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Quantum Trajectories, State Diffusion, and Time-Asymmetric Eventum Mechanics

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Abstract

We show that the quantum stochastic Langevin model for continuous in time measurements provides an exact formulation of the von Neumann uncertainty error-disturbance principle. Moreover, as it was shown in the 1980s, this Markov model induces all stochastic linear and nonlinear equations of the phenomenological informational dynamics such as quantum state diffusion and spontaneous localization by a simple quantum filtering method. Here we prove that the quantum Langevin equation is equivalent to a Dirac-type boundary-value problem for the second quantized input “offer waves from future” in one extra dimension, and to a reduction of the algebra of the consistent histories of past events to an Abelian subalgebra for the “trajectories of the output particles.” This result supports the wave-particle duality in the form of the thesis of Eventum Mechanics that everything in the future is constituted by quantized waves, everything in the past by trajectories of the recorded particles. We demonstrate how this time arrow can be derived from the principle of quantum causality for nondemolition continuous in time measurements.

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References

  • Barchielli, A. and Belavkin, V. P. (1991). Measurements continuous in time and a posteriori states in quantum mechanics. Journal of Physics A: Mathematical and General 24, 1495-1514.

    Google Scholar 

  • Belavkin, V. P. (1999). Optimal Measurement and Control in Quantum Dynamical Systems, Preprint No. 411, Institute of Physics, Nicholas Copernicus University, Torun', 1979, Feb. [Published in Rep. Math. Phys. 43(3), 405-425

    Google Scholar 

  • Belavkin, V. P. (1980). Quantum filtering of Markov signals with Wight quantum noise. Radiotechnika and Electronika 25, 1445-1453. [Full English translation in Quantum Communications and Measurement. Belavkin, V. P. et al., eds., Plenum Press, New York, pp. 381–392, 1994.

    Google Scholar 

  • Belavkin, V. P. (1985). In: Information Complexity and Control in Quantum Physics, A. Blaquière ed., Springer-Verlag, Udine, pp. 331-336.

    Google Scholar 

  • Belavkin, V. P. (1988a). In: Modelling and Control of Systems, A. Blaquière, ed., Lecture Notes in Control and Information Sciences, Springer, Vol. 121, pp. 245-265.

  • Belavkin, V. P. (1988b). A New Form and a *-Algebraic Structure of Quantum Stochastic Integrals in Fock Space. Rediconti del Sem. Mat. e Fis. di Milano LVIII, 177-193.

    Google Scholar 

  • Belavkin, V. P. (1989). A new wave equation for a continuous nondemolition measurement. Physics Letters A 140, 355-358.

    Google Scholar 

  • Belavkin, V. P. (1990a). In: Stochastic Methods in Experimental Sciences, W. Kasprzak and A. Weron, eds., World Scientific, Singapore, pp. 26-42.

    Google Scholar 

  • Belavkin, V. P. (1990b). A Posterior Schrödinger Equation for Continuous Nondemolition Measurement. Journal of Mathematics and Physics 31, 2930-2934.

    Google Scholar 

  • Belavkin, V. P. (1990c). Stochastic posterior equations for quantum nonlinear filtering. In Prob. Theory and Math. Stat., B. Grigelionis et al. eds., VSP/Mokslas, Vilnius 1, pp. 91-109.

    Google Scholar 

  • Belavkin, V. P. (1991). A quantum nonadapted itô formula and stochastic analysis in fock scale. J. Func. Anal. 102(2), 414-447.

    Google Scholar 

  • Belavkin, V. P. (1992a). Quantum stochastic calculus and quantum nonlinear filtering. Journal of Multivariate Analysis 42(2), 171-201.

    Google Scholar 

  • Belavkin, V. P. (1992). Quantum continual measurements and a posteriori collapse on CCR. Communications in Mathematical Physics 146, 611-635.

    Google Scholar 

  • Belavkin, V. P. (1993). The unified itô formula has the pseudo-poisson structure. Journal of Mathematics and Physics 34(4), 1508-1518.

    Google Scholar 

  • Belavkin, V. P. (1994). Nondemolition principle of quantum measurement theory. Foundations of Physics 24(5), 685-714.

    Google Scholar 

  • Belavkin, V. P. (1995a). A dynamical theory of quantum continuous measurements and spontaneous localizations. Russian Journal of Mathematical Physics 3(1), 3-24.

    Google Scholar 

  • Belavkin, V. P. (1995b). On stochastic generators of completely positive cocycles. Russian Journal of Mathematics and Physics 3, 523-528.

    Google Scholar 

  • Belavkin, V. P. (1997). Quantum stochastic positive evolutions: Charcterization, construction, dilation. Communications in Mathematical Physics 184, 533-566.

    Google Scholar 

  • Belavkin, V. P. (1998). On quantum itô algebras and their decompositions. Letters in Mathematical Physics 45, 131-145.

    Google Scholar 

  • Belavkin, V. P. (2000a, April). In New Development of Infinite-Dimensional Analysis and Quantum Probability, RIMS, Kokyuroku, Vol. 1139, pp. 54-73.

    Google Scholar 

  • Belavkin, V. P. (2000b). Quantum stochastics, dirac boundary value problem, and the inductive stochastic limit. Rep. Math. Phys. 46(3), 359-382.

    Google Scholar 

  • Belavkin, V. P. (2001). In Evolution Equations and Their Applications in Physical and Life Sciences, G. Lumer and L. Weis, eds., Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc, New York, Vol. 215, pp. 311-327.

    Google Scholar 

  • Belavkin, V. P. (2002). Quantum causality, stochastics, trajectories and information. Reports on Progress in Physics 65, 353-420.

    Google Scholar 

  • Belavkin, V. P. and Kolokoltsov, V. N. (2001). Stochastic evolutions as boundary value problems. In Infinite Dimensional Analysis and Quantum Probability, Research Institute for Mathematical Studies, Kokyuroku, Vol. 1227, pp. 83-95.

    Google Scholar 

  • Belavkin, V. P. and Melsheimer, O. (1996). A stochastic Hamiltonian approach for quantum Jumps, spontaneous localizations and continuous trajectories. Quantum and Semiclassical Optics 8, 167-187.

    Google Scholar 

  • Belavkin, V. P. and Staszewski, P. (1989). A quantum particle undergoing continuous observation. Physical Letters A 140, 359-362.

    Google Scholar 

  • Belavkin, V. P. and Staszewski, P. (1992). Nondemolition observation of a free Quantum particle. Physical Review A 45(3), 1347-1356.

    Google Scholar 

  • Carmichael, H. J. (1993). An Open System Approach to Quantum Optics, Lecture Notes in Physics, m18, Springer, Berlin.

    Google Scholar 

  • Carmichael, H. J. (1994). In Quantum Optics VI, J. D. Harvey and D. F. Walls, eds., Springer, Berlin.

    Google Scholar 

  • Chruscinski, D. and Staszewski, P. (1992). On the asymptotic solutions of the Belavkin's stochastic wave Equation. Physica Scripta. 45, 193-199.

    Google Scholar 

  • Cramer, J. G. (1986). Review of Modern Physics 58, 647-687.

    Google Scholar 

  • Diosi, L. (1988). Physical Review A 40, 1165-1174.

    Google Scholar 

  • Dunn, R., Parkins, A. S., Zoller, P., and Gardiner, C. W. (1992). Physical Review A 46, 4382.

    Google Scholar 

  • Gardiner, C. W. (1991). Quntum Noise, Springer-Verlag, Berlin.

    Google Scholar 

  • Gardiner, C. W. and Collett, M. J. (1985). Physical Review A 31, 3761.

    Google Scholar 

  • Gardiner, C. W., Parkins, A. S., and Zoller, P. (1992). Physical Review A 46, 4363.

    Google Scholar 

  • Gisin, N. (1984). Quantum Measurement and Stochastic Proceses. Physical Review Letters 52(19), 1657-1660.

    Google Scholar 

  • Gisin, N. (1989). Helvetica Physica Acta 62, 363.

    Google Scholar 

  • Gisin, N. and Percival, I. C. (1992). The quantum state diffusion model applied to open systems. Journal of Physics A: Mathematical and General 25, 5677-5691.

    Google Scholar 

  • Gisin, N. and Percival, I. C. (1993). The quantum state diffusion picture of physical processes. Journal of Physics A: Mathematical and General 26, 2245-2260.

    Google Scholar 

  • Goetsch, P. and Graham, R. (1993). Quantum trajectories for nonlinear optical processes. Annalen der Physik 2, 708-719.

    Google Scholar 

  • Goetsch, P. and Graham, R. (1994). Linear stochastic wave equation for continuously measured quantum systems. Physical Review A 50, 5242-5255.

    Google Scholar 

  • Goetsch, P., Graham, R., and Haake, F. (1995). Schrödinger cat and single runs for the damped harmonic oscillator. Physical Review A 51(1), 136-142.

    Google Scholar 

  • Hudson, R. L. and Parthasarathy, K. R. (1984). Quantum itô's formula and stochastic evolution. Communications in Mathematical Physics 93, 301-323.

    Google Scholar 

  • Itô, K. (1951). On a formula concerning stochastic differentials. Nagoya Mathematical Journal 3, 55-65.

    Google Scholar 

  • Kolokoltsov, V. N. (1995). Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate. Journal of Mathematics and Physics 36(6), 2741-2760.

    Google Scholar 

  • Lindblad, G. (1976). On the generators of quantum stochastic semigroups. Communications of Mathematical Physics 48, 119-130.

    Google Scholar 

  • Persival, I. (1999). Quantum State Diffusion, Cambridge University Press, Cambridge, UK.

    Google Scholar 

  • Schrödinger, E. (1931). Sitzberg Press Akad. Wiss. Phys-Math. Kl. 144-153.

  • Stratonovich, R. L. (1966). Conditional Markov Processes and Their Applications to Optimal Control. Moscow State University, Moscow.

    Google Scholar 

  • Wigner, E. (1963). American Journal of Physics 31, 6.

    Google Scholar 

  • Wiseman, H. M. and Milburn, G. J. (1993). Physical Review A 47, 642.

    Google Scholar 

  • Wiseman, H. M. and Milburn, G. J. (1994). Physical Review A 49, 1350.

    Google Scholar 

Download references

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Authors and Affiliations

  1. Mathematics Department, University of Nottingham, Nottingham, NG7 2RD, Nottinghamshire, United Kingdom

    V. P. Belavkin

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Belavkin, V.P. Quantum Trajectories, State Diffusion, and Time-Asymmetric Eventum Mechanics. International Journal of Theoretical Physics 42, 2461–2485 (2003). https://doi.org/10.1023/B:IJTP.0000005969.92111.b5

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