Bounding and Estimating the Classical Information Rate of Quantum Channels with Memory

TL;DR

This work considers the scenario of classical communication over a finite-dimensional quantum channel with memory using a separable-state input ensemble and local output measurements, and proposes algorithms for estimating the information rate and extensions of their counterparts for (classical) finite-state-machine channels.

Abstract

We consider the scenario of classical communication over a finite-dimensional quantum channel with memory using a separable-state input ensemble and local output measurements. We propose algorithms for estimating the information rate of such communication setups, along with algorithms for bounding the information rate based on so-called auxiliary channels. Some of the algorithms are extensions of their counterparts for (classical) finite-state-machine channels. Notably, we discuss suitable graphical models for doing the relevant computations. Moreover, the auxiliary channels are learned in a data-driven approach; i.e., only input/output sequences of the true channel are needed, but not the channel model of the true channel.

Figures (19)

• Figure 1: Interpretations of quantum channels with memory.
• Figure 2: Classical communications over quantum channels.
• Figure 3: Classical communication over a quantum channel with memory using a separable ensemble and local measurements.
• Figure 4: Channel with a classical state: closing the top box yields the input process $Q^{(n)}$, closing the bottom box yields the joint channel law $W({\bm{y}}_1^n|{\bm{x}}_1^n)$.
• Figure 5: Representation of $\{W^{y|x}\}_{x,y}$ using an NFG.

Theorems & Definitions (10)

• Example 1: Gilbert--Elliott channels
• Definition 3: Quantum-State Channel
• Proposition 4
• proof
• Example 5: BCJR bahl1974optimal decoding for CC-QSCs
• Proposition 6
• proof
• Definition 7
• Theorem 8
• proof