Non-Markovian Quantum Feedback Networks I: Quantum Transmission Lines, Lossless Bounded Real Property and Limit Markovian Channels

TL;DR

This paper explores modeling non-Markovian quantum transmission lines, establishing their properties, and connecting them to Markovian models for quantum communication and control.

Contribution

It extends quantum network rules to non-Markovian channels, introduces an analytic scattering matrix, and links stability to the lossless bounded real property.

Findings

Analytic scattering matrix for non-Markovian models

Stability characterized by lossless bounded real property

Limit to Markovian models enables standard quantum filtering

Abstract

The purpose of this paper is to set out the problems of modeling quantum communication and signal processing where the communication between systems via a non-Markovian channel. This is a general feature of quantum transmission lines. Our ultimate objective is to extend the networks rules that have been developed for Markovian models. To this end we recall the Hamiltonian description of such non-Markov models of transmission lines and their quantization. These have occurred in the context of non-quilibrium thermodynamics, but our interest is in the transmission lines as carriers of information rather than heat baths. We show that there is an analytic scattering matrix associated with these models and that stability may be formulated in terms of the lossless bounded real property. Noting that the input and output fields do not separately satisfy a non-self- demolition principle, we discuss the rigorous limit in which such models appear Markov and so amenable to standard approaches of quantum filtering and control