Concavity of the Auxiliary Function for Classical-Quantum Channels

The auxiliary function of a classical channel appears in two fundamental quantities, the random coding exponent and the sphere-packing exponent, which yield upper and lower bounds on the error probability of decoding, respectively. A crucial property of the auxiliary function is its concavity, and this property consequently leads to several important results in finite blocklength analysis. In this paper, we prove that the auxiliary function of a classical-quantum channel also enjoys the same concavity property, extending an earlier partial result to its full generality. We also prove that the auxiliary function satisfies the data-processing inequality, among various other important properties. Furthermore, we show that the concavity property of the auxiliary function enables a geometric interpretation of the random coding exponent and the sphere-packing exponent of a classical-quantum channel. The key component in our proof is an important result from the theory of matrix geometric means.

[1]  Y. Lim,et al.  Metric convexity of symmetric cones , 2007 .

[2]  F. Hiai,et al.  Introduction to Matrix Analysis and Applications , 2014 .

[3]  Jaspal Singh Aujla,et al.  Some inequalities for unitarily invariant norm , 2012 .

[4]  N. Sloane,et al.  Lower Bounds to Error Probability for Coding on Discrete Memoryless Channels. I , 1993 .

[5]  H. Araki.  On an inequality of Lieb and Thirring , 1990 .

[6]  F. Hiai.  Concavity of certain matrix trace and norm functions. II , 2012, 1507.00853.

[7]  Alexander S. Holevo,et al.  Reliability function of general classical-Quantum channel , 1999, IEEE Trans. Inf. Theory.

[8]  Elliott H. Lieb,et al.  Monotonicity of a relative Rényi entropy , 2013, ArXiv.

[9]  Alain Joye,et al.  Spectral Transition for Random Quantum Walks on Trees , 2012, 1212.6078.

[10]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[11]  R. Gallager.  Information Theory and Reliable Communication , 1968 .

[12]  Horacio Porta,et al.  Convexity of the geodesic distance on spaces of positive operators , 1994 .

[13]  Aaron B. Wagner,et al.  Moderate Deviations in Channel Coding , 2012, IEEE Transactions on Information Theory.

[14]  A. Holevo,et al.  On Reliability Function of Quantum Communication Channel , 1997 .

[15]  Tomohiro Ogawa,et al.  Strong converse to the quantum channel coding theorem , 1999, IEEE Trans. Inf. Theory.

[16]  A. Holevo.  Bounds for the quantity of information transmitted by a quantum communication channel , 1973 .

[17]  Elwyn R. Berlekamp,et al.  Lower Bounds to Error Probability for Coding on Discrete Memoryless Channels. II , 1967, Inf. Control..

[18]  E. Carlen.  TRACE INEQUALITIES AND QUANTUM ENTROPY: An introductory course , 2009 .

[19]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[20]  Marco Dalai,et al.  Lower Bounds on the Probability of Error for Classical and Classical-Quantum Channels , 2012, IEEE Transactions on Information Theory.

[21]  T. Andô,et al.  Means of positive linear operators , 1980 .

[22]  R. Subramanian,et al.  Inequalities between means of positive operators , 1978, Mathematical Proceedings of the Cambridge Philosophical Society.

[23]  Robert G. Gallager,et al.  A simple derivation of the coding theorem and some applications , 1965, IEEE Trans. Inf. Theory.

[24]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[25]  R. Bhatia.  Positive Definite Matrices , 2007 .

[26]  Jun Ichi Fujii,et al.  Concavity of the auxiliary function appearing in quantum reliability function , 2005, IEEE Transactions on Information Theory.

[27]  Fumio Hiai,et al.  Log-majorizations and norm inequalities for exponential operators , 1997 .

[28]  R. Bhatia,et al.  Norm inequalities related to the matrix geometric mean , 2012, 1502.04497.