Hybrid quantum-classical model of quantum measurements

Abstract

Two types of Hamiltonian hybrid quantum-classical theories are considered as potential models ofthe 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. The first phenomenological model of the measurement process in quantum mechanics (QM), which is mathematically explicit and rigorous, was given by von Neumann [1]. The model postulates mathematical form of the change undergone by the state vector of the system composed by the measurement apparatus (A) and the measured system (S), when an ideal measurement of a physical quantity, represented by a Hermitian operator, is performed. The type of change experienced by the system -\-apparatus (SA) state vector in the measurement process is called the collapse of tbe state vector and is qualitatively different from any other dynamical process occurring in an isolated quantum system. If the state vector is interpreted as a real property of the SA system, and with other standard axioms of the von Neumann type, the special status ofthe collapse process among all other physical processes demands an explanation. Many attempts have been made to provide at least an approximate description of the state vector collapse in terms of usual dynamical processes in a SA system [2,3]. One type of attempt to model quantum measurement, if not explain it, considers the SA system as a novel kind of so-called hybrid quantum-classical systems. Such models start with an isolated quantum system S and an isolated classical system A, which are then allowed to interact and form the hybrid SA system, with its own type of states and the corresponding hybrid dynamics. Hybrid systems are interesting independent of their application in modeling the measurement, and several hybrid theories have been proposed (for a recent review, see Ref.[4]). Some of the suggested hybrid theories are mathematically inconsistent, and "no-go" theorems have been formulated [5], suggesting that no consistent hybrid theory can be formulated. Nevertheless, mathematically consistent hybrid theories exist [4,6-8].The goal of this communication is to compare descriptions of the quantum measurement in two mathematically consistent theories of hybrid quantum-classical systems. The dynamics in the two theories is the same and is described in terms of Hamiltonian dynamical systems, as for example in Refs. [4,[9][10][11]. However, the association of states ofthe hybrid SA system with the states of the quantum SA system is different in the two hybrid theories, and due to this difference one of the hybrid theories gives a good model of the quantum *buric@ipb.ac.rs me...

Some not-so-common ideas about gravity

“…In all these approaches one starts with initially separated purely classical and quantum sectors and then makes them interact in order to analyze the outcome. Without pretending to be exhaustive, we can classify these approaches in the following categories: (1) approaches that try to maintain the use of quantum states (or density matrices) to describe the quantum sector and trajectories for the classical sector [4,5], (2) those that first formulate the classical sector as a quantum theory [6,7,8] and then work with a formally completely quantum system [9,10,11,12,13,14,15,16,17], (3) conversely, those that first formulate the quantum sector as a classical theory [18] and then work with a formally completely classical system [19,20,21,22], and (4) approaches that take the quantum and the classical sectors to a common language and then extend it to a single framework in the presence of interactions, for instance, using Hamilton-Jacobi statistical theory for the classical sector and Madelung representation for the quantum sector [23,24,25] or modeling classical and quantum dynamics starting from Ehrenfest equations [26]. This classification is not sharp and in some cases it is subject to interpretation, but it may be useful as a way to organize the possible procedures and conceptual viewpoints in the enterprise of constructing a hybrid theory.…”

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Phase space theory of quantum–classical systems with nonlinear and stochastic dynamics

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Entropy and canonical ensemble of hybrid quantum classical systems

“…This possibility arises when there are two different energy or mass scales, as it happens, for instance, in molecular and condensed matter systems where the nuclei are heavy and slow, while the electrons are light and fast. Hybrid models have also been proposed to explain the measurement process [1,2]: the measurement device is modeled as a classical system coupled to the quantum system to be measured. In field theory, hybrid quantum-classical systems have also been considered as candidates to describe quantum matter fields interacting with a (classical) gravitational field, as a semiclassical approximation or even a fundamental theory (see Refs.…”