A Path Integral for Classical Dynamics, Entanglement, and Jaynes-Cummings Model at the Quantum-Classical Divide

Elze, Hans‐Thomas; Gambarotta, G; Vallone, Fabio

doi:10.1142/s021974991100723x

Cited by 18 publications

(27 citation statements)

References 22 publications

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“…This remark has allowed us, through standard dynamical analyses, not only to find 111 I fight and emerge 112 She is agitated by the stream, but does not sink 113 2) brings the magnetic dot into its metastable paramagnetic state. The truncation eliminates the off-diagonal blocks of the density matrix of S + A describing the full set of measurements.…”

Section: Individual Processes Ordinary Probabilities and Born Rulementioning

confidence: 99%

“…This scheme is based on coupled quantum and classical ensembles. A related set-up of hybrid dynamics is proposed by Elze and coworkers based on a pathintegral formulation [113], see also [114]. If Hall and Reginato's claim is true that such schemes can circumvent no-go theorems [109,112], it should be interesting to look again at the features of quantum measurements from the perspective of an explicitly classical apparatus: Bohr's program can still be opened!…”

Section: Dutch Proverbmentioning

confidence: 99%

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Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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The quantum measurement problem, to wit, understanding why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After an introduction we review the many dynamical models proposed over the years for elucidating quantum measurements. The approaches range from standard quantum theory, relying for instance on quantum statistical mechanics or on decoherence, to quantum-classical methods, to consistent histories and to modifications of the theory. Next, a flexible and rather realistic quantum model is introduced, describing the measurement of the z-component of a spin through interaction with a magnetic memory simulated by a Curie-Weiss magnet, including N 1 spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out, exhibiting several time scales. Conditions on the parameters of the model are found, which ensure that the process satisfies all the features of ideal measurements. Various imperfections of the measurement are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrixD(t). Its off-diagonal blocks in a basis selected by the spin-pointer coupling, rapidly decay owing to the many degrees of freedom of the pointer. Recurrences are ruled out either by some randomness of that coupling, or by the interaction with the bath. On a longer time scale, the trend towards equilibrium of the magnet produces a final stateD(t f ) that involves correlations between the system and the indications of the pointer, thus ensuring registration. AlthoughD(t f ) has the form expected for ideal measurements, it only describes a large set of runs. Individual runs are approached by analyzing the final states associated with all possible subensembles of runs, within a specified version of the statistical interpretation. There the difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrixD(t f ) into a sum of sub-matrices, so that one cannot infer from its sole determination the states that would describe small subsets of runs. This difficulty is overcome by dynamics due to suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance of the pointer. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Finally, pedagogical exercises are proposed and lessons for future works on m...

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Section: Individual Processes Ordinary Probabilities and Born Rulementioning

confidence: 99%

Section: Dutch Proverbmentioning

confidence: 99%

Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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“…For the optical tomogram of the hybrid-system state, we obtained the new inequality extending the entropic uncertainty relations known for quantum systems. We will study the other examples of dynamical maps of the tomographic probability distributions and their connections with other models of the hybrid systems and their evolution suggested by Elze et al [2,3,[23][24][25] in the future publication.…”

Section: Discussionmentioning

confidence: 98%

Entropy of conditional tomographic probability distributions for classical and quantum systems

Man’ko

2013

J. Phys.: Conf. Ser.

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Abstract. The possibility to describe hybrid systems containing classical and quantum subsystems by means of conditional tomographic probability distributions (tomograms) is discussed. Tomographic Shannon and Rényi entropies associated with the tomograms are studied, and new tomographic uncertainty relations are obtained. IntroductionThe possibility to describe the hybrid systems containing both classical and quantum subsystems was considered, for example, in [1][2][3][4][5]. The problems presented in the process of considering combined systems are related to a difference in the description of classical and quantum states. The classical states are associated with the probability density on the phase space. The quantum states are associated either with the wave function or with the density matrix [6,7].Recently the probability representation of quantum states was introduced using the tomographic approach [8] (see also the review [9]). Within the framework of this approach, the states in quantum mechanics are identified with the probability density of the particle position (called the tomogram) measured in a rotated and scaled reference frame in the phase space. It is important that the states in classical mechanics can also be identified with tomograms [10,11].In view of the mentioned above, the difference in the concept of classical and quantum states in the tomographic approach is reduced to the difference of the sets where the classical and quantum tomograms are located. This circumstance was used to make an attempt to describe a hybrid system consisted of the classical and quantum subsystems by a joint tomographic probability distribution depending on the positions of the classical and quantum particles. Also it was introduced that the tomograms can be interpreted as conditional probability distributions [12,13]. For example, the optical tomogram w(X, θ), being dependent on a random homodyne quadrature X and the local oscillator phase θ [14,15], which is a measurable parameter (see, e.g., [16,17]), can be interpreted as the conditional probability density w(X, θ) = w(X | θ).In view of Bayesian formula, the conditional probability distribution is connected with a joint probability distribution P (X, θ) of the both random variables.

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“…It has been shown by Primas [35] and also, e.g., Man'ko and Man'ko [45] and Elze et al [46], that there are physical systems with both classical and quantal properties. Regarding our considerations of Section 2.2., we intend here to include in this class also biological systems, tentatively already at the cellular level, genetical dynamics.…”

Section: Epistemology and Physics Of The Elementary Projections As Thmentioning

confidence: 95%

The Biological “Invariant of Motion” vs. “Struggle for Life”? On the Possible Quantum Mechanical Origin and Evolution of Semiotic Controls in Biology

Balázs

2013

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A novel, alternative and deeper view to the "selfish gene" paradigm is presented, describable as the "selfish code" frame. Introducing it, we put forth a quantum mechanical algorithm as a new description of the intracellular protein synthetizing machinery. The successive steps of the algorithm are, tentatively, semiotic constraints of the well-known quantum mechanical molecular "internal measurement" type. It is proposed that this molecular algorithm mediates a quantum mechanical time reversed dynamics with a primordial special version of this latter molecular measurement type ("mixed measurement") as its origin. It is furthermore suggested that this intracellular regressive algorithmical dynamics is a component of biological "motion", the other, strongly coupled component being the macroscopic phenotypic motion. The biological "invariant of motion" of this hierarchically coupled overall generalized dynamics is suggested to be the evolutionally converged invariant genetic code vocabulary. It forms, possibly, the underlying internal "driving force" of evolution, as being "struggle for life".

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A Path Integral for Classical Dynamics, Entanglement, and Jaynes-Cummings Model at the Quantum-Classical Divide

Cited by 18 publications

References 22 publications

Understanding quantum measurement from the solution of dynamical models

Understanding quantum measurement from the solution of dynamical models

Entropy of conditional tomographic probability distributions for classical and quantum systems

The Biological “Invariant of Motion” vs. “Struggle for Life”? On the Possible Quantum Mechanical Origin and Evolution of Semiotic Controls in Biology

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