Computable Rényi mutual information: Area laws and correlations

Samuel O. Scalet; Álvaro M. Alhambra; Georgios Styliaris; J. Ignacio Cirac

doi:10.22331/q-2021-09-14-541

Computable Rényi mutual information: Area laws and correlations

Samuel O. Scalet, Álvaro M. Alhambra, Georgios Styliaris, and J. Ignacio Cirac

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

Published:	2021-09-14, volume 5, page 541
Eprint:	arXiv:2103.01709v2
Doi:	https://doi.org/10.22331/q-2021-09-14-541
Citation:	Quantum 5, 541 (2021).

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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

Quantum systems of many particles are notoriously difficult to handle due to the large number of parameters needed for their description. This complexity is often related to the structure of their correlations. Due to this, there is a pressing need for efficient and physically meaningful methods of quantifying these correlations. This is often done with different versions of the so-called mutual information. However, these previously considered versions are either impossible to calculate efficiently, or exhibit various pathological features.

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.

► BibTeX data

@article{Scalet2021computablerenyi,
  doi = {10.22331/q-2021-09-14-541},
  url = {https://doi.org/10.22331/q-2021-09-14-541},
  title = {Computable {R}{\'{e}}nyi mutual information: {A}rea laws and correlations},
  author = {Scalet, Samuel O. and Alhambra, {\'{A}}lvaro M. and Styliaris, Georgios and Cirac, J. Ignacio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {541},
  month = sep,
  year = {2021}
}

► References

[1] J. Eisert, arXiv:1308.3318 [quant-ph].
arXiv:1308.3318

[2] I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, arXiv:2011.12127 [quant-ph].
arXiv:2011.12127

[3] F. Verstraete and J. I. Cirac, Phys. Rev. B 73, 094423 (2006).
https://doi.org/10.1103/PhysRevB.73.094423

[4] M. B. Hastings, J. Stat. Mech. 2007, P08024 (2007).
https://doi.org/10.1088/1742-5468/2007/08/p08024

[5] M. B. Hastings, Phys. Rev. B 73, 085115 (2006).
https://doi.org/10.1103/PhysRevB.73.085115

[6] A. Molnar, N. Schuch, F. Verstraete, and J. I. Cirac, Phys. Rev. B 91, 045138 (2015).
https://doi.org/10.1103/PhysRevB.91.045138

[7] T. Kuwahara, A. M. Alhambra, and A. Anshu, Phys. Rev. X 11, 011047 (2021).
https://doi.org/10.1103/PhysRevX.11.011047

[8] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).
https://doi.org/10.1103/PhysRevLett.91.147902

[9] B. Groisman, S. Popescu, and A. Winter, Phys. Rev. A 72, 032317 (2005).
https://doi.org/10.1103/PhysRevA.72.032317

[10] M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Phys. Rev. Lett. 100, 070502 (2008).
https://doi.org/10.1103/PhysRevLett.100.070502

[11] F. Alcaraz and M. Rajabpour, Phys. Rev. B 90, 075132 (2014).
https://doi.org/10.1103/physrevb.90.075132

[12] J.-M. Stéphan, Phys. Rev. B 90, 045424 (2014).
https://doi.org/10.1103/physrevb.90.045424

[13] R. R. P. Singh, M. B. Hastings, A. B. Kallin, and R. G. Melko, Phys. Rev. Lett. 106, 135701 (2011).
https://doi.org/10.1103/PhysRevLett.106.135701

[14] M. C. Bañuls, N. Y. Yao, S. Choi, M. D. Lukin, and J. I. Cirac, Phys. Rev. B 96, 174201 (2017).
https://doi.org/10.1103/PhysRevB.96.174201

[15] M. Kormos and Z. Zimborás, J. Phys. A: Math. Theor. 50, 264005 (2017).
https://doi.org/10.1088/1751-8121/aa70f6

[16] M. Tomamichel, Quantum Information Processing with Finite Resources (Springer International Publishing, 2016).
https://doi.org/10.1007/978-3-319-21891-5

[17] S. Khatri and M. M. Wilde, arXiv:2011.04672 [quant-ph].
arXiv:2011.04672

[18] M. Berta, M. Christandl, and R. Renner, Commun. Math. Phys. 306, 579 (2011).
https://doi.org/10.1007/s00220-011-1309-7

[19] A. Anshu, V. K. Devabathini, and R. Jain, Phys. Rev. Lett. 119, 120506 (2017).
https://doi.org/10.1103/PhysRevLett.119.120506

[20] M. M. Wilde, A. Winter, and D. Yang, Commun. Math. Phys. 331, 593 (2014).
https://doi.org/10.1007/s00220-014-2122-x

[21] M. Mosonyi, IEEE Trans. Inf. Theory 61, 2997–3012 (2015).
https://doi.org/10.1109/tit.2015.2417877

[22] F. Leditzky, M. M. Wilde, and N. Datta, J. Math. Phys. 57, 082202 (2016).
https://doi.org/10.1063/1.4960099

[23] M. Mosonyi and T. Ogawa, Commun. Math. Phys. 355, 373–426 (2017).
https://doi.org/10.1007/s00220-017-2928-4

[24] D. Ding and M. M. Wilde, Probl. Inf. Transm. 54, 1–19 (2018).
https://doi.org/10.1134/s0032946018010015

[25] K. Fang and H. Fawzi, arXiv:1909.05758 [quant-ph].
arXiv:1909.05758

[26] M. Mosonyi and T. Ogawa, Commun. Math. Phys. 334, 1617–1648 (2014).
https://doi.org/10.1007/s00220-014-2248-x

[27] M. Hayashi and M. Tomamichel, J. Math. Phys. 57, 102201 (2016).
https://doi.org/10.1063/1.4964755

[28] D. Petz, Rep. Math. Phys. 23, 57 (1986).
https://doi.org/10.1016/0034-4877(86)90067-4

[29] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, J. Math. Phys. 54, 122203 (2013).
https://doi.org/10.1063/1.4838856

[30] 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.
https://doi.org/10.1007/978-981-13-2487-1

[31] N. Datta, IEEE Trans. Inf. Theory 55, 2816–2826 (2009).
https://doi.org/10.1109/tit.2009.2018325

[32] K. M. R. Audenaert and N. Datta, Journal of Mathematical Physics 56, 022202 (2015).
https://doi.org/10.1063/1.4906367

[33] H. Fawzi and O. Fawzi, Quantum 5, 387 (2021).
https://doi.org/10.22331/q-2021-01-26-387

[34] P. Calabrese and J. Cardy, J. Stat. Mech. 2004, P06002 (2004).
https://doi.org/10.1088/1742-5468/2004/06/p06002

[35] P. Calabrese and J. Cardy, J. Phys. A: Math. 42, 504005 (2009).
https://doi.org/10.1088/1751-8113/42/50/504005

[36] C. T. Asplund and A. Bernamonti, Phys. Rev. D 89, 066015 (2014).
https://doi.org/10.1103/PhysRevD.89.066015

[37] C. A. Agón and T. Faulkner, J. High Energy Phys. 2016 (8), 1.
https://doi.org/10.1007/JHEP08(2016)118

[38] B. Chen, P.-X. Hao, and W. Song, J. High Energy Phys. 2019 (10).
https://doi.org/10.1007/jhep10(2019)037

[39] N. Lashkari, Phys. Rev. Lett. 113, 051602 (2014).
https://doi.org/10.1103/PhysRevLett.113.051602

[40] J. Zhang, P. Ruggiero, and P. Calabrese, Phys. Rev. Lett. 122, 141602 (2019).
https://doi.org/10.1103/PhysRevLett.122.141602

[41] H. Bernigau, M. J. Kastoryano, and J. Eisert, J. Stat. Mech. 2015, P02008 (2015).
https://doi.org/10.1088/1742-5468/2015/02/p02008

[42] B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete, New J. Phys. 12, 025012 (2010).
https://doi.org/10.1088/1367-2630/12/2/025012

[43] J. I. Cirac and G. Sierra, Phys. Rev. B 81, 104431 (2010).
https://doi.org/10.1103/PhysRevB.81.104431

[44] M. B. Hastings, I. González, A. B. Kallin, and R. G. Melko, Phys. Rev. Lett. 104, 157201 (2010).
https://doi.org/10.1103/PhysRevLett.104.157201

[45] S. Humeniuk and T. Roscilde, Phys. Rev. B 86, 235116 (2012).
https://doi.org/10.1103/PhysRevB.86.235116

[46] T. Grover, Phys. Rev. Lett. 111, 130402 (2013).
https://doi.org/10.1103/PhysRevLett.111.130402

[47] R. Renner, Int. J. Quantum Inf. 06, 1 (2008).
https://doi.org/10.1142/s0219749908003256

[48] S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
https://doi.org/10.1103/PhysRevLett.69.2863

[49] M. Kliesch, D. Gross, and J. Eisert, Phys. Rev. Lett. 113, 160503 (2014).
https://doi.org/10.1103/PhysRevLett.113.160503

[50] U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005).
https://doi.org/10.1103/RevModPhys.77.259

[51] U. Schollwöck, Annals of Physics 326, 96 (2011), january 2011 Special Issue.
https://doi.org/10.1016/j.aop.2010.09.012

[52] Z. Landau, U. Vazirani, and T. Vidick, Nature Physics 11, 566 (2015).
https://doi.org/10.1038/nphys3345

[53] M. Block, J. Motruk, S. Gazit, M. P. Zaletel, Z. Landau, U. Vazirani, and N. Y. Yao, Phys. Rev. B 103, 195122 (2021).
https://doi.org/10.1103/PhysRevB.103.195122

[54] D. Aharonov, D. Gottesman, S. Irani, and J. Kempe, Communications in Mathematical Physics 287, 41 (2009).
https://doi.org/10.1007/s00220-008-0710-3

[55] 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.
https://projecteuclid.org/euclid.bsmsp/1200512181

[56] M. Berta, O. Fawzi, and M. Tomamichel, Lett. Math. Phys. 107, 2239 (2017).
https://doi.org/10.1007/s11005-017-0990-7

[57] D. Pérez-García and A. Pérez-Hernández, arXiv:2004.10516 [math-ph].
arXiv:2004.10516

[58] H. Araki, Commun. Math. Phys. 14, 120 (1969).
https://doi.org/10.1007/BF01645134

[59] 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).
https://doi.org/10.1007/978-3-662-02313-6

[60] D. A. Abanin, W. De Roeck, and F. m. c. Huveneers, Phys. Rev. Lett. 115, 256803 (2015).
https://doi.org/10.1103/PhysRevLett.115.256803

[61] I. Arad, T. Kuwahara, and Z. Landau, J. Stat. Mech. 2016, 033301 (2016).
https://doi.org/10.1088/1742-5468/2016/03/033301

[62] T. Kuwahara, T. Mori, and K. Saito, Ann. Phys. 367, 96–124 (2016).
https://doi.org/10.1016/j.aop.2016.01.012

[63] G. Bouch, J. Math. Phys. 56, 123303 (2015).
https://doi.org/10.1063/1.4936209

[64] A. Avdoshkin and A. Dymarsky, Phys. Rev. Research 2, 043234 (2020).
https://doi.org/10.1103/PhysRevResearch.2.043234

[65] F. Verstraete and J. I. Cirac, arXiv:cond-mat/0407066.
arXiv:arXiv:cond-mat/0407066

[66] G. De las Cuevas, N. Schuch, D. Pérez-García, and J. I. Cirac, New J. Phys. 15, 123021 (2013).
https://doi.org/10.1088/1367-2630/15/12/123021

[67] G. De las Cuevas, T. S. Cubitt, J. I. Cirac, M. M. Wolf, and D. Pérez-García, J. Math. Phys. 57, 071902 (2016).
https://doi.org/10.1063/1.4954983

[68] F. Hiai, M. Ohya, and M. Tsukada, Pacific J. Math. 96, 99 (1981).
https://doi.org/10.2140/pjm.1981.96.99

[69] G. L. Gilardoni, IEEE Trans. Inf. Theory 56, 5377 (2010).
https://doi.org/10.1109/TIT.2010.2068710

[70] J. Guth Jarkovský, A. Molnár, N. Schuch, and J. I. Cirac, PRX Quantum 1, 010304 (2020).
https://doi.org/10.1103/PRXQuantum.1.010304

[71] V. P. Belavkin and P. Staszewski, Ann. Inst. Henri Poincare A 37, 51 (1982).
http://www.numdam.org/item/AIHPA_1982__37_1_51_0

[72] S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems (Cambridge University Press, 2017).
https://doi.org/10.1017/9781316882603

[73] O. Perron, Math. Ann. 64, 248 (1907).
https://doi.org/10.1007/bf01449896

[74] M. Lenci and L. Rey-Bellet, J. Stat. Phys. 119, 715–746 (2005).
https://doi.org/10.1007/s10955-005-3015-3

[75] D. Ruelle, Statistical mechanics: Rigorous results (Benjamin, 1969).

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