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

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.

I. Introduction

We consider the transmission rate of classical information over a finite-dimensional quantum channel with memory [1]–[3]. Recall that in the memoryless case, given an input system and an output system , described by some Hilbert spaces and , respectively, a memoryless quantum channel can be modeled as a completely positive trace-preserving (CPTP) map from the set of density operators on to the set of density operators on [4], [5]; such a quantum channel is said to be finite-dimensional if both and are of finite dimension. A quantum channel with memory is a quantum channel equipped with a memory system ; namely it is a CPTP map from the set of density operators on to the set of density operators on , where is the Hilbert space describing , and stands for the tensor product. The system can be understood either as a state of the channel (as illustrated in Fig. 1(a)), or as a part of the environment that does not decay between consecutive channel uses (as illustrated in Fig. 1(b)). Interesting examples of quantum channels with memory include spin chains [6] and fiber optic links [7]. Fig. 1.

Interpretations of quantum channels with memory.