Non-signaling Assisted Capacity of a Classical Channel with Causal CSIT

TL;DR

The paper identifies the NS-assisted capacity of a classical channel with causal CSIT as $C^{NS,ca}=\max_{P_{X|S}} I(X;Y|S)$, showing it matches both the NS-assisted non-causal CSIT capacity and the classical/NS capacity when CSIR is available. The achievability uses a novel NS-assisted coding scheme that enforces a fixed-state-type transformation and employs an authentication mechanism to enable reliable decoding, yielding a rate arbitrarily close to $I(X;Y|S)$. Additionally, the work proves that CSIR can strictly improve finite-length performance under NS assistance for causal CSIT, via a $Z_0/Z_1$ channel counterexample, indicating the teleportation intuition from the non-causal setting does not fully carry over to the causal case. These results provide a clean capacity characterization under NS resources and illuminate finite-blocklength gains in NS-assisted communication with causal CSIT. The combination of a closed-form capacity expression and an explicit NS-based coding construction advances understanding of non-signaling resources in classical information theory and their relation to CSIT causality and CSIR access.

Abstract

The non-signaling (NS) assisted capacity of a classical channel with causal channel state information at the transmitter (CSIT) is shown to be $C^{NS,ca}=\max_{P_{X|S}}I(X;Y\mid S)$, where $X, Y, S$ correspond to the input, output and state of the channel. Remarkably, this is the same as the capacity of the channel in the NS-assisted non-causal CSIT setting, $C^{NS,nc}=\max_{P_{X|S}}I(X;Y\mid S)$, which was previously established, and also matches the (either classical or with NS assistance) capacity of the channel where the state is available not only (either causally or non-causally) to the transmitter but also to the receiver. While the capacity remains unchanged, the optimal probability of error for fixed message size and blocklength, in the NS-assisted causal CSIT setting can be further improved if channel state is made available to the receiver. This is in contrast to corresponding NS-assisted non-causal CSIT setting where it was previously noted that the optimal probability of error cannot be further improved by providing the state to the receiver.

Figures (4)

• Figure 1: NS-assisted coding scheme with causal CSIT
• Figure 2: The $Z_0/Z_1$ channel acts as the channel $Z_0$ when the state is $s=0$, and as the channel $Z_1$ when the state is $s=1$. The state is assumed equally likely to be $0$ or $1$.
• Figure 3: Running instances of Algorithm \ref{['alg:type_mapping']}. $\mathcal{A}= \{0,1\}$.
• Figure 4: Illustration of the scheme construction. We assume the channel state alphabet $\mathcal{S} = \{1,2,\cdots, |\mathcal{S}|\}$. $\Lambda$ is an independent classical random variable internally generated by the scheme with $\Pr(\Lambda =1) = \lambda$ and $\Pr(\lambda =0) = 1-\lambda$.

Theorems & Definitions (10)

• Remark 1
• Remark 2
• Remark 3
• Theorem 1
• Definition 1: Channel with CSIR
• Remark 4: CSIR
• Definition 2: $Z_0/Z_1$ channel
• Theorem 2
• Remark 5
• Remark 6