Communications in Mathematical Physics

, Volume 306, Issue 1, pp 165–186 | Cite as

Min- and Max-Entropy in Infinite Dimensions

  • Fabian Furrer
  • Johan ÅbergEmail author
  • Renato Renner
Open Access


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.


Separable Hilbert Space Projected State Strong Operator Topology Trace Class Operator Positive Operator Value Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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).

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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.


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© The Author(s) 2011

Open AccessThis is an open access article 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.

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsLeibniz Universität HannoverHannoverGermany
  2. 2.Institute for Theoretical PhysicsETH ZurichZurichSwitzerland

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