Hybrid classical-quantum formulations ask for hybrid notions

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

doi:10.1103/physreva.86.042120

Cited by 54 publications

(77 citation statements)

References 41 publications

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“…Observe that the classical Hilbert space can be partitioned into equivalence classes |ψ ∼ e iφ |ψ , where each class corresponds to a single density ρ. The evolution equations preserve the equivalence classes because there is no interaction [4].…”

Section: Interpolating Abstract Systems and Hybrid Modelsmentioning

confidence: 99%

“…Peres and Terno [3] analyzed consistency of the Koopmanvon Neumann-Sudarshan (KNS) hybrid dynamics with the quantum-quantum and the classical-classical limits for the case of linear interaction between harmonic oscillators. Some aspects of the KNS formalism for a hybrid system with specific interaction have also been studied in [4]. The authors investigated the role of unphysical variables which are called unobservables because they do not influence the evolution of the physical observables of the quantum or the classical part if there is no quantum-classical interaction.…”

Section: Introductionmentioning

confidence: 99%

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Ehrenfest principle and unitary dynamics of quantum-classical systems with general potential interaction

et al. 2014

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Representation of classical dynamics by unitary transformations has been used to develop a unified description of hybrid classical-quantum systems with a particular type of interaction, and to formulate abstract systems interpolating between classical and quantum ones. We solved the problem of a unitary description of two interpolating systems with general potential interaction. The general solution is used to show that with arbitrary potential interaction between the two interpolating systems the evolution of the so-called unobservable variables is decoupled from that of the observable ones if and only if the interpolation parameters in the two interpolating systems are equal.

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Section: Interpolating Abstract Systems and Hybrid Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Ehrenfest principle and unitary dynamics of quantum-classical systems with general potential interaction

et al. 2014

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“…In fact, there are hybrid theories which take into account classical as well as quantum degrees of freedom (see for instance [1][2][3][4][5][6][7]), but will not be considered here. Concerning the classical limit of quantum mechanics, in Ref.…”

Section: Introductionmentioning

confidence: 98%

Classical and quantum behavior of the harmonic and the quartic oscillators

Brizuela

2014

Phys. Rev. D

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In a previous paper a formalism to analyze the dynamical evolution of classical and quantum probability distributions in terms of their moments was presented. Here the application of this formalism to the system of a particle moving on a potential is considered in order to derive physical implications about the classical limit of a quantum system. The complete set of harmonic potentials is considered, which includes the particle under a uniform force, as well as the harmonic and the inverse harmonic oscillators. In addition, as an example of anharmonic system, the pure quartic oscillator is analyzed. Classical and quantum moments corresponding to stationary states of these systems are analytically obtained without solving any differential equation. Finally, dynamical states are also considered in order to study the differences between their classical and quantum evolution.

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“…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][7][8].…”

mentioning

confidence: 99%

Hybrid quantum-classical model of quantum measurements

et al. 2013

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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...

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Hybrid classical-quantum formulations ask for hybrid notions

Cited by 54 publications

References 41 publications

Ehrenfest principle and unitary dynamics of quantum-classical systems with general potential interaction

Ehrenfest principle and unitary dynamics of quantum-classical systems with general potential interaction

Classical and quantum behavior of the harmonic and the quartic oscillators

Hybrid quantum-classical model of quantum measurements

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