# Classical Information Storage in an *n*-Level Quantum System

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## Abstract

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.

## Keywords

Mutual Information Convex Hull Density Matrice Channel Matrix Stochastic Matrix## Preview

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## References

- 1.Bapat R.B.: Mixed discriminants of positive semidefinite matrices. Linear Algebra Appl.
**126**, 107–124 (1989)zbMATHMathSciNetCrossRefGoogle Scholar - 2.Bengtsson I., Ericsson Å.: Mutually unbiased bases and the complementarity polytope. Open Syst. Inf. Dyn.
**12**, 107–120 (2005)zbMATHMathSciNetCrossRefGoogle Scholar - 3.Bennett C.H., Wiesner S.J.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett.
**69**, 2881–2884 (1992)zbMATHMathSciNetCrossRefADSGoogle Scholar - 4.Buhrman, H., Cleve, R., Watrous, J., de Wolf, R.: Quantum fingerprinting. Phys. Rev. Lett.
**87**, (2001)Google Scholar - 5.Fawzi, H., Gouveia, J., Parrilo, P.A., Robinson, R.Z., Thomas, R.R.: Positive semidefinite rank. Math. Program. (to appear). arXiv:1407.4095
- 6.Frenkel P.E., Weiner M.: On vector configurations that can be realized in the cone of positive matrices. Linear Alg. Appl.
**459**, 465–474 (2014)zbMATHMathSciNetCrossRefGoogle Scholar - 7.Holevo A.S.: Bounds for the quantity of information transmitted by a quantum communication channel. Problems Inform. Transm.
**9**, 177–183 (1973)MathSciNetGoogle Scholar - 8.Holevo A.S.: Probabilistic and Statistical Aspects of Quantum Theory. Springer, Berlin (2011)zbMATHCrossRefGoogle Scholar
- 9.Klobuchar, A.J.: Classical capacities of a qubit. Notes, BSM fall semester (2010). http://www.renyi.hu/~mweiner/qubit
- 10.Laurent, M., Piovesan, T.: Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone. arXiv:1312.6643v4
- 11.Lovász L., Plummer M.D.: Matching Theory. North-Holland, Amsterdam (1986)zbMATHGoogle Scholar
- 12.Ohya, M., Petz, D.: Quantum Entropy and Its Use, 2nd edn. Springer, Berlin (1993)Google Scholar
- 13.Sasaki, M., Barnett, S.M., Jozsa, R., Osaki, M., Hirota, O.: Phys. Rev. A
**59**, 3325 (1999)Google Scholar - 14.Weiner M.: A gap for the maximum number of mutually unbiased bases. Proc. Am. Math. Soc.
**141**, 1963–1969 (2013)zbMATHMathSciNetCrossRefGoogle Scholar - 15.Wilde, M.M.: Quantum Information Theory. Cambridge University Press, Cambridge (2013)zbMATHCrossRefGoogle Scholar