Generalized relative entropies and the capacity of classical-quantum channels

We provide lower and upper bounds on the information transmission capacity of one single use of a classical-quantum channel. The lower bound is expressed in terms of the Hoeffding capacity, which we define similarly to the Holevo capacity but replacing the relative entropy with the Hoeffding distance. Similarly, our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max-relative entropy in the definition of the divergence radius of a channel.

[1]  E. Lieb Convex trace functions and the Wigner-Yanase-Dyson conjecture , 1973 .

[2]  Masahito Hayashi Error exponent in asymmetric quantum hypothesis testing and its application to classical-quantum channel coding , 2006, quant-ph/0611013.

[3]  Nilanjana Datta,et al.  One-shot quantum capacities of quantum channels , 2009 .

[4]  D. Petz Quasi-entropies for finite quantum systems , 1986 .

[5]  D. Petz,et al.  Quantum Entropy and Its Use , 1993 .

[6]  Michael D. Westmoreland,et al.  Sending classical information via noisy quantum channels , 1997 .

[7]  Thomas M. Cover,et al.  Elements of Information Theory: Cover/Elements of Information Theory, Second Edition , 2005 .

[8]  Tomohiro Ogawa,et al.  Strong converse and Stein's lemma in quantum hypothesis testing , 2000, IEEE Trans. Inf. Theory.

[9]  F. Hiai,et al.  The proper formula for relative entropy and its asymptotics in quantum probability , 1991 .

[10]  Alexander S. Holevo,et al.  The Capacity of the Quantum Channel with General Signal States , 1996, IEEE Trans. Inf. Theory.

[11]  M. Nussbaum,et al.  A lower bound of Chernoff type for symmetric quantum hypothesis testing , 2006 .

[12]  M. Nussbaum,et al.  Asymptotic Error Rates in Quantum Hypothesis Testing , 2007, Communications in Mathematical Physics.

[13]  M. Nussbaum,et al.  THE CHERNOFF LOWER BOUND FOR SYMMETRIC QUANTUM HYPOTHESIS TESTING , 2006, quant-ph/0607216.

[14]  F. Hiai,et al.  Asymptotic distinguishability measures for shift-invariant quasifree states of fermionic lattice systems , 2008 .

[15]  Milan Mosonyi Hypothesis testing for Gaussian states on bosonic lattices , 2009 .

[16]  I. Csiszár $I$-Divergence Geometry of Probability Distributions and Minimization Problems , 1975 .

[17]  Roger Colbeck,et al.  Simple channel coding bounds , 2009, 2009 IEEE International Symposium on Information Theory.

[18]  F. Hiai,et al.  Error exponents in hypothesis testing for correlated states on a spin chain , 2007, 0707.2020.

[19]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[20]  K. Audenaert,et al.  Discriminating States: the quantum Chernoff bound. , 2006, Physical review letters.

[21]  I. Bjelakovic,et al.  An Ergodic Theorem for the Quantum Relative Entropy , 2003, quant-ph/0306094.

[22]  F. Hiai,et al.  Entropy Densities for Algebraic States , 1994 .

[23]  Nilanjana Datta,et al.  The Quantum Capacity of Channels With Arbitrarily Correlated Noise , 2009, IEEE Transactions on Information Theory.

[24]  Masahito Hayashi,et al.  General formulas for capacity of classical-quantum channels , 2003, IEEE Transactions on Information Theory.

[25]  Renato Renner,et al.  The Single-Serving Channel Capacity , 2006, 2006 IEEE International Symposium on Information Theory.

[26]  Robert S. Kennedy,et al.  Optimum testing of multiple hypotheses in quantum detection theory , 1975, IEEE Trans. Inf. Theory.

[27]  Masanori Ohya,et al.  On capacities of quantum channels , 2008 .

[28]  Tomohiro Ogawa,et al.  Making Good Codes for Classical-Quantum Channel Coding via Quantum Hypothesis Testing , 2007, IEEE Transactions on Information Theory.

[29]  Nilanjana Datta,et al.  Min- and Max-Relative Entropies and a New Entanglement Monotone , 2008, IEEE Transactions on Information Theory.

[30]  F. Hiai,et al.  Large deviations and Chernoff bound for certain correlated states on a spin chain , 2007, 0706.2141.

[31]  Robert König,et al.  The Operational Meaning of Min- and Max-Entropy , 2008, IEEE Transactions on Information Theory.

[32]  H. Nagaoka The Converse Part of The Theorem for Quantum Hoeffding Bound , 2006, quant-ph/0611289.

[33]  A. Uhlmann Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory , 1977 .