Abstract
We consider an extension of the conditional min- and max-entropies to infinite-dimensional separable Hilbert spaces. We show that these satisfy characterizing properties known from the finite-dimensional case, and retain information-theoretic operational interpretations, e.g., the min-entropy as maximum achievable quantum correlation, and the max-entropy as decoupling accuracy. We furthermore generalize the smoothed versions of these entropies and prove an infinite-dimensional quantum asymptotic equipartition property. To facilitate these generalizations we show that the min- and max-entropy can be expressed in terms of convergent sequences of finite-dimensional min- and max-entropies, which provides a convenient technique to extend proofs from the finite to the infinite-dimensional setting.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Shannon, C.E.: A Mathematical Theory of Communication. Bell System Technical Journal 27, 379–423 and 623–656 (1948)
Rényi, A.: On measures of entropy and information. Proc. of the 4th Berkley Symp. on Math. Statistics and Prob. 1, Berkeley, CA: Univ. of Calif. Press, 1961, pp. 547–561
Barnum H., Nielsen M.A., Schumacher B.: Information transmission through a noisy quantum channel. Phys. Rev. A 57, 4153–4175 (1998)
Schumacher B.: Quantum coding. Phys. Rev. A 51, 2738–2747 (1995)
Renner, R.: Security of Quantum Key Distribution. Ph.D. thesis, Swiss Fed. Inst. of Technology, Zurich, 2005, Available at http://arXiv.org/abs/quant-ph/0512258v2, 2006
Renner, R., Wolf, S.: Smooth Renyi entropy and applications Proc. 2004 IEEE International Symposium on Information Theory, Piscataway, NJ: IEEE, 2004, p. 233
Renes, J. M., Renner, R.: One-Shot Classical Data Compression with Quantum Side Information and the Distillation of Common Randomness or Secret Keys. http://arXiv.org/abs/1008.0452v2 [quant-ph], 2010
Renner, R., Wolf, S., Wullschleger, J.: The Single-Serving Channel Capacity Proc. 2006 IEEE International Symposium on Information Theory, Piscataway, NJ: IEEE, 2006, pp. 1424–1427
Tomamichel M., Colbeck R., Renner R.: A Fully Quantum Asymptotic Equipartition Property. IEEE Trans. Inf. Th. 55, 5840–5847 (2009)
Berta, M., Christandl, M., Renner, R.: A Conceptually Simple Proof of the Quantum Reverse Shannon Theorem. http://arXiv.org/abs/0912.3805v1 [quant-ph], 2009
Berta M., Christandl M., Colbeck R., Renes J.M., Renner R.: The uncertainty principle in the presence of quantum memory. Nature Physics 6, 659–662 (2010)
Datta N., Renner R.: Smooth Entropies and the Quantum Information Spectrum. IEEE Trans. Inf. Theor. 55, 2807–2815 (2009)
Han T.S.: Information-Spectrum Methods in Information Theory. Springer-Verlag, New York (2002)
Han T.S., Verdu S.: Approximation theory of output statistics. IEEE Trans. Inform. Theory 39, 752–772 (1993)
Datta N.: Min- and Max-Relative Entropies and a New Entanglement Monotone. IEEE Trans. Inf. Theor. 55, 2816–2826 (2009)
Brandão F.G.S.L., Datta N.: One-shot rates for entanglement manipulation under non-entangling maps. IEEE Trans. Inf. Theor. 57, 1754 (2011)
Buscemi F., Datta N.: Entanglement Cost in Practical Scenarios. Phys. Rev. Lett 106, 130503 (2011)
Mosonyi M., Datta N.: Generalized relative entropies and the capacity of classical-quantum channels. J. Math. Phys. 50, 072104 (2009)
Dahlsten O.C.O., Renner R., Rieper E., Vedral V.: Inadequacy of von Neumann entropy for characterizing extractable work. New J. Phys. 13, 053015 (2011)
del Rio L., Åberg J., Renner R., Dahlsten O., Vedral V.: The thermodynamic meaning of negative entropy. Nature 474, 61–63 (2011)
Scarani V., Bechmann-Pasquinucci H., Cerf N.J., Dušek M., Lütkenhaus N., Peev M.: The security of practical quantum key distribution. Rev. Mod. Phys. 81, 1301–1350 (2009)
Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer-Verlag, New York (1979)
König R., Renner R., Schaffner C.: The Operational Meaning of Min- and Max-Entropy. IEEE Trans. Inf. Th. 55, 4337–4347 (2009)
Tomamichel M., Colbeck R., Renner R.: Duality Between Smooth Min- and Max-Entropies. IEEE Trans. Inf. Th. 56, 4674–4681 (2010)
Lieb E.H., Ruskai M.B.: A Fundamental Property of Quantum-Mechanical Entropy. Phys. Rev. Lett. 30, 434–436 (1973)
Lieb E.H.: Convex trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math. 11, 267–288 (1973)
Lieb E.H., Ruskai M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14, 1938–1941 (1973)
Owari M., Braunstein S.L., Nemoto K., Murao M.: Epsilon-convertibility of entangled states and extension of Schmidt rank in infinite-dimensional systems. Quant. Inf. and Comp. 8, 30–52 (2008)
Kraus, K.: Lecture Notes in Physics 190, States, Effects, and Operations. Berlin Heidelberg: Springer-Verlag, 1983
Nielsen M.L., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Uhlmann A.: The transition probability in the state space of a *-algebra. Rep. Math. Phys. 9, 273–279 (1976)
Holenstein H., Renner R.: On the Randomness of Independent Experiments. IEEE Trans. Inf. Theor. 57(4), 1865–1871 (2011)
Kuznetsova A.A.: Quantum conditional entropy for infinite-dimensional systems. Theory Probab. Appl. 55, 782–790 (2010)
Klein O.: Zur quantenmechanischen Begründung des zweiten Hauptsatzes der Wärmelehre. Z. f. Phys. A 72, 767–775 (1931)
Lindblad G.: Entropy, Information and Quantum Measurements. Commun. Math. Phys. 33, 305–322 (1973)
Lindblad G.: Expectations and Entropy Inequalities for Finite Quantum Systems. Commun. Math. Phys. 39, 111–119 (1974)
Holevo A.S., Shirokov M.E.: Mutual and Coherent Information for Infinite-Dimensional Quantum Channels. Probl. Inf. Transm. 46, 201–217 (2010)
Cover, T.M., Thomas, J.A.: Elements of Information Theory. 2nd ed. New York: Wiley, 2006
Alicki R., Fannes M.: Continuity of quantum conditional information. J. Phys. A 37, L55–L57 (2004)
Horodecki M., Oppenheim J., Winter A.: Partial quantum information. Nature 436, 673–676 (2005)
Berta, M.: Single-shot Quantum State Merging. Diploma thesis, ETH Zurich, February 2008, available at http://arXiv.org/abs/0912.4495v1 [quant-ph], 2009
Grümm H.R.: Two theorems about \({\mathcal{C}_{p}}\). Rep. Math. Phys. 4, 211–215 (1973)
Simon, B.: Trace Ideals and Their Applications. 2nd ed. Providence, RI: Amer. Math. Soc., 2005
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. I: Functional Analysis. New York: Academic Press, 1978
Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups. Providence, RI: Amer. Math. Soc., 1957
Acknowledgements
We thank Roger Colbeck and Marco Tomamichel for helpful comments and discussions, and an anonymous referee for very valuable suggestions. Fabian Furrer acknowledges support from the Graduiertenkolleg 1463 of the Leibniz University Hannover. We furthermore acknowledge support from the Swiss National Science Foundation (grant No. 200021-119868).
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M.B. Ruskai
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Furrer, F., Åberg, J. & Renner, R. Min- and Max-Entropy in Infinite Dimensions. Commun. Math. Phys. 306, 165–186 (2011). https://doi.org/10.1007/s00220-011-1282-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-011-1282-1