Consistent dynamics which couples classical and quantum degrees of freedom exists. This dynamics is linear in the hybrid state, completely positive and trace preserving. Starting from completely positive classical-quantum master equations, we derive a general path integral representation for such dynamics in terms of a classical-quantum action, which includes the necessary and sufficient conditions for complete positivity and trace preservation. The path integral we study is a generalization of the Feynman path integral for quantum systems, and the stochastic path integral used to study classical stochastic processes, allowing for interaction between the classical and quantum systems. When the classical-quantum Hamiltonian is at most quadratic in the momenta we are able to derive a configuration space path integral, providing a map between master equations and covariant classical-quantum path integrals.
Consistent dynamics which couples classical and quantum degrees of freedom exists. This dynamics is linear in the hybrid state, completely positive and trace preserving. Starting from completely positive classical-quantum master equations, we derive a general path integral representation for such dynamics in terms of a classical-quantum action, which includes the necessary and sufficient conditions for complete positivity and trace preservation. The path integral we study is a generalization of the Feynman path integral for quantum systems, and the stochastic path integral used to study classical stochastic processes, allowing for interaction between the classical and quantum systems. When the classical-quantum Hamiltonian is at most quadratic in the momenta we are able to derive a configuration space path integral, providing a map between master equations and covariant classical-quantum path integrals.
Consistent dynamics which couples classical and quantum degrees of freedom exists, provided it is stochastic. This dynamics is linear in the hybrid state, completely positive and trace preserving. One application of this is to study the back-reaction of quantum fields on space-time which does not suffer from the pathologies of the semi-classical equations. Here we introduce several toy models in which to study hybrid classical-quantum evolution, including a qubit coupled to a particle in a potential, and a quantum harmonic oscillator coupled to a classical one. We present an unravelling approach to calculate the dynamics, and provide code to numerically simulate it. Unlike the purely quantum case, the trajectories (or histories) of this unravelling can be unique, conditioned on the classical degrees of freedom for discrete realisations of the dynamics, when different jumps in the classical degrees of freedom are accompanied by the action of unique operators on the quantum system. As a result, the “measurement postulate'' of quantum theory is not needed; quantum systems become classical because they interact with a fundamentally classical field.
A concise and self-contained derivation of hybrid quantum-classical dynamics is given in terms of Markovian master equations. Many previously known results are rederived and revised and some of them are completed or corrected. Using a method as simple as possible, our goal is a brief introduction to the state of the art of hybrid dynamics, with a limited discussion of the implications for foundations and without discussion of further relevance in the measurement problem, quantum gravity, chemistry, numeric methods, etc. Hybrid dynamics is defined as a special case of composite quantum dynamics where the observables of one of the two subsystems are restricted to the commuting set of diagonal operators in a fixed basis. With this restriction, the derivation of hybrid dynamical equations is clear conceptually and simple technically. Jump and diffusive dynamics follow in the form of hybrid master equations. Their stochastic interpretation (called unravelings) is derived. We discuss gauge-type ambiguities, problems of uniqueness, and covariance of the diffusive master equation. Also conditions of minimum noise and of monitoring the quantum trajectory are derived. We conclude that the hybrid formalism is equivalent to the standard Markovian theory of time-continuous quantum measurement (monitoring) on the one hand and is a motivating alternative formalism on the other.
Coupling between quantum and classical systems is consistent, provided the evolution is linear in the state space, preserves the split of systems into quantum and classical degrees of freedom, and preserves probabilities. The evolution law must be a completely positive and norm preserving map. We prove that if the dynamics is memoryless, there are two classes of these dynamics, one which features finite sized jumps in the classical phase space and one which is continuous. We find the most general form of each class of classical-quantum master equation. This is achieved by applying the complete positivity conditions using a generalized Cauchy-Schwartz inequality applicable to classical-quantum systems. The key technical result is a generalisation of the Pawula theorem.
Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.
We develop a Bayesian formulation of diffusive quantum-classical dynamics by treating the wave function and classical variables as components of an ordinary stochastic process. The joint probability density P(psi,x,t) obeys a classical Fokker-Planck equation, while the quantum state appears as its second moment. Requiring autonomous second-moment evolution and trace preservation, together with a full-column-rank quantum-classical noise correlation matrix, forces the ensemble dynamics into hybrid Lindblad form. The same result also determines the general class of diffusive stochastic unravelings compatible with this hybrid evolution. Notably, this construction makes positivity immediate and does not take complete positivity or an underlying unitary quantum dilation as assumptions, but instead relies on the presence of a sufficiently coupled classical sector.
We formulate quantum measurement, filtering and smoothing of hybrid quantum-classical dynamics as standard Bayesian inference and show how to correlation between the quantum and classical noise fully explains the differences between quantum smoothing and smoothing in classical time series models. We recover the stochastic master equation from the Kushner-Stratonovich equation with correlated noise and show how the quantum effect operator is related to the Bayesian backward message through the adjoint dynamics of the linear unraveling. The Bayesian posterior also defines a smoothed density matrix and, more generally, a posterior distribution over latent quantum-classical trajectories.
These quantities can be approximated with standard particle filtering and smoothing methods. Numerical examples show that smoothing improves reconstruction of a hidden quantum-classical trajectory and that the full trajectory posterior can retain structure, such as multimodality, that is absent from its density-matrix second moment.
Coupling between quantum and classical systems is consistent, provided the evolution is linear in the state space, preserves the split of systems into quantum and classical degrees of freedom, and preserves probabilities. The evolution law must be a completely positive and norm preserving map. We prove that if the dynamics is memoryless there are two classes of these dynamics, one which features finite jumps in the classical phase space and one which is continuous. We derive the most general form of the dynamics. This is achieved by applying the complete positivity conditions using a Cauchy-Schwarz inequality applicable to classical-quantum systems. The key technical result is a generalization of the Pawula theorem.