The purpose of this paper is to present simple and general algebraic methods for describing series connections in quantum networks. These methods build on and generalize existing methods for series (or cascade) connections by allowing for more genera
l interfaces, and by introducing an efficient algebraic tool, the series product. We also introduce another product, which we call the concatenation product, that is useful for assembling and representing systems without necessarily having connections. We show how the concatenation and series products can be used to describe feedforward and feedback networks. A selection of examples from the quantum control literature are analyzed to illustrate the utility of our network modeling methodology.
A quantum network is an open system consisting of several component Markovian input-output subsystems interconnected by boson field channels carrying quantum stochastic signals. Generalizing the work of Chebotarev and Gregoratti, we formulate the mod
el description by prescribing a candidate Hamiltonian for the network including details the component systems, the field channels, their interconnections, interactions and any time delays arising from the geometry of the network. (We show that the candidate is a symmetric operator and proceed modulo the proof of self-adjointness.) The model is non-Markovian for finite time delays, but in the limit where these delays vanish we recover a Markov model and thereby deduce the rules for introducing feedback into arbitrary quantum networks. The type of feedback considered includes that mediated by the use of beam splitters. We are therefore able to give a system-theoretic approach to introducing connections between quantum mechanical state-based input-output systems, and give a unifying treatment using non-commutative fractional linear, or Mobius, transformations.
We show that it is possible to construct closed quantum systems governed by a bilinear Hamiltonian depending on an arbitrary input signal. This is achieved by coupling the system to a quantum input field and performing a feedback of the output field
back into the system to cancel out the stochastic effects, with the signal being added to the field between these events and later subtracted. Here we assume the zero time delay limit between the various connections and operations.
The mathematical theory of quantum feedback networks has recently been developed for general open quantum dynamical systems interacting with bosonic input fields. In this article we show, for the special case of linear dynamical systems Markovian sys
tems with instantaneous feedback connections, that the transfer functions can be deduced and agree with the algebraic rules obtained in the nonlinear case. Using these rules, we derive the the transfer functions for linear quantum systems in series, in cascade, and in feedback arrangements mediated by beam splitter devices.
