Quantum-Classical Hybrid Systems and their Quasifree Transformations

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.

arXiv comment63 pages, 6 figures

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• Hybrid quantum-classical systems: Quasi-free Markovian dynamics
• Markovian master equations for quantum-classical hybrid systems
• The Schmidt rank for the commuting operator framework
• Markovian dynamics for a quantum/classical system and quantum trajectories
• Fourier multipliers and their applications to PDE on the quantum Euclidean space
• Convergence of Dynamics on Inductive Systems of Banach Spaces
• Exact Model Reduction for Continuous-Time Open Quantum Dynamics
• $L^p-L^q$ boundedness of Fourier multipliers on quantum Euclidean spaces
• Sobolev inequality and its applications to nonlinear PDE on noncommutative Euclidean spaces
• Benchmarks for quantum communication via gravity
• Restoring the second law to classical-quantum dynamics