A Holevo-Type Bound for a Hilbert Schmidt Distance Measure

DOI: 10.4236/jqis.2015.54015   PDF   HTML   XML   3,654 Downloads   4,144 Views   Citations

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.

Share and Cite:

Tamir, B. and Cohen, E. (2015) A Holevo-Type Bound for a Hilbert Schmidt Distance Measure. Journal of Quantum Information Science, 5, 127-133. doi: 10.4236/jqis.2015.54015.

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

  
comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.