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.
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.
 M. B. Hastings, J. Stat. Mech. 2007, P08024 (2007).
 M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Phys. Rev. Lett. 100, 070502 (2008).
 R. R. P. Singh, M. B. Hastings, A. B. Kallin, and R. G. Melko, Phys. Rev. Lett. 106, 135701 (2011).
 A. Anshu, V. K. Devabathini, and R. Jain, Phys. Rev. Lett. 119, 120506 (2017).
 K. Matsumoto, in Reality and Measurement in Algebraic Quantum Theory, edited by M. Ozawa, J. Butterfield, H. Halvorson, M. Rédei, Y. Kitajima, and F. Buscemi (Springer Singapore, Singapore, 2018) pp. 229–273.
 P. Calabrese and J. Cardy, J. Stat. Mech. 2004, P06002 (2004).
 P. Calabrese and J. Cardy, J. Phys. A: Math. 42, 504005 (2009).
 N. Lashkari, Phys. Rev. Lett. 113, 051602 (2014).
 J. Zhang, P. Ruggiero, and P. Calabrese, Phys. Rev. Lett. 122, 141602 (2019).
 H. Bernigau, M. J. Kastoryano, and J. Eisert, J. Stat. Mech. 2015, P02008 (2015).
 B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete, New J. Phys. 12, 025012 (2010).
 M. B. Hastings, I. González, A. B. Kallin, and R. G. Melko, Phys. Rev. Lett. 104, 157201 (2010).
 T. Grover, Phys. Rev. Lett. 111, 130402 (2013).
 M. Kliesch, D. Gross, and J. Eisert, Phys. Rev. Lett. 113, 160503 (2014).
 A. Rényi, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics (University of California Press, Berkeley, Calif., 1961) pp. 547–561.
 O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics: Volume 1: C*-and W*-Algebras. Symmetry Groups. Decomposition of States (Springer Verlag Berlin Heidelberg, 1979).
 D. A. Abanin, W. De Roeck, and F. m. c. Huveneers, Phys. Rev. Lett. 115, 256803 (2015).
 I. Arad, T. Kuwahara, and Z. Landau, J. Stat. Mech. 2016, 033301 (2016).
 A. Avdoshkin and A. Dymarsky, Phys. Rev. Research 2, 043234 (2020).
 G. De las Cuevas, N. Schuch, D. Pérez-García, and J. I. Cirac, New J. Phys. 15, 123021 (2013).
 V. P. Belavkin and P. Staszewski, Ann. Inst. Henri Poincare A 37, 51 (1982).
 D. Ruelle, Statistical mechanics: Rigorous results (Benjamin, 1969).
 Gilles Parez, Riccarda Bonsignori, and Pasquale Calabrese, "Exact quench dynamics of symmetry resolved entanglement in a free fermion chain", arXiv:2106.13115.
 Andreas Bluhm, Ángela Capel, and Antonio Pérez-Hernández, "Exponential decay of mutual information for Gibbs states of local Hamiltonians", arXiv:2104.04419.
The above citations are from SAO/NASA ADS (last updated successfully 2021-09-17 18:03:55). The list may be incomplete as not all publishers provide suitable and complete citation data.
On Crossref's cited-by service no data on citing works was found (last attempt 2021-09-17 18:03:54).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.