
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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.
Featured image: The mutual information measures correlations in bipartite systems. In a thermal state of a local Hamiltonian the mutual information only scales with the size of the boundary. In this work, the maximal Rényi divergence as well as measured Rényi divergences are used to define a computable mutual information.
Popular summary
In this work, we propose an alternative definition, a so-called Rényi mutual information, which does not have either of these problems. First, it can be computed in practice using standard numerical techniques from many-body physics. Second, it satisfies all the desirable properties of a measure of correlations that previous ones did not, such as upper bounding all correlation functions. In addition, we show its significance for the ubiquitous thermal states by proving an area law: the quantity evaluated for two regions in a thermal state only grows with the size of their boundary.
With these results, we provide a new tool for the study of correlations of strongly coupled many-body systems. This is a subject of crucial importance, since many such models are becoming increasingly relevant due to the possibility of simulating them in leading quantum platforms.
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Cited by
[1] Gilles Parez, Riccarda Bonsignori, and Pasquale Calabrese, “Exact quench dynamics of symmetry resolved entanglement in a free fermion chain”, arXiv:2106.13115.
[2] Andreas Bluhm, Ángela Capel, and Antonio Pérez-Hernández, “Exponential decay of mutual information for Gibbs states of local Hamiltonians”, arXiv:2104.04419.
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