Quantum concentration inequalities. (English) Zbl 1501.81002
Summary: We establish Transportation Cost Inequalities (TCIs) with respect to the quantum Wasserstein distance by introducing quantum extensions of well-known classical methods: First, we generalize the Dobrushin uniqueness condition to prove that Gibbs states of 1D commuting Hamiltonians satisfy a TCI at any positive temperature and provide conditions under which this first result can be extended to non-commuting Hamiltonians. Next, using a non-commutative version of Ollivier’s coarse Ricci curvature, we prove that high temperature Gibbs states of commuting Hamiltonians on arbitrary hypergraphs \(H=(V,E)\) satisfy a TCI with constant scaling as \(O(|V|)\). Third, we argue that the temperatuSobolevre range for which the TCI holds can be enlarged by relating it to recently established modified logarithmic inequalities. Fourth, we prove that the inequality still holds for fixed points of arbitrary reversible local quantum Markov semigroups on regular lattices, albeit with slightly worsened constants, under a seemingly weaker condition of local indistinguishability of the fixed points. Finally, we use our framework to prove Gaussian concentration bounds for the distribution of eigenvalues of quasi-local observables and argue the usefulness of the TCI in proving the equivalence of the canonical and microcanonical ensembles and an exponential improvement over the weak Eigenstate Thermalization Hypothesis.
MSC:
| 81P16 | Quantum state spaces, operational and probabilistic concepts |
| 82B30 | Statistical thermodynamics |
| 80A10 | Classical and relativistic thermodynamics |
| 60E15 | Inequalities; stochastic orderings |
| 47D07 | Markov semigroups and applications to diffusion processes |
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
References:
| [1] | Raginsky, M., Sason, I.: Concentration of measure inequalities in information theory, communications and coding. arXiv preprint arXiv:1212.4663 (2012) · Zbl 1278.94031 |
| [2] | Boucheron, S.; Lugosi, G.; Massart, P., Concentration Inequalities: A Nonasymptotic Theory of Independence (2013), Oxford: Oxford University Press, Oxford · Zbl 1279.60005 |
| [3] | Kontorovich, A.; Raginsky, M.; Carlen, E.; Madiman, M.; Werner, E., Concentration of measure without independence: a unified approach via the martingale method, Convexity and Concentration, 183-210 (2017), New York, NY: Springer, New York, NY · Zbl 1387.60037 |
| [4] | Marton, K.: A simple proof of the blowing-up lemma (corresp.). IEEE Trans. Inf. Theory 32(3), 445-446 (1986) · Zbl 0594.94003 |
| [5] | Bobkov, SG; Götze, F., Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., 163, 1, 1-28 (1999) · Zbl 0924.46027 |
| [6] | Dobrushin, RL, Prescribing a system of random variables by conditional distributions, Theory Probab. Appl., 15, 3, 458-486 (1970) · Zbl 0264.60037 |
| [7] | Marton, K., Logarithmic Sobolev inequalities in discrete product spaces, Comb. Probab. Comput., 28, 6, 919-935 (2019) · Zbl 1434.60189 |
| [8] | Dobrushin, RL; Shlosman, SB, Completely analytical interactions: constructive description, J. Stat. Phys., 46, 5, 983-1014 (1987) · Zbl 0683.60080 |
| [9] | Külske, C., Concentration inequalities for functions of Gibbs fields with application to diffraction and random Gibbs measures, Commun. Math. Phys., 239, 1, 29-51 (2003) · Zbl 1023.60080 |
| [10] | Junge, M.; Zeng, Q., Noncommutative martingale deviation and Poincaré type inequalities with applications, Probab. Theory Relat. Fields, 161, 3-4, 449-507 (2014) · Zbl 1327.46062 |
| [11] | Tropp, J.A.: An introduction to matrix concentration inequalities. Found. Trends® Mach. Learn. 8(1-2), 1-230 (2015) · Zbl 1391.15071 |
| [12] | Huang, D.; Tropp, JA, From Poincaré inequalities to nonlinear matrix concentration, Bernoulli, 27, 3, 1724-1744 (2021) · Zbl 1485.60021 |
| [13] | Rouzé, C.; Datta, N., Concentration of quantum states from quantum functional and transportation cost inequalities, J. Math. Phys., 60, 1 (2019) · Zbl 1447.60051 |
| [14] | De Palma, G.; Marvian, M.; Trevisan, D.; Lloyd, S., The quantum Wasserstein distance of order 1, IEEE Trans. Inf. Theory, 67, 10, 6627-6643 (2021) · Zbl 1487.81026 |
| [15] | De Palma, G.; Trevisan, D., Quantum optimal transport with quantum channels, Ann. Henri Poincaré, 22, 3199-3234 (2021) · Zbl 1508.81355 |
| [16] | Golse, F.; Mouhot, C.; Paul, T., On the mean field and classical limits of quantum mechanics, Commun. Math. Phys., 343, 1, 165-205 (2016) · Zbl 1418.81021 |
| [17] | Cole, S., Eckstein, M., Friedland, S., Życzkowski, K.: Quantum optimal transport. arXiv preprint arXiv:2105.06922 (2021) |
| [18] | Carlen, EA; Maas, J., Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance, J. Funct. Anal., 273, 5, 1810-1869 (2017) · Zbl 1386.46057 |
| [19] | Bardet, I., Capel, Á., Gao, L., Lucia, A., Pérez-García, D., Rouzé, C.: Entropy decay for Davies semigroups of a one dimensional quantum lattice (2021) |
| [20] | Bardet, I., Capel, Á., Gao, L., Lucia, A., Pérez-García, D., Rouzé, C.: Rapid thermalization of spin chain commuting Hamiltonians (2021) |
| [21] | Nielsen, M.A., Chuang, I.: Quantum computation and quantum information (2002) · Zbl 1049.81015 |
| [22] | Wilde, MM, Quantum Information Theory (2013), Cambridge: Cambridge University Press, Cambridge · Zbl 1296.81001 |
| [23] | Holevo, AS, Quantum Systems, Channels, Information: A Mathematical Introduction (2012), Berlin: Walter de Gruyter, Berlin · Zbl 1332.81003 |
| [24] | Donald, MJ, On the relative entropy, Commun. Math. Phys., 105, 1, 13-34 (1986) · Zbl 0594.46059 |
| [25] | Petz, D., Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Commun. Math. Phys., 105, 1, 123-131 (1986) · Zbl 0597.46067 |
| [26] | Hiai, F.; Petz, D., The proper formula for relative entropy and its asymptotics in quantum probability, Commun. Math. Phys., 143, 1, 99-114 (1991) · Zbl 0756.46043 |
| [27] | Berta, M.; Fawzi, O.; Tomamichel, M., On variational expressions for quantum relative entropies, Lett. Math. Phys., 107, 12, 2239-2265 (2017) · Zbl 1386.94053 |
| [28] | Junge, M.; Renner, R.; Sutter, D.; Wilde, MM; Winter, A., Universal recovery maps and approximate sufficiency of quantum relative entropy, Ann. Henri Poincaré, 19, 2955-2978 (2018) · Zbl 1401.81025 |
| [29] | Sutter, D.; Berta, M.; Tomamichel, M., Multivariate trace inequalities, Commun. Math. Phys., 352, 1, 37-58 (2016) · Zbl 1365.15027 |
| [30] | Hayden, P.; Jozsa, R.; Petz, D.; Winter, A., Structure of states which satisfy strong subadditivity of quantum entropy with equality, Commun. Math. Phys., 246, 2, 359-374 (2004) · Zbl 1126.82004 |
| [31] | Tomamichel, M., Quantum Information Processing with Finite Resources: Mathematical Foundations (2015), Berlin: SpringerBriefs in Mathematical Physics. Springer International Publishing, Berlin · Zbl 1325.81003 |
| [32] | Ollivier, Y., Ricci curvature of Markov chains on metric spaces, J. Funct. Anal., 256, 3, 810-864 (2009) · Zbl 1181.53015 |
| [33] | Eldan, R.; Lee, JR; Lehec, J.; Loebl, M.; Nešetřil, J.; Thomas, R., Transport-entropy inequalities and curvature in discrete-space Markov chains, A Journey Through Discrete Mathematics, 391-406 (2017), Cham: Springer, Cham · Zbl 1391.60173 |
| [34] | Griffiths, RB, Correlations in ising ferromagnets, III. Commun. Math. Phys., 6, 2, 121-127 (1967) |
| [35] | Kastoryano, MJ; Temme, K., Quantum logarithmic Sobolev inequalities and rapid mixing, J. Math. Phys., 54, 5 (2013) · Zbl 1379.81021 |
| [36] | Gao, L., Rouzé, C.: Complete entropic inequalities for quantum Markov chains. arXiv preprint arXiv:2102.04146 (2021) · Zbl 1501.46053 |
| [37] | Carlen, EA; Maas, J., Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems, J. Stat. Phys., 178, 2, 319-378 (2019) · Zbl 1445.46049 |
| [38] | Capel, Á., Rouzé, C., França, D.S.: The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions. arXiv preprint arXiv:2009.11817 (2020) |
| [39] | Cubitt, TS; Lucia, A.; Michalakis, S.; Perez-Garcia, D., Stability of local quantum dissipative systems, Commun. Math. Phys., 337, 1275-1315 (2015) · Zbl 1316.81062 |
| [40] | Kastoryano, MJ; Brandao, FG, Quantum Gibbs samplers: the commuting case, Commun. Math. Phys., 344, 3, 915-957 (2016) · Zbl 1386.82011 |
| [41] | Brandão, FGSL; Kastoryano, MJ, Finite correlation length implies efficient preparation of quantum thermal states, Commun. Math. Phys., 365, 1, 1-16 (2019) · Zbl 1406.81017 |
| [42] | Junge, M., LaRacuente, N., Rouzé, C.: Stability of logarithmic Sobolev inequalities under a noncommutative change of measure. arXiv preprint arXiv:1911.08533 (2019) |
| [43] | Kuwahara, T.; Saito, K., Gaussian concentration bound and ensemble equivalence in generic quantum many-body systems including long-range interactions, Ann. Phys., 421 (2020) · Zbl 1448.81498 |
| [44] | Matsumoto, K.; Ozawa, M.; Butterfield, J.; Halvorson, H.; Rédei, M.; Kitajima, Y.; Buscemi, F., A new quantum version of f-divergence, Nagoya Winter Workshop: Reality and Measurement in Algebraic Quantum Theory, 229-273 (2015), Singapore: Springer, Singapore · Zbl 1414.81053 |
| [45] | Anshu, A., Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems, New J. Phys., 18, 8 (2016) · Zbl 1456.81043 |
| [46] | Kuwahara, T.: Connecting the probability distributions of different operators and generalization of the Chernoff-Hoeffding inequality. J. Stat. Mech.: Theory Exp. 2016(11), 113103 (2016) · Zbl 1456.82593 |
| [47] | Kuwahara, T.; Saito, K., Eigenstate thermalization from the clustering property of correlation, Phys. Rev. Lett., 124, 20 (2020) |
| [48] | Kliesch, M.; Gogolin, C.; Kastoryano, MJ; Riera, A.; Eisert, J., Locality of temperature. Phys. Rev. X, 4 (2014) |
| [49] | Lima, R., Equivalence of ensembles in quantum lattice systems, Annales de l’IHP Physique théorique, 15, 1, 61-68 (1971) |
| [50] | Lima, R., Equivalence of ensembles in quantum lattice systems: states, Commun. Math. Phys., 24, 3, 180-192 (1972) |
| [51] | Müller, MP; Adlam, E.; Masanes, L.; Wiebe, N., Thermalization and canonical typicality in translation-invariant quantum lattice systems, Commun. Math. Phys., 340, 2, 499-561 (2015) · Zbl 1335.82007 |
| [52] | Brandão, F.G.S.L., Cramer, M.: Equivalence of statistical mechanical ensembles for non-critical quantum systems. arXiv:1502.03263 [cond-mat, physics:quant-ph], February 2015 |
| [53] | Deutsch, JM, Quantum statistical mechanics in a closed system, Phys. Rev. A, 43, 4, 2046 (1991) |
| [54] | Srednicki, M., Chaos and quantum thermalization, Phys. Rev. E, 50, 2, 888-901 (1994) |
| [55] | Gogolin, C.; Eisert, J., Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems, Rep. Prog. Phys., 79, 5 (2016) |
| [56] | Reimann, P., Dynamical typicality approach to eigenstate thermalization, Phys. Rev. Lett., 120, 23 (2018) |
| [57] | De Palma, G.; Serafini, A.; Giovannetti, V.; Cramer, M., Necessity of eigenstate thermalization, Phys. Rev. Lett., 115, 22 (2015) |
| [58] | Biroli, G.; Kollath, C.; Läuchli, AM, Effect of rare fluctuations on the thermalization of isolated quantum systems, Phys. Rev. Lett., 105, 25 (2010) |
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