A Holevo-Type Bound for a Hilbert Schmidt Distance Measure

Boaz Tamir^1*, Eliahu Cohen^2,3

¹Faculty of Interdisciplinary Studies, Bar-Ilan University, Ramat-Gan, Israel.
²H.H. Wills Physics Laboratory, University of Bristol, Bristol, UK.
³School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, Israel.

DOI: 10.4236/jqis.2015.54015                

Abstract

We prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kullback-Leibler divergence. Suppose Alice is sending classical information to Bob by using a quantum channel while Bob is performing some projective measurements. We bound the classical mutual information in terms of the Hilbert-Schmidt norm by its quantum Hilbert-Schmidt counterpart. This constitutes a Holevo-type upper bound on the classical information transmission rate via a quantum channel. The resulting inequality is rather natural and intuitive relating classical and quantum expressions using the same measure.

Keywords

Holevo Bound, Hilbert-Schmidt Norm, Entanglement Measures

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Holevo. A.S. (1973) Bounds for the Quantity of Information Transmitted by a Quantum Communication Channel. Problemy Peredachi Informatsii, 9, 3-11.
[2]	Nielsen, M.A. and Chuang, I.L. (2010) Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, UK. http://dx.doi.org/10.1017/CBO9780511976667
[3]	Cerf, N.J. and Adami, C. (1996) Accessible Information in Quantum Measurement. http://arxiv.org/abs/quant-ph/9611032
[4]	Winter, A. (1999) Coding Theorem and Strong Converse for Quantum Channels. IEEE Transactions on Information Theory, 45, 2481-2485. http://dx.doi.org/10.1109/18.796385
[5]	Buzek, V. and Hillery, M. (1996) Quantum Copying: Beyond the No-Cloning Theorem. Physical Review A, 54, 1844-1852. http://dx.doi.org/10.1103/PhysRevA.54.1844
[6]	Vedral, V. and Plenio, M.B. (1998) Entanglement Measures and Purification Procedures. Physical Review A, 57, 1619-1633. http://dx.doi.org/10.1103/PhysRevA.57.1619
[7]	Perez-Garcia, D., Wolf, M.M., Petz, D. and Ruskai, M.B. (2006) Contractivity of Positive and Trace-Preserving Maps under Lp Norms. Journal of Mathematical Physics, 47, Article ID: 083506. http://dx.doi.org/10.1063/1.2218675
[8]	Dllerman, E. (2013) Information as Distinction: New Foundation for Information Theory. http://arxiv.org/abs/1301.5607
[9]	Wolf, M.M., Verstraete, F., Hasting, M.B. and Cirac, J.I. (2008) Area Law in Quantum Systems: Mutual Information and Correlations. Physical Review Letters, 100, Article ID: 070502. http://dx.doi.org/10.1103/PhysRevLett.100.070502
[10]	Witte, C. and Trucks, M. (1999) A New Entanglement Measure Induced by the Hilbert-Schmidt Norm. Physics Letters A, 257, 14-20. http://dx.doi.org/10.1016/S0375-9601(99)00279-0
[11]	Ozawa, M. (2000) Entanglement Measures and the Hilbert-Schmidt Distance. Physics Letters A, 268, 158-160. http://dx.doi.org/10.1016/S0375-9601(00)00171-7
[12]	Tsallis, C. (1988) Possible Generalization of Boltzmann-Gibbs Statistics. Journal of Statistical Physics, 52, 479-487. http://dx.doi.org/10.1007/BF01016429
[13]	Peters, N.A., Wei, T.C. and Kwiat, P.G. (2004) Mixed State Sensitivity of Several Quantum Information Benchmarks. Physical Review A, 70, Article ID: 052309. http://dx.doi.org/10.1103/PhysRevA.70.052309
[14]	Paulsen, V. (2002) Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge, UK.
[15]	Matsumoto, K., Shimono, T. and Winter, A. (2004) Remarks on Additivity of the Holevo Channel Capacity and of the Entanglement of Formation. Communications in Mathematical Physics, 246, 427-442. http://dx.doi.org/10.1007/s00220-003-0919-0
[16]	Rastegin, A.E. (2010) Some General Properties of Unified Entropies. Journal of Statistical Physics, 143, 1120-1135. http://dx.doi.org/10.1007/s10955-011-0231-x
[17]	Tamir, B. and Cohen, E. (2014) Logical Entropy for Quantum States. http://arxiv.org/abs/1412.0616