Saddle Point in the Minimax Converse for Channel Coding

A minimax metaconverse has recently been proposed as a simultaneous generalization of a number of classical results and a tool for the nonasymptotic analysis. In this paper, it is shown that the order of optimizing the input and output distributions can be interchanged without affecting the bound. In the course of the proof, a number of auxiliary results of separate interest are obtained. In particular, it is shown that the optimization problem is convex and can be solved in many cases by the symmetry considerations. As a consequence, it is demonstrated that in the latter cases, the (multiletter) input distribution in information-spectrum (Verdú-Han) converse bound can be taken to be a (memoryless) product of single-letter ones. A tight converse for the binary erasure channel is rederived by computing the optimal (nonproduct) output distribution. For discrete memoryless channels, a conjecture of Poor and Verdú regarding the tightness of the information spectrum bound on the error exponents is resolved in the negative. Concept of the channel symmetry group is established and relations with the definitions of symmetry by Gallager and Dobrushin are investigated.National Science Foundation (U.S.) (Center for Science of Information, under Grant CCF-0939370

On the calculation of the minimax-converse of the channel coding problem

A minimax-converse has been suggested for the general channel coding problem by Polyanskiy etal. This converse comes in two flavors. The first flavor is generally used for the analysis of the coding problem with non-vanishing error probability and provides an upper bound on the rate given the error probability. The second flavor fixes the rate and provides a lower bound on the error probability. Both converses are given as a min-max optimization problem of an appropriate binary hypothesis testing problem. The properties of the first converse were studies by Polyanskiy and a saddle point was proved. In this paper we study the properties of the second form and prove that it also admits a saddle point. Moreover, an algorithm for the computation of the saddle point, and hence the bound, is developed. In the DMC case, the algorithm runs in a polynomial time.Comment: Extended version of a submission to ISIT 201

A Minimax Converse for Quantum Channel Coding

We prove a one-shot "minimax" converse bound for quantum channel coding assisted by positive partial transpose channels between sender and receiver. The bound is similar in spirit to the converse by Polyanskiy, Poor, and Verdu [IEEE Trans. Info. Theory 56, 2307-2359 (2010)] for classical channel coding, and also enjoys the saddle point property enabling the order of optimizations to be interchanged. Equivalently, the bound can be formulated as a semidefinite program satisfying strong duality. The convex nature of the bound implies channel symmetries can substantially simplify the optimization, enabling us to explicitly compute the finite blocklength behavior for several simple qubit channels. In particular, we find that finite blocklength converse statements for the classical erasure channel apply to the assisted quantum erasure channel, while bounds for the classical binary symmetric channel apply to both the assisted dephasing and depolarizing channels. This implies that these qubit channels inherit statements regarding the asymptotic limit of large blocklength, such as the strong converse or second-order converse rates, from their classical counterparts. Moreover, for the dephasing channel, the finite blocklength bounds are as tight as those for the classical binary symmetric channel, since coding for classical phase errors yields equivalently-performing unassisted quantum codes.Comment: merged with arXiv:1504.04617 version 1 ; see version

Orthogonal Codes for Robust Low-Cost Communication

Orthogonal coding schemes, known to asymptotically achieve the capacity per unit cost (CPUC) for single-user ergodic memoryless channels with a zero-cost input symbol, are investigated for single-user compound memoryless channels, which exhibit uncertainties in their input-output statistical relationships. A minimax formulation is adopted to attain robustness. First, a class of achievable rates per unit cost (ARPUC) is derived, and its utility is demonstrated through several representative case studies. Second, when the uncertainty set of channel transition statistics satisfies a convexity property, optimization is performed over the class of ARPUC through utilizing results of minimax robustness. The resulting CPUC lower bound indicates the ultimate performance of the orthogonal coding scheme, and coincides with the CPUC under certain restrictive conditions. Finally, still under the convexity property, it is shown that the CPUC can generally be achieved, through utilizing a so-called mixed strategy in which an orthogonal code contains an appropriate composition of different nonzero-cost input symbols.Comment: 2nd revision, accepted for publicatio

Secure Transmission with Multiple Antennas II: The MIMOME Wiretap Channel

The capacity of the Gaussian wiretap channel model is analyzed when there are multiple antennas at the sender, intended receiver and eavesdropper. The associated channel matrices are fixed and known to all the terminals. A computable characterization of the secrecy capacity is established as the saddle point solution to a minimax problem. The converse is based on a Sato-type argument used in other broadcast settings, and the coding theorem is based on Gaussian wiretap codebooks. At high signal-to-noise ratio (SNR), the secrecy capacity is shown to be attained by simultaneously diagonalizing the channel matrices via the generalized singular value decomposition, and independently coding across the resulting parallel channels. The associated capacity is expressed in terms of the corresponding generalized singular values. It is shown that a semi-blind "masked" multi-input multi-output (MIMO) transmission strategy that sends information along directions in which there is gain to the intended receiver, and synthetic noise along directions in which there is not, can be arbitrarily far from capacity in this regime. Necessary and sufficient conditions for the secrecy capacity to be zero are provided, which simplify in the limit of many antennas when the entries of the channel matrices are independent and identically distributed. The resulting scaling laws establish that to prevent secure communication, the eavesdropper needs 3 times as many antennas as the sender and intended receiver have jointly, and that the optimimum division of antennas between sender and intended receiver is in the ratio of 2:1.Comment: To Appear, IEEE Trans. Information Theor

The third-order term in the normal approximation for singular channels

For a singular and symmetric discrete memoryless channel with positive dispersion, the third-order term in the normal approximation is shown to be upper bounded by a constant. This finding completes the characterization of the third-order term for symmetric discrete memoryless channels. The proof method is extended to asymmetric and singular channels with constant composition codes, and its connection to existing results, as well as its limitation in the error exponents regime, are discussed.Comment: Submitted to IEEE Trans. Inform. Theor