Quantum-Classical Hybrid Systems and their Quasifree Transformations

authored by
Lars Dammeier, Reinhard F. Werner

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. The data defining a quasifree channel are, first, a linear map from the output phase space to the input phase space, which describes how the Weyl operators are connected. The second element is a scalar “noise factor”, which is usually needed to make the channel completely positive. Channels with noise factor 1 are called noiseless. These are homomorphisms in the Heisenberg picture. For any quasifree channel, the admissible noise functions are in one-to-one correspondence to states on a certain hybrid system. Since many basic tasks (e.g., joint measurement, cloning, or teleportation) are encoded in the linear phase space map, this gives a compact characterization of the possible noises for channels implementing the task. We establish a general Stinespring-like decomposition of any quasifree channel into the expansion by an additional system followed by a noiseless operation. The additional system is itself a hybrid and in the state characterizing the noise. This allows a clear distinction between classical and quantum noise of the channel. Technically, our main contribution is the clarification of the functional analysis of the spaces of states observables and channels. This required the resolution of a mismatch in the standard approaches to classical and quantum systems, respectively, which would have bogged down the theory with many case distinctions. In the scheme that we propose all hybrid systems and quasifree operations are treated in a uniform manner. For example, the noise analysis of dense coding and teleportation become virtually identical. All quasifree channels can equivalently be considered in the Schrödinger picture or in a variety of Heisenberg pictures differing by the degree of smoothness demanded of the observables.

Institute of Theoretical Physics
CRC 1227 Designed Quantum States of Matter (DQ-mat)
No. of pages
Publication date
Publication status
Peer reviewed
ASJC Scopus subject areas
Atomic and Molecular Physics, and Optics, Physics and Astronomy (miscellaneous)
Electronic version(s)
https://doi.org/10.22331/q-2023-07-26-1068 (Access: Open)