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arXiv:1206.7036 · Physical Review A

Hybrid classical-quantum formulations ask for hybrid notions

Carlos Barceló, Raúl Carballo-Rubio, Luis J. Garay, Ricardo Gómez-Escalante

Published Oct 22, 2012• 52 citations
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Abstract

We reappraise some of the hybrid classical-quantum models proposed in the literature with the goal of retrieving some of their common characteristics. In particular, first, we analyze in detail the Peres-Terno argument regarding the inconsistency of hybrid quantizations of the Sudarshan type. We show that to accept such hybrid formalism entails the necessity of dealing with additional degrees of freedom beyond those in the straight complete quantization of the system. Second, we recover a similar enlargement of degrees of freedom in the so-called statistical hybrid models. Finally, we use Wigner's quantization of a simple model to illustrate how in hybrid systems the subsystems are never purely classical or quantum. A certain degree of quantumness (classicality) is being exchanged between the different sectors of the theory, which in this particular unphysical toy model makes them undistinguishable.

DOI
10.1103/physreva.86.042120

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We reappraise some of the hybrid classical-quantum models proposed in the literature with the goal of retrieving some of their common characteristics. In particular, first, we analyze in detail the Peres-Terno argument regarding the inconsistency of hybrid quantizations of the Sudarshan type. We show that to accept such hybrid formalism entails the necessity of dealing with additional degrees of freedom beyond those in the straight complete quantization of the system. Second, we recover a similar enlargement of degrees of freedom in the so-called statistical hybrid models. Finally, we use Wigner's quantization of a simple model to illustrate how in hybrid systems the subsystems are never purely classical or quantum. A certain degree of quantumness (classicality) is being exchanged between the different sectors of the theory, which in this particular unphysical toy model makes them undistinguishable.
Four questions are discussed which may be addressed to any proposal of a quantum-classical hybrid theory which concerns the direct coupling of classical and quantum mechanical degrees of freedom. In particular, we consider the formulation of hybrid dynamics presented recently in Ref. [1]. This linear theory differs from the nonlinear ensemble theory of Hall and Reginatto, but shares with it to fulfil all consistency requirements discussed in the literature, while earlier attempts failed. - Here, we additionally ask: Does the theory allow for superposition, separable, and entangled states originating in the quantum mechanical sector? Does it allow for "Free Will", as introduced, in this context, by Diósi [2]? Is it local? Does it provide hints for the emergence of quantum mechanics from dynamics beneath?
In this work we provide a complete model of semiclassical theories by including back-reaction and correlation into the picture. We specially aim at the interaction between light and a two-level atom, and we also illustrate it via the coupling of two harmonic oscillators. Quantum and classical systems are treated on the same grounds via the Wigner-Weyl phase-space correspondence of the quantum theory. We show that this model provides a suitable mixture of the quantum and classical degrees of freedom, including the fact that the evolution transfers nonclassical features to the classical subsystem and nonquantum behavior to the quantum subsystem. In that sense, we can no longer distinguish between classical and quantum variables and we need to talk about a hybrid model.
Two types of Hamiltonian hybrid quantum-classical theories are considered as potential models of the quantum measurement process. The two theories have the same Hamiltonian dynamics but differ in the association of states of the quantum system with states of the hybrid model. In the first type of association pure quantum states are modeled by pure states of the hybrid, while in the second type the pure quantum states are modeled by statistical mixtures of the hybrid pure states. It is shown that the second of the two theories describes correctly the quantum measurement while the first provides only an averaged description.
We present a consistent formalism to describe the dynamics of hybrid systems with mixed classical and quantum degrees of freedom. The probability function of the system, which, in general, will be a combination of the classical distribution function and the quantum density matrix, is described in terms of its corresponding moments. We then define a hybrid Poisson bracket, such that the dynamics of the moments is ruled by an effective Hamiltonian. In particular, a closed formula for the Poisson brackets between any two moments for an arbitrary number of degrees of freedom is presented, which corrects previous expressions derived in the literature for the purely quantum case. This formula is of special relevance for practical applications of the formalism. Finally, we study the dynamics of a particular hybrid system given by two coupled oscillators, one being quantum and the other classical. Due to the coupling, specific quantum and classical properties are transferred between different sectors. In particular, the quantum sector is allowed to violate the uncertainty relation, though we explicitly show that there exists a minimum positive bound of the total uncertainty of the hybrid system.
The notion of hybrid quantum-classical control system was introduced as a control dynamical system which combined classical and quantum degrees of freedom. Classical and quantum objects were combined within a geometrical description of both types of systems. We also considered the notion of hybrid quantum-classical controllability by means of the usual definitions of geometric control theory, and we discussed how the different concepts associated to quantum controllability are lost in the hybrid context because of the nonlinearity of the dynamics. We also considered several examples of physically relevant problems, such as the spin-boson model or the notion of hybrid spline.
Quantum mechanics is among the most successful physical theories, yet its formulation and empirical testing rely on classical structures. Following Lev Landau and Niels Bohr, this reliance is not merely pragmatic: quantum observables acquire empirical meaning only relative to classical reference frames, and, in practice, quantization starts from classical models. At the same time, the two domains display forms of mutual irreducibility: intrinsically quantum features (that is, spin and exchange statistics) have no counterpart in the phase-space ontology of classical point-particle mechanics, while classical trajectory chaos does not arise straightforwardly from unitary quantum evolution in closed systems. A hierarchy is commonly established between classical and quantum theories, namely, a claim of ontological and explanatory priority according to which quantum mechanics is fundamental and classical mechanics is only a limiting case. This claim is less secure than is often assumed; therefore, the traditional hierarchy deserves to be examined. In this paper, we argue that a quantum–classical framework provides an effective and structurally faithful representation of empirically accessible physical systems in regimes where quantum and classical degrees of freedom coexist within a single, consistent effective dynamical description. To give this point of view a firm theoretical basis, we discuss the quasi-Lie formal structure underlying quantum–classical hybrid dynamics, with applications ranging from gravity and condensed matter to open, driven systems in biology and complex media.