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arXiv:2304.14219 · IEEE Transactions on Information Theory

The Mutual Information in the Vicinity of Capacity-Achieving Input Distributions

Barış Nakiboğlu, Hao–Chung Cheng

Published Apr 17, 2025• 0 citations
arXivHTMLPDF

Abstract

The mutual information is bounded from above by a decreasing affine function of the square of the distance between the input distribution and the set of all capacity-achieving input distributions ΠA, on small enough neighborhoods of ΠA, using an identity due to Topsøe and the Pinsker’s inequality, assuming that the input set of the channel is finite and the constraint set A is polyhedral, i.e., can be described by (possibly multiple but) finitely many linear constraints. Counterexamples demonstrating nonexistence of such a quadratic bound are provided for the case of infinitely many linear constraints and the case of infinite input sets. Using Taylor’s theorem with the remainder term, rather than the Pinsker’s inequality and invoking Moreau’s decomposition theorem the exact characterization of the slowest decrease of the mutual information with the distance to ΠAis determined on small neighborhoods of ΠA. Corresponding results for classical-quantum channels are established under separable output Hilbert space assumption for the quadratic bound and under finite-dimensional output Hilbert space assumption for the exact characterization. Implications of these observations for the channel coding problem and applications of the proof techniques to related problems are discussed.

DOI
10.1109/tit.2025.3562098

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The mutual information is bounded from above by a decreasing affine function of the square of the distance between the input distribution and the set of all capacity-achieving input distributions ΠA, on small enough neighborhoods of ΠA, using an identity due to Topsøe and the Pinsker’s inequality, assuming that the input set of the channel is finite and the constraint set A is polyhedral, i.e., can be described by (possibly multiple but) finitely many linear constraints. Counterexamples demonstrating nonexistence of such a quadratic bound are provided for the case of infinitely many linear constraints and the case of infinite input sets. Using Taylor’s theorem with the remainder term, rather than the Pinsker’s inequality and invoking Moreau’s decomposition theorem the exact characterization of the slowest decrease of the mutual information with the distance to ΠAis determined on small neighborhoods of ΠA. Corresponding results for classical-quantum channels are established under separable output Hilbert space assumption for the quadratic bound and under finite-dimensional output Hilbert space assumption for the exact characterization. Implications of these observations for the channel coding problem and applications of the proof techniques to related problems are discussed.
One of the main recent achievements of quantum information theory [1] is the proof of the equivalence of several (sub-, super-) additivity conjectures for finite-dimensional quantum channels and systems [2] (see the survey in [3]). The main purpose of this paper is to give a generalization of this result to the infinite-dimensional case. Let H and H ′ be separable Hilbert spaces. A quantum channel is a linear completely positive trace-preserving map Φ: T(H ) 7→ T(H ′), where T(H ) is the ideal of all trace class operators on H . In particular, Φ generates an affine map of the convex set S(H ) of states (that is, density operators) on the space H to the set S(H ′) of states on the space H ′ [1]. Important characteristics of a quantum channel are the output entropy HΦ(ρ) = H(Φ(ρ)), a lower-semicontinuous concave function on the input state space with values in [0,+∞], and its convex closure (see [4]), denoted by b HΦ(ρ) and called the b H-function of the channel Φ. It was shown in [5] that the convex closure of the output entropy of an arbitrary quantum channel Φ is given by the expression b HΦ(ρ) = inf μ Z
One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity:
Abstract We obtain continuity bounds for basic information characteristics of quantum channels depending on their input dimension (if it is finite) and on the input energy bound (if the input dimension is infinite). We pay special attention to the case in which a multimode quantum oscillator is an input system. First, we prove continuity bounds for the output conditional mutual information for a single channel and for copies of a channel. Then we obtain estimates for the variation of the output Holevo quantity with respect to simultaneous variations of a channel and of an input ensemble. As a result, tight and close-to-tight continuity bounds for basic capacities of quantum channels, which depend on the input dimension, are obtained. They complement the Leung–Smith continuity bounds, which depend on the output dimension. Finally, we obtain tight and close-to-tight continuity bounds for basic capacities of infinite-dimensional energy-constrained channels with respect to the energy-constrained Bures distance generating the strong convergence of quantum channels.
Following initial work by JaJa, and Ahlswede and Cai, and inspired by a recent renewed surge in interest in deterministic identification (DI) via noisy channels, we consider the problem in its generality for memoryless channels with finite output, but arbitrary input alphabets. Such a channel is essentially given by (the closure of) the subset of its output distributions in the probability simplex. Our main findings are that the maximum number of messages thus identifiable scales super-exponentially as2Rnlog⁡n2^{Rn\log n}2Rnlognwith the block lengthnnn, and that the optimal rateRRRis upper and lower bounded in terms of the covering (aka Minkowski, or Kolmogorov, or entropy) dimensiondddof a certain algebraic transformation of the output set:14d≤R≤12d\frac{1}{4}d\leq R\leq\frac{1}{2}d41​d≤R≤21​d, Along the way, we present a Hypothesis Testing Lemma that shows it is sufficient to ensure pairwise reliable distinguishability of the output distributions to construct a DI code. Although we do not know the exact capacity formula, we can conclude that the DI capacity exhibits super-activation: there exist channels whose capacity is zero, but whose product has positive capacity. These results are then generalised to classical-quantum channels with finite-dimensional output quantum system (but arbitrary input alphabet), and in particular to quantum channels on finite-dimensional quantum systems under the constraint that the identification code can only use tensor product inputs.
Following initial work by JaJa, Ahlswede and Cai, and inspired by a recent renewed surge in interest in deterministic identification (DI) via noisy channels, we consider the problem in its generality for memoryless channels with finite output, but arbitrary input alphabets. Such a channel is essentially given by its output distributions as a subset in the probability simplex. Our main findings are that the maximum length of messages thus identifiable scales superlinearly asR nlog⁡nR\,n\log nRnlognwith the block length n, and that the optimal rate R is bounded in terms of the covering (aka Minkowski, or Kolmogorov, or entropy) dimension d of a certain algebraic transformation of the output set:14d≤R≤12d\frac {1}{4} d \leq R \leq \frac {1}{2} d41​d≤R≤21​d. Remarkably, both the lower and upper Minkowski dimensions play a role in this result. Along the way, we present a Hypothesis Testing Lemma showing that it is sufficient to ensure pairwise reliable distinguishability of the output distributions to construct a DI code. Although we do not know the exact capacity formula, we can conclude that the DI capacity exhibits superactivation: there exist channels whose capacities individually are zero, but whose product has positive capacity. We also generalise these results to classical-quantum channels with finite-dimensional output quantum system, in particular to quantum channels on finite-dimensional quantum systems under the constraint that the identification code can only use tensor product inputs.
The mismatched communication channel with an infinite-dimensional real separable Hilbert space as input and output spaces is considered. To study communication in the presence of jamming, a two person zero sum game with mutual information of input source and output source as the payoff function can be formulated. The coder’s goal is to make mutual information as large as possible, and the jammer’s goal is to make mutual information as small as possible. The optimal mutual information under appropriate constraints of coders and jammers is obtained.