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

TL;DR

This paper establishes that the non-signaling assisted capacity of a classical channel with causal CSIT equals that with non-causal CSIT, and shows potential error probability improvements when the receiver also has access to the channel state.

Contribution

It proves the capacity equivalence between causal and non-causal CSIT settings under non-signaling assistance and explores error probability improvements with receiver state knowledge.

Findings

Capacity with causal CSIT matches non-causal CSIT under NS assistance.

Providing state to the receiver can improve error probabilities in causal CSIT.

No capacity difference between causal and non-causal CSIT with NS assistance.

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.