The Classical-Quantum Channel with Random State Parameters Known to the Sender

TL;DR

This paper investigates the capacity of a quantum channel with random state parameters known to the sender, distinguishing between causal and non-causal information, and provides single-letter and regularized capacity formulas.

Contribution

It offers the first single-letter capacity formula for causal state information and introduces two regularized formulas for non-causal information, highlighting the complexity change.

Findings

Single-letter capacity formula for causal case

Regularized capacity expressions for non-causal case

Complexity jump from trivial to difficult when switching from causal to non-causal information

Abstract

We study an analog of the well-known Gel'fand Pinsker Channel which uses quantum states for the transmission of the data. We consider the case where both the sender's inputs to the channel and the channel states are to be taken from a finite set (cq-channel with state information at the sender). We distinguish between causal and non-causal channel state information at the sender. The receiver remains ignorant, throughout. We give a single-letter description of the capacity in the first case. In the second case we present two different regularized expressions for the capacity. It is an astonishing and unexpected result of our work that a simple change from causal to non-causal channel state information at the encoder causes the complexity of a numerical computation of the capacity formula to change from trivial to seemingly difficult. Still, even the non-single letter formula allows one to draw nontrivial conclusions, for example regarding continuity of the capacity with respect to changes in the system parameters. The direct parts of both coding theorems are based on a special class of POVMs which are derived from orthogonal projections onto certain representations of the symmetric groups. This approach supports a reasoning that is inspired by the classical method of types. In combination with the non-commutative union bound these POVMs yield an elegant method of proof for the direct part of the coding theorem in the first case.