Interaction between classical and quantum systems: A new approach to quantum measurement.I

Sherry, T. N.; Sudarshan, E. C. G.

doi:10.1103/physrevd.18.4580

Cited by 67 publications

(52 citation statements)

References 6 publications

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“…On the other hand compound classical systems are described on the Cartesian product of the component's phase spaces. Attempts to formulate a consistent dynamical theory of interacting quantumclassical, commonly called hybrid, systems are numerous as is illustrated by the following rather partial list of references [1][2][3][4][5][6][7][8]. Current technologies are sufficiently developed to enable experimental studies of the interaction between typically quantum and typically classical objects [9,10], but such experiments require detailed preliminary theoretical models.…”

Section: Introductionmentioning

confidence: 99%

Hybrid quantum-classical models as constrained quantum systems

2012

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Constrained Hamiltonian description of the classical limit is utilized in order to derive consistent dynamical equations for hybrid quantum-classical systems. Starting with a compound quantum system in the Hamiltonian formulation conditions for classical behavior are imposed on one of its subsystems and the corresponding hybrid dynamical equations are derived. The presented formalism suggests that the hybrid systems have properties that are not exhausted by those of quantum and classical systems.PACS numbers: 03.65. Fd, 03.65.Sq A. IntroductionFundamental assumption of quantum mechanics is that the evolution of an isolated quantum system is given by the linear Schröedinger equation. On the other hand, all macroscopic systems usually obey nonlinear evolution equations of classical mechanics to an excellent approximation. The classical and the quantum theory have developed different formalism to successfully describe interactions between systems belonging to their respective domains. Correlations between quantum objects are mathematically captured by the direct product structure of the Hilbert spaces. On the other hand compound classical systems are described on the Cartesian product of the component's phase spaces. Attempts to formulate a consistent dynamical theory of interacting quantumclassical, commonly called hybrid, systems are numerous as is illustrated by the following rather partial list of references [1][2][3][4][5][6][7][8]. Current technologies are sufficiently developed to enable experimental studies of the interaction between typically quantum and typically classical objects [9,10], but such experiments require detailed preliminary theoretical models.In this work the framework of the theory of Hamiltonian dynamical systems is used to treat the hybrid quantum-classical systems and to develop a description of the interactions within such systems which is consistent with the main physically justified requirements. In fact, it is well known [6,[11][12][13][14][15][16][17] that quantum mechanics can be formalized as a Hamiltonian dynamical system with the corresponding phase space and with the quantum observables described by functions which are quadratic forms of the canonical variables. More general functions on the quantum phase space do not have any physical interpretation. This formalism is used in [8] to develop a description of the hybrid classical-quantum systems by treating both, quantum and classical, formally as Hamiltonian systems described in the Hamiltonian language. The coupling between the systems is introduced somewhat ad hoc as if both systems were classical, just because they are both described in the framework of the * buric@ipb.ac.rs Hamiltonian dynamical systems. This assumption about the treatment of compound systems is not trivially obvious. For example, such treatment of coupling between two quantum systems, both separately described in the Hamiltonian framework, would be incorrect. In this paper we start with the total compound quantum system in the geometric Hamiltonian framewo...

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Section: Introductionmentioning

confidence: 99%

Hybrid quantum-classical models as constrained quantum systems

2012

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“…Let us mention that in a series of papers Sudarshan first proposed this hybrid formalism as a way of understanding quantum measurement [7][8][9]. In doing that he restricted the types of interaction terms that one can consistently introduce.…”

Section: Introductionmentioning

confidence: 99%

“…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 [2,3], (2) those that first formulate the classical sector as a quantum theory [4][5][6] and then work with a formally completely quantum system [7][8][9][10][11], (3) conversely, those that first formulate the quantum sector as a classical theory [12] and then work with a formally completely classical system [13][14][15][16], 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 [17][18][19] or modeling classical and quantum dynamics starting from Ehrenfest equations [20]. This classification is not sharp and in some cases 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.…”

Section: Introductionmentioning

confidence: 99%

Hybrid classical-quantum formulations ask for hybrid notions

et al. 2012

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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.

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“…However, as we have seen, the dynamical equations for classical and quantum systems are different. It is precisely this point, largely glossed over, that is addressed by Sudarshan's approach [4,112,113,114,115]. As we have seen, he proposed that the KvNS formalism can be looked upon as an embedding of the classical system in a quantum system with a continuum of superselection sectors.…”

Section: Realism and Non-contextualitymentioning

confidence: 99%

The Unfinished Search for Wave-Particle and Classical-Quantum Harmony

Ghose¹

2015

j adv phys

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The main purpose of this paper is to review the progress that has taken place so far in the search for a single unifying principle that harmonizes (i) the wave and particle natures of matter and radiation, both at the quantum and the classical levels, on the one hand and (ii) the classical and quantum theories of matter and radiation on the other hand. In the author's opinion, the Koopman-von Neumann-Sudarshan (KvNS) Hilbert space theory based on complex wave functions underlying particle trajectories in classical phase space, is an important step forward in that direction. To appreciate the similarities and differences between classical and quantum wave functions, it is important first to review the famous paradoxes of quantum theory arising mainly from the dual character of matter and radiation and the mysterious nature of measurements in quantum theory. They have given rise to the suspicion that quantum mechanics is an incomplete theory and to multiple interpretations of quantum mechanics as well as no-go theorems ruling out certain types of hidden variable theories introduced to 'complete' quantum mechanics in some way. It is also important to point out that experimental verifications of the predictions of the double-prism experiment and the observation of average single photon trajectories in weak measurements appear to favour the spacetime picture of particles favoured by Einstein, de Broglie and Bohm over the complementarity approach favoured by Bohr. The KvN theory of classical mechanics provides a clear and beautiful harmony of classical waves and particles. Sudarshan has given an alternative but equivalent formulation that shows that classical mechanics can be regarded as a quantum theory with essentially hidden noncommuting variables. An extension of KvNS theory to classical electrodynamics provides a sound Hilbert space foundation to it and satisfactorily accounts for entanglement and Bell-CHSH-like violations already observed in classical polarization optics. An important new insight that has been obtained through these developments is that entanglement and Bell-like inequality violations are neither unique signatures of quantumness nor of non-locality-they are rather signatures of non-separability. Another new insight is the interpretation of the Wigner function as a KvNS wave function, i.e. a probability amplitude which need not be positive everywhere. This has important implications for simulating certain types of quantum information processes using classical polarization optics. Finally, Sudarshan's proposed solution to the measurement problem using KvNS theory for the measuring apparatus is sketched to show to what extent wave and particles can be harmonized in quantum theory.

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Interaction between classical and quantum systems: A new approach to quantum measurement.I

Cited by 67 publications

References 6 publications

Hybrid quantum-classical models as constrained quantum systems

Hybrid quantum-classical models as constrained quantum systems

Hybrid classical-quantum formulations ask for hybrid notions

The Unfinished Search for Wave-Particle and Classical-Quantum Harmony

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