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The purpose of this paper is to set out the problems of modeling quantumcommunication and signal processing where the communication between systems viaa non-Markovian channel. This is a general feature of quantum transmissionlines. Our ultimate objective is to extend the networks rules that have beendeveloped for Markovian models. To this end we recall the Hamiltoniandescription of such non-Markov models of transmission lines and theirquantization. These have occurred in the context of non-quilibriumthermodynamics, but our interest is in the transmission lines as carriers ofinformation rather than heat baths. We show that there is an analyticscattering matrix associated with these models and that stability may beformulated in terms of the lossless bounded real property. Noting that theinput and output fields do not separately satisfy a non-self- demolitionprinciple, we discuss the rigorous limit in which such models appear Markov andso amenable to standard approaches of quantum filtering and control

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