Path summation and von Neumann–like quantum measurements

Sokolovski, D.; Mayato, R. Sala

doi:10.1103/physreva.71.042101

Cited by 11 publications

(20 citation statements)

References 42 publications

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“…We will pay special attention to the importance of the quantum uncertainty principle and emphasize the role that quantum interference plays in the loss of information about the system's past. Throughout the paper, we will use the simplest case of a two-level system as an example and refer the reader to [8][9][10][11][12][13][14][15] for a more general analysis.…”

Section: Introductionmentioning

confidence: 99%

Quantum Measurements, Stochastic Networks, the Uncertainty Principle, and the Not So Strange “Weak Values”

Sokolovski

2016

Mathematics

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Suppose we make a series of measurements on a chosen quantum system. The outcomes of the measurements form a sequence of random events, which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become "real", and the mean values of a quantity (a functional) are directly related to the frequencies with which the paths are traveled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities' amplitudes. For pre-and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one from distinguishing between interfering alternatives, while leaving the interference between them intact.

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Section: Introductionmentioning

confidence: 99%

Quantum Measurements, Stochastic Networks, the Uncertainty Principle, and the Not So Strange “Weak Values”

Sokolovski

2016

Mathematics

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“…It is convenient to choose an othonormal basis {|n }, n = 1, 2..N corresponding to the "position" operator [18] n ≡ ∑ N n=1 |n n n|. In general, the transition amplitude can be written as a sum over all virtual paths n(t) which take the values 1, 2...N at any given time t [19,20]. Since Ĥ = 0, there are only N constant paths n(t) = 1, 2...N and the path decomposition of the transition amplitudes takes a simple form…”

Section: Quantum Measurements and Virtual Pathsmentioning

confidence: 99%

Feynman-path analysis of Hardy's paradox: Measurements and the uncertainty principle

2008

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Hardy's paradox is analysed within Feynman's formulation of quantum mechanics. A transition amplitude is represented as a sum over virtual paths which different intermediate measurements convert into different sets of real pathways. Contradictions arise if conflicting statements are applied to the same statistical ensemble. Usefulness of "strange" weak values for resolving the paradox is disputed.

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“…A formalism for measuring the time average of any dynamical quantity on individual Feynman histories is developed in [12,13] . Also see [14] for the general relation between restricted paths sums and von-Neumann-like quantum measurements. A final note regarding the meaning of a quantum measurement: In this article we adopt the view from [6] regarding the distinction between a quantum measurement and an observation.…”

Section: Introductionmentioning

confidence: 99%

Time-extended measurement of the position of a particle

Roig

2006

Phys. Rev. A

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The von Neumann interaction between a particle and an apparatus, both of arbitrary mass, has been considered in the measurement of the position of a simple harmonic oscillator acted on by an external force. When the measurement has finite duration, both the motion of the pointer and the oscillator influence the result of the measurement. Provided that the oscillator is in an eigenstate of its position at the start of the measurement, the pointer will indicate the arithmetic average between the initial and final position of the particle with an added term which depends on the duration of the measurement and the frequency of the oscillator. This additional term is determined by the external force which also causes the appearance of a phase factor in the wave function at the end of the measurement. This phase factor depends on the average of the initial and final positions of the particle. Furthermore, the probability that the pointer indicator variable will correlate with a given average value is equal to the transition probability for the undisturbed free oscillator to experience the change in position. If the initial state of the pointer is a narrow wavepacket, then for any initial state of the oscillator, the measurement yields, approximately, the undisturbed probability distribution for the position of the free oscillator at the end of the measurement. The transition probability for the pointer to experience a change in its position has also been evaluated.

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Path summation and von Neumann–like quantum measurements

Cited by 11 publications

References 42 publications

Quantum Measurements, Stochastic Networks, the Uncertainty Principle, and the Not So Strange “Weak Values”

Quantum Measurements, Stochastic Networks, the Uncertainty Principle, and the Not So Strange “Weak Values”

Feynman-path analysis of Hardy's paradox: Measurements and the uncertainty principle

Time-extended measurement of the position of a particle

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