Communications in Mathematical Physics

, Volume 340, Issue 2, pp 563–574 | Cite as

Classical Information Storage in an n-Level Quantum System

  • Péter E. Frenkel
  • Mihály WeinerEmail author


A game is played by a team of two—say Alice and Bob—in which the value of a random variable x is revealed to Alice only, who cannot freely communicate with Bob. Instead, she is given a quantum n-level system, respectively a classical n-state system, which she can put in possession of Bob in any state she wishes. We evaluate how successfully they managed to store and recover the value of x by requiring Bob to specify a value z and giving a reward of value f (x,z) to the team. We show that whatever the probability distribution of x and the reward function f are, when using a quantum n-level system, the maximum expected reward obtainable with the best possible team strategy is equal to that obtainable with the use of a classical n-state system. The proof relies on mixed discriminants of positive matrices and—perhaps surprisingly—an application of the Supply–Demand Theorem for bipartite graphs. As a corollary, we get an infinite set of new, dimension dependent inequalities regarding positive operator valued measures and density operators on complex n-space. As a further corollary, we see that the greatest value, with respect to a given distribution of x, of the mutual information I (x; z) that is obtainable using an n-level quantum system equals the analogous maximum for a classical n-state system.


Mutual Information Convex Hull Density Matrice Channel Matrix Stochastic Matrix 
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.


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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Algebra and Number TheoryEötvös Loránd UniversityBudapestHungary
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  3. 3.Department of AnalysisBudapest University of Technology and EconomicsBudapestHungary

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