# Exponential decay of mutual information for Gibbs states of local Hamiltonians

Andreas Bluhm1, Ángela Capel2,3,4, and Antonio Pérez-Hernández5,6

1QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
2Fachbereich Mathematik, Universität Tübingen, 72076 Tübingen, Germany
3Zentrum Mathematik, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany
4Munich Center for Quantum Science and Technology (MCQST), München, Germany
5Departamento de Matemática Aplicada I, Escuela Técnica Superior de Ingenieros Industriales, Universidad Nacional de Educación a Distancia, calle Juan del Rosal 12, 28040 Madrid (Ciudad Universitaria), Spain

### Abstract

The thermal equilibrium properties of physical systems can be described using Gibbs states. It is therefore of great interest to know when such states allow for an easy description. In particular, this is the case if correlations between distant regions are small. In this work, we consider 1D quantum spin systems with local, finite-range, translation-invariant interactions at any temperature. In this setting, we show that Gibbs states satisfy uniform exponential decay of correlations and, moreover, the mutual information between two regions decays exponentially with their distance, irrespective of the temperature. In order to prove the latter, we show that exponential decay of correlations of the infinite-chain thermal states, exponential uniform clustering and exponential decay of the mutual information are equivalent for 1D quantum spin systems with local, finite-range interactions at any temperature. In particular, Araki's seminal results yields that the three conditions hold in the translation-invariant case. The methods we use are based on the Belavkin-Staszewski relative entropy and on techniques developed by Araki. Moreover, we find that the Gibbs states of the systems we consider are superexponentially close to saturating the data-processing inequality for the Belavkin-Staszewski relative entropy.

Quantum spin chains are used to model atoms sitting on a line. If such systems have translation-invariant short-ranged interactions and are in thermal equilibrium, it was shown by Araki decades ago that correlations between separated regions decrease rapidly (exponentially fast) with the distance between these regions. This result is independent of the temperature and implies that such systems are easy to describe.

Our work studies a stronger measure of correlation than the one studied by Araki, namely the mutual information, which has a clear operational interpretation. We show that also the mutual information between distant regions decays exponentially fast, also independently of the temperature. To derive these results, we use a quantity that has not been used in quantum information theory so much, the Belavkin-Staszewski relative entropy, which gives one way to quantify the difference between quantum states. Moreover, we show that states which model the thermal equilibrium on quantum spin chains with translation-invariant short-ranged interactions almost factorize into smaller states, which should be of independent interest.

### ► References

[1] Y. Aragonés-Soria, J. Aberg, C.-Y. Park, and M. J. Kastoryano. Classical restrictions of generic matrix product states are quasi-locally Gibbsian. J. Math. Phys., 62: 093511, 2021. 10.1063/​5.0040256.
https:/​/​doi.org/​10.1063/​5.0040256

[2] H. Araki. Gibbs states of the one-dimensional quantum spin chain. Commun. Math. Phys., 14: 120–157, 1969. 10.1007/​BF01645134.
https:/​/​doi.org/​10.1007/​BF01645134

[3] I. Bardet, Á. Capel, L. Gao, A. Lucia, D. Pérez-García, and C. Rouzé. Entropy decay for Davies semigroups of a one dimensional quantum lattice. arXiv preprint, arXiv:2112.00601, 2021a. URL https:/​/​arxiv.org/​abs/​2112.00601.
arXiv:2112.00601

[4] I. Bardet, Á. Capel, L. Gao, A. Lucia, D. Pérez-García, and C. Rouzé. Rapid thermalization of spin chain commuting Hamiltonians. arXiv preprint, arXiv:2112.00593, 2021b. URL https:/​/​arxiv.org/​abs/​2112.00593.
arXiv:2112.00593

[5] I. Bardet, Á. Capel, A. Lucia, D. Pérez-García, and C. Rouzé. On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems. J. Math. Phys., 62: 061901, 2021c. 10.1063/​1.5142186.
https:/​/​doi.org/​10.1063/​1.5142186

[6] V. P. Belavkin and P. Staszewski. $C^*$-algebraic generalization of relative entropy and entropy. Ann. Inst. Henri Poincaré, section A, 37 (1): 51–58, 1982. URL http:/​/​www.numdam.org/​item/​?id=AIHPA_1982__37_1_51_0.
http:/​/​www.numdam.org/​item/​?id=AIHPA_1982__37_1_51_0

[7] F. Benatti. Dynamics, Information and Complexity in Quantum Systems. Theoretical and Mathematical Physics. Springer, 2009. 10.1007/​978-1-4020-9306-7.
https:/​/​doi.org/​10.1007/​978-1-4020-9306-7

[8] R. Bhatia. Matrix Analysis, volume 169 of Graduate texts in mathematics. Springer, 1997. 10.1007/​978-1-4612-0653-8.
https:/​/​doi.org/​10.1007/​978-1-4612-0653-8

[9] A. Bluhm and Á. Capel. A strengthened data processing inequality for the Belavkin-Staszewski relative entropy. Rev. Math. Phys., 32 (2): 2050005, 2020. 10.1142/​S0129055X20500051.
https:/​/​doi.org/​10.1142/​S0129055X20500051

[10] A. Bluhm, Á. Capel, and A. Pérez-Hernández. Weak quasi-factorization for the Belavkin-Staszewski relative entropy. In 2021 IEEE International Symposium on Information Theory (ISIT), pages 118–123, 2021. 10.1109/​ISIT45174.2021.9517893.
https:/​/​doi.org/​10.1109/​ISIT45174.2021.9517893

[11] F. G. S. L. Brandão and M. Horodecki. An area law for entanglement from exponential decay of correlations. Nat. Phys., 9: 721–726, 2013. 10.1038/​nphys2747.
https:/​/​doi.org/​10.1038/​nphys2747

[12] F. G. S. L. Brandão and M. Horodecki. Exponential decay of correlations implies area law. Commun. Math. Phys., 333: 761–798, 2015. 10.1007/​s00220-014-2213-8.
https:/​/​doi.org/​10.1007/​s00220-014-2213-8

[13] F. G. S. L. Brandão and M. J. Kastoryano. Finite correlation length implies efficient preparation of quantum thermal states. Commun. Math. Phys., 365: 1–16, 2019. 10.1007/​s00220-018-3150-8.
https:/​/​doi.org/​10.1007/​s00220-018-3150-8

[14] O. Bratteli and D. W. Robinson. Operator algebras and quantum-statistical mechanics I. C$^\ast$ and W$^\ast$-algebras. Symmetry groups. Decompositions of states. Texts and Monographs in Physics. Springer, 1979. 10.1007/​978-3-662-02313-6.
https:/​/​doi.org/​10.1007/​978-3-662-02313-6

[15] O. Bratteli and D. W. Robinson. Operator algebras and quantum-statistical mechanics II. Equilibrium states. Models in quantum statistical mechanics. Texts and Monographs in Physics. Springer, 1981. 10.1007/​978-3-662-09089-3.
https:/​/​doi.org/​10.1007/​978-3-662-09089-3

[16] A. Capel, C. Rouzé, and D. Stilck França. The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions. arXiv preprint, arXiv:2009.11817, 2020. URL https:/​/​arxiv.org/​abs/​2009.11817.
arXiv:2009.11817

[17] C.-F. Chen, K. Kato, and F. G. S. L. Brandão. Matrix Product Density Operators: when do they have a local parent Hamiltonian? arXiv preprint, arXiv:2010.14682, 2020. URL https:/​/​arxiv.org/​abs/​2010.14682.
arXiv:2010.14682

[18] J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete. Matrix product states and projected entangled pair states: Concepts, symmetries, and theorems. Rev. Mod. Phys., 93: 045003, 2021. 10.1103/​RevModPhys.93.045003.
https:/​/​doi.org/​10.1103/​RevModPhys.93.045003

[19] K. Fang and H. Fawzi. Geometric Rényi divergence and its applications in quantum channel capacities. Commun. Math. Phys., 384: 1615–1677, 2021. 10.1007/​s00220-021-04064-4.
https:/​/​doi.org/​10.1007/​s00220-021-04064-4

[20] O. Fawzi and R. Renner. Quantum conditional mutual information and approximate Markov chains. Commun. Math. Phys., 340: 575–611, 2015. 10.1007/​s00220-015-2466-x.
https:/​/​doi.org/​10.1007/​s00220-015-2466-x

[21] B. Groisman, S. Popescu, and A. Winter. Quantum, classical, and total amount of correlations in a quantum state. Phys. Rev. A, 72: 032317, 2005. 10.1103/​PhysRevA.72.032317.
https:/​/​doi.org/​10.1103/​PhysRevA.72.032317

[22] L. Gross. Decay of correlations in classical lattice models at high temperature. Commun. Math. Phys., 68: 9–27, 1979. 10.1007/​BF01562538.
https:/​/​doi.org/​10.1007/​BF01562538

[23] F. Hansen and G. K. Pedersen. Jensen's Operator Inequality. Bull. London Math. Soc., 35: 553–564, 2003. 10.1112/​S0024609303002200.
https:/​/​doi.org/​10.1112/​S0024609303002200

[24] A. Harrow, S. Mehraban, and M. Soleimanifar. Classical algorithms, correlation decay, and complex zeros of partition functions of quantum many-body systems. In STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 378–386, 2020. 10.1145/​3357713.3384322.
https:/​/​doi.org/​10.1145/​3357713.3384322

[25] M. B. Hastings. Quantum belief propagation: an algorithm for thermal quantum systems. Phys. Rev. B, 76 (20): 201102, 2007a. 10.1103/​PhysRevB.76.201102.
https:/​/​doi.org/​10.1103/​PhysRevB.76.201102

[26] M. B. Hastings. Entropy and entanglement in quantum ground states. Phys. Rev. B, 76: 035114, 2007b. 10.1103/​PhysRevB.76.035114.
https:/​/​doi.org/​10.1103/​PhysRevB.76.035114

[27] P. Hayden, D. Leung, P. W. Shor, and A. Winter. Randomizing quantum states: Constructions and applications. Commun. Math. Phys., 250: 371–391, 2004. 10.1007/​s00220-004-1087-6.
https:/​/​doi.org/​10.1007/​s00220-004-1087-6

[28] F. Hiai and M. Mosonyi. Different quantum f-divergencies and the reversibility of quantum operations. Rev. Math. Phys., 29 (7): 1750023, 2017. 10.1142/​S0129055X17500234.
https:/​/​doi.org/​10.1142/​S0129055X17500234

[29] M. J. Kastoryano and J. Eisert. Rapid mixing implies exponential decay of correlations. J. Math. Phys., 54: 102201, 2013. 10.1063/​1.4822481.
https:/​/​doi.org/​10.1063/​1.4822481

[30] K. Kato and F. G. S. L. Brandão. Quantum approximate Markov chains are thermal. Commun. Math. Phys., 370: 117–149, 2019. 10.1007/​s00220-019-03485-6.
https:/​/​doi.org/​10.1007/​s00220-019-03485-6

[31] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert. Locality of temperature. Phys. Rev. X, 4: 031019, 2014. 10.1103/​PhysRevX.4.031019.
https:/​/​doi.org/​10.1103/​PhysRevX.4.031019

[32] S. Kullback and R. A. Leibler. On information and sufficiency. Annals of Math. Stat., 22 (1): 79–86, 1951. 10.1214/​aoms/​1177729694.
https:/​/​doi.org/​10.1214/​aoms/​1177729694

[33] T. Kuwahara, K. Kato, and F. G. S. L. Brandão. Clustering of conditional mutual information for quantum gibbs states above a threshold temperature. Phys. Rev. Lett., 124: 220601, 2020. 10.1103/​PhysRevLett.124.220601.
https:/​/​doi.org/​10.1103/​PhysRevLett.124.220601

[34] T. Kuwahara, Á. M. Alhambra, and A. Anshu. Improved thermal area law and quasilinear time algorithm for quantum gibbs states. Phys. Rev. X, 11: 011047, 2021. 10.1103/​PhysRevX.11.011047.
https:/​/​doi.org/​10.1103/​PhysRevX.11.011047

[35] E. H. Lieb and D. W. Robinson. The finite group velocity of quantum spin systems. Commun. Math. Phys., 28: 251–257, 1972. 10.1007/​BF01645779.
https:/​/​doi.org/​10.1007/​BF01645779

[36] K. Matsumoto. A new quantum version of $f$-divergence. In Reality and Measurement in Algebraic Quantum Theory, pages 229–273. Springer, 2018. 10.1007/​978-981-13-2487-1_10.
https:/​/​doi.org/​10.1007/​978-981-13-2487-1_10

[37] A. Molnar, N. Schuch, F. Verstraete, and J. I. Cirac. Approximating Gibbs states of local Hamiltonians efficiently with projected entangled pair states. Phys. Rev. B, 91: 045138, 2015. 10.1103/​PhysRevB.91.045138.
https:/​/​doi.org/​10.1103/​PhysRevB.91.045138

[38] M. Ohya and D. Petz. Quantum Entropy and Its Use. Texts and Monographs in Physics. Springer, 1993.

[39] Y. M. Park and H. J. Yoo. Uniqueness and Clustering Properties of Gibbs States for Classical and Quantum Unbounded Spin Systems. J. Stat. Phys., 80: 223–271, 1995. 10.1007/​BF02178359.
https:/​/​doi.org/​10.1007/​BF02178359

[40] M. S. Pinsker. Information and Information Stability of Random Variables and Processes. Holden Day, 1964.

[41] D. Pérez-García and A. Pérez-Hernández. Locality estimates for complex-time evolution in 1D. arXiv preprint, arXiv:2004.10516, 2020. URL https:/​/​arxiv.org/​abs/​2004.10516.
arXiv:2004.10516

[42] S. O. Scalet, Á. M. Alhambra, G. Styliaris, and J. I. Cirac. Computable Rényi mutual information: Area laws and correlations. Quantum, 5: 541, 2021. 10.22331/​q-2021-09-14-541.
https:/​/​doi.org/​10.22331/​q-2021-09-14-541

[43] D. Sutter and R. Renner. Necessary criterion for approximate recoverability. Ann. Henri Poincaré, 19: 3007–3029, 2018. 10.1007/​s00023-018-0715-1.
https:/​/​doi.org/​10.1007/​s00023-018-0715-1

[44] D. Ueltschi. Cluster expansions and correlation functions. Mosc. Math. J., 4 (2): 511–522, 2004. 10.17323/​1609-4514-2004-4-2-511-522.
https:/​/​doi.org/​10.17323/​1609-4514-2004-4-2-511-522

[45] H. Umegaki. Conditional expectation in an operator algebra, IV (entropy and information). Kodai Math. Sem. Rep., 14 (2): 59–85, 1962. 10.2996/​kmj/​1138844604.
https:/​/​doi.org/​10.2996/​kmj/​1138844604

[46] M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac. Area laws in quantum systems: Mutual information and correlations. Phys. Rev. Lett., 100: 070502, 2008. 10.1103/​PhysRevLett.100.070502.
https:/​/​doi.org/​10.1103/​PhysRevLett.100.070502

### Cited by

[1] Tomotaka Kuwahara and Keiji Saito, "Exponential Clustering of Bipartite Quantum Entanglement at Arbitrary Temperatures", Physical Review X 12 2, 021022 (2022).

[2] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé, "Rapid thermalization of spin chain commuting Hamiltonians", arXiv:2112.00593.

[3] Álvaro M. Alhambra and J. Ignacio Cirac, "Locally Accurate Tensor Networks for Thermal States and Time Evolution", PRX Quantum 2 4, 040331 (2021).

[4] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé, "Entropy decay for Davies semigroups of a one dimensional quantum lattice", arXiv:2112.00601.

The above citations are from Crossref's cited-by service (last updated successfully 2022-10-04 19:42:28) and SAO/NASA ADS (last updated successfully 2022-10-04 19:42:30). The list may be incomplete as not all publishers provide suitable and complete citation data.