Quantum-Classical Hybrid Systems and their Quasifree Transformations

TL;DR

This paper develops a unified framework for quantum-classical hybrid systems using quasifree transformations, encompassing Gaussian operations, noise models, and information protocols, with rigorous mathematical treatment.

Contribution

It introduces a comprehensive, rigorous approach to quasifree operations in hybrid quantum-classical systems, extending Gaussian frameworks to more general noise and dynamics.

Findings

Characterization of quasifree operations via phase-space translations

Inclusion of all states as quasifree within the framework

Analysis of quantum information protocols like cloning and teleportation

Abstract

We study continuous variable systems, in which quantum and classical degrees of freedom are combined and treated on the same footing. Thus all systems, including the inputs or outputs to a channel, may be quantum-classical hybrids. This allows a unified treatment of a large variety of quantum operations involving measurements or dependence on classical parameters. The basic variables are given by canonical operators with scalar commutators. Some variables may commute with all others and hence generate a classical subsystem. We systematically study the class of "quasifree" operations, which are characterized equivalently either by an intertwining condition for phase-space translations or by the requirement that, in the Heisenberg picture, Weyl operators are mapped to multiples of Weyl operators. This includes the well-known Gaussian operations, evolutions with quadratic Hamiltonians, and "linear Bosonic channels", but allows for much more general kinds of noise. For example, all states are quasifree. We sketch the analysis of quasifree preparation, measurement, repeated observation, cloning, teleportation, dense coding, the setup for the classical limit, and some aspects of irreversible dynamics, together with the precise salient tradeoffs of uncertainty, error, and disturbance. Although the spaces of observables and states are infinite dimensional for every non-trivial system that we consider, we treat the technicalities related to this in a uniform and conclusive way, providing a calculus that is both easy to use and fully rigorous.