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Lecture notes in mathematics

An intrinsic approach to the evolution of quantum observables in terms of stochastic processes on phase space

J. Bertrand, Guy Rideau

Published Jan 1, 1985• 3 citations
Publisher
DOI
10.1007/bfb0101536

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A specific model of quantum measurement [D. F. Walls, M. J. Collet, and G. J. Milburn, Phys. Rev. D 32, 3208 (1985)] based on a generic system-meter-environment measurement scheme is investigated in terms of probability distributions on projective Hilbert space. Starting from the unitary evolution of the total system a stochastic process for the wave function of the reduced system-meter system is derived by elimination of the degrees of freedom of the environment. The stochastic process is interpreted in terms of the direct detection of the photoelectrons created by the meter quanta in the environment. Two general variances are introduced which represent, respectively, the average intrinsic quantum fluctuations and the statistical fluctuations of the quantum expectation of the measured observable. Involving fourth moments of the wave function, these variances allow to monitor the transformation from quantum to classical probabilities during the process of measurement. \textcopyright{} 1996 The American Physical Society.

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The Black–Scholes pricing formula in the quantum context

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A natural explanation for extreme irregularities in the evolution of prices in financial markets is provided by quantum effects. The lack of simultaneous observability of relevant variables and the interference of attempted observation with the values of these variables represent such effects. These characteristics have been noted by traders and economists and appear intrinsic to market dynamics. This explanation is explored here in terms of a corresponding generalization of the Wiener process and its role in the Black-Scholes-Merton theory. The differentiability of the Wiener process as a sesquilinear form on a dense domain in the Hilbert space of square-integrable functions over Wiener space is shown and is extended to the quantum context. This provides a basis for a corresponding generalization of the Ito theory of stochastic integration. An extension of the Black-Scholes option pricing formula to the quantum context is deduced.

Proceedings of the National Academy of Sciences•Mar 31, 1998•86 citations
Contents 1 Introduction 2 2 Quantum Interference and the Construction of Nonclassical States 3 2.1 Nonclassical Light, Quadratures and Phase--Sensitive Observables 3 2.2 Phase--Space and Quasi--Probabilities . . . . . . . . . . . . . . . . 13 2.3 Quantum Noise as a Stochastic Process . . . . . . . . . . . . . . 15 2.4 Marginal distributions and reconstruction of Wigner functions . . 18 2.5 Joint Measurements of p, q in Phase Space . . . . . . . . . . . . . 20 2.6 Phase--Space Sampling Entropies . . . . . . . . . . . . . . . . . . 27 2.7 Mutual Information and Entropic Uncertainty . . . . . . . . . . . 28 2.8 Building Nonclassical States by Quantum Interference . . . . . . 30 3 Dissipation Versus Coherent Evolution: Decoherence 36 4 Unravelling the Master Equation 37 5 Quantum Jumps 43 5.1 Intrinsic or Extrinsic Jumps . . . . . . . . . . . . . . . . . . . . . 43 5.2 Observation of Quantum Jumps . . . . . . . . . . . . . .
A specific model of quantum measurement [D. F. Walls, M. J. Collet, and G. J. Milburn, Phys. Rev. D 32, 3208 (1985)] based on a generic system-meter-environment measurement scheme is investigated in terms of probability distributions on projective Hilbert space. Starting from the unitary evolution of the total system a stochastic process for the wave function of the reduced system-meter system is derived by elimination of the degrees of freedom of the environment. The stochastic process is interpreted in terms of the direct detection of the photoelectrons created by the meter quanta in the environment. Two general variances are introduced which represent, respectively, the average intrinsic quantum fluctuations and the statistical fluctuations of the quantum expectation of the measured observable. Involving fourth moments of the wave function, these variances allow to monitor the transformation from quantum to classical probabilities during the process of measurement. \textcopyright{} 1996 The American Physical Society.
A natural explanation for extreme irregularities in the evolution of prices in financial markets is provided by quantum effects. The lack of simultaneous observability of relevant variables and the interference of attempted observation with the values of these variables represent such effects. These characteristics have been noted by traders and economists and appear intrinsic to market dynamics. This explanation is explored here in terms of a corresponding generalization of the Wiener process and its role in the Black-Scholes-Merton theory. The differentiability of the Wiener process as a sesquilinear form on a dense domain in the Hilbert space of square-integrable functions over Wiener space is shown and is extended to the quantum context. This provides a basis for a corresponding generalization of the Ito theory of stochastic integration. An extension of the Black-Scholes option pricing formula to the quantum context is deduced.