Abstract
We propose an extension of the sandwiched Rényi relative \(\alpha \)-entropy to normal positive functionals on arbitrary von Neumann algebras, for the values \(\alpha >1\). For this, we use Kosaki’s definition of noncommutative \(L_p\)-spaces with respect to a state. We show that these extensions coincide with the previously defined Araki–Masuda divergences (Berta et al. in arXiv:1608.05317, 2016) and prove some of their properties, in particular the data processing inequality with respect to positive normal unital maps. As a consequence, we obtain monotonicity of the Araki relative entropy with respect to such maps, extending the results of Müller-Hermes and Reeb. (Ann. Henri Poincaré 18:1777–1788, 2017) to arbitrary von Neumann algebras. It is also shown that equality in data processing inequality characterizes sufficiency (reversibility) of quantum channels.
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References
Araki, H.: Relative entropy of states of von Neumann algebras. Publ. RIMS Kyoto Univ. 11, 809–833 (1976)
Araki, H., Masuda, T.: Positive cones and \(L_p\)-spaces for von Neumann algebras. Publ. RIMS Kyoto Univ. 18, 339–411 (1982)
Audenaert, K.M.R., Datta, N.: \(\alpha -z\)-Rényi relative entropies. J. Math. Phys. 56, 022202 (2015)
Audenaert, K.M.R., Nussbaum, M., Szkola, A., Verstraete, F.: Asymptotic error rates in quantum hypothesis testing. Commun. Math. Phys. 279, 251–283 (2008). arXiv:0708.4282
Beigi, S.: Sandwiched Rényi divergence satisfies data processing inequality. J. Math. Phys. 54, 122202 (2013). arXiv:1306.5920 [quant-ph]
Bergh, J., Löfström, J.: Interpolation Spaces. Springer, New York (1976)
Berta, M., Scholz, V.B., Tomamichel, M.: Rényi divergences as weighted non-commutative vector valued \(L_p\)-spaces. arXiv:1608.05317 (2016)
Calderón, A.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)
Csiszár, I.: Generalized cutoff rates and Rényi information measures. IEEE Trans. Inf. Theory 41, 26–34 (1995)
Diestel, J.: Geometry of Banach Spaces-Selected Topics. Lecture Notes in Mathematics. Springer, Berlin (1975)
Frank, R.L., Lieb, E.H.: Monotonicity of a relative Rényi entropy. J. Math. Phys. 54, 122201 (2013). arXiv:1306.5358 [math-ph]
Haagerup, U.: \(L_p\)-spaces associated with an arbitrary von Neumann algebra. In: Algébres d’opérateurs et leurs applications en physique mathématique (Proc. Colloq. Marseille, 1977), Volume 274 of Colloq. Internat. CNRS, pp. 175–184. CNRS (1979)
Hiai, F.: Sandwiched Rényi divergences in von Neumann algebras. Unpublished Notes (2017)
Hiai, F., Mosonyi, M., Ogawa, T.: Error exponents in hypothesis testing for correlated states on a spin chain. J. Math. Phys. 49, 032112 (2008). arXiv:0707.2020
Hiai, F., Mosonyi, M., Petz, D., Beny, C.: Quantum f-divergences and error correction. Rev. Math. Phys. 23, 691–747 (2011). arXiv:1008.2529
Jaksic, V., Ogata, Y., Pautrat, Y., Pillet, C.A.: Entropic fluctuations in quantum statistical mechanics. An introduction. In: Quantum Theory from Small to Large Scales: Lecture Notes of the Les Houches Summer School, vol. 95. Oxford University Press (2012)
Jaksic, V., Ogata, Y., Pillet, C.A., Seiringer, R.: Quantum hypothesis testing and non-equilibrium statistical mechanics. Rev. Math. Phys. 24, 1230002 (2012)
Jenčová, A.: Preservation of a quantum Rényi relative entropy implies existence of a recovery map. J. Phys. A Math. Theor. 50, 085303 (2017)
Jenčová, A.: Rényi relative entropies and noncommutative \(L_p\)-spaces II. arXiv:1707.00047 (2017)
Jenčová, A., Petz, D.: Sufficiency in quantum statistical inference. Commun. Math. Phys. 263, 259–276 (2006). arXiv:math-ph/0412093
Junge, M., Xu, Q.: Noncommutative Burkholder/Rosenthal inequalities. Ann. Probab. 31, 948–995 (2003)
Kosaki, H.: Positive cones and \(L_p\)-spaces associated with a von Neumann algebra. J. Oper. Theory 6, 13–23 (1981)
Kosaki, H.: Applications of the complex interpolation method to a von Neumann algebra: non-commutative \(L_p\)-spaces. J. Funct. Anal. 56, 26–78 (1984)
Kosaki, H.: Applications of uniform convexity of noncommutative \(L^{p}\)-spaces. Trans. Am. Math. Soc. 283, 265–282 (1984)
Kosaki, H.: An inequality of Araki–Lieb–Thirring (von Neumann algebra case). Proc. Am. Math. Soc. 114, 477–481 (1992)
Kümmerer, B., Nagel, R.: Mean ergodic semigroups on W*-algebras. Acta Sci. Math. 41, 151–155 (1979)
Mosonyi, M., Hiai, F.: On the quantum Rényi relative entropies and related capacity formulas. IEEE Trans. Inf. Theory 57, 2474–2487 (2011). arXiv:0912.1286 [quant-ph]
Mosonyi, M., Ogawa, T.: Strong converse exponent for classical-quantum channel coding. Commun. Math. Phys. 355(1), 373–426 (2017)
Mosonyi, M., Ogawa, T.: Quantum hypothesis testing and the operational interpretation of the quantum Rényi relative entropies. Commun. Math. Phys. 334, 1617–1648 (2015). arXiv:1309.3228 [quant-ph]
Müller-Hermes, A., Reeb, D.: Monotonicity of the quantum relative entropy under positive maps. Annales Henri Poincaré 18, 1777–1788 (2017). arXiv:1512.06117
Müller Lennert, M., Dupuis, F., Szehr, O., Fehr, S., Tomamichel, M.: On quantum Rényi entropies: a new generalization and some properties. J. Math. Phys. 54, 122203 (2013). arXiv:1306.3142 [quant-ph]
Ohya, M., Petz, D.: Quantum Entropy and Its Use. Springer, Heidelberg (1993)
Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2002)
Petz, D.: Quasi-entropies for finite quantum systems. Rep. Math. Phys. 23, 57–65 (1984)
Petz, D.: Quasi-entropies for states of a von Neumann algebra. Publ. RIMS Kyoto Univ. 21, 787–800 (1985)
Petz, D.: Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Commun. Math. Phys. 105, 123–131 (1986)
Petz, D.: Sufficiency of channels over von Neumann algebras. Quart. J. Math. Oxf. 39, 97–108 (1988)
Pisier, G., Xu, Q.: Non-commutative \(L_p\)-spaces. Handb. Geom. Banach Spaces 2, 1459–1517 (2003)
Rényi, A.: On measures of information and entropy. In: Proceedings of the Symposium on Mathematical Statistics and Probability, pp. 547–561. University of California Press (1961)
Stratila, S., Zhidó, L.: Lectures on von Neumann Algebras. Editura Academiei, Bucharest (1979)
Takesaki, M.: Theory of Operator Algebras II. Springer, Berlin (2003)
Terp, M.: \(L_p\) spaces associated with von Neumann algebras. Notes, Copenhagen University (1981)
Terp, M.: Interpolation spaces between a von Neumann algebra and its predual. J. Oper. Theory 8, 327–360 (1982)
Trunov, N.V.: A noncommutative analogue of the space \(L_p\). Izvestiya VUZ Matematika 23, 69–77 (1979)
Uhlmann, A.: Relative entropy and Wigner–Yanase–Dyson–Lieb concavity in an interpolation theory. Commun. Math. Phys. 54, 21–32 (1977)
Wilde, M.M., Winter, A., Yang, D.: Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy. Commun. Math. Phys. 331, 593–622 (2014). arXiv:1306.1586 [quant-ph]
Zolotarev, A.A.: \(L_p\)-spaces with respect to a state on a von Neumann algebra, and interpolation. Izvestiya VUZ Matematika 26, 36–43 (1982)
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Communicated by Claude Alain Pillet.
Research was supported by the Grants VEGA 2/0069/16 and APVV-16-0073.
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Jenčová, A. Rényi Relative Entropies and Noncommutative \(L_p\)-Spaces. Ann. Henri Poincaré 19, 2513–2542 (2018). https://doi.org/10.1007/s00023-018-0683-5
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DOI: https://doi.org/10.1007/s00023-018-0683-5