Abstract
We give a common mathematical characterization of relevant stochastic evolution schemes built up in the literatute to attack the quantum measurement problem. This characterization is based on two hypotheses, namely, (i) the trace conservation with probability one and (ii) the existence of a complex phase determining a linear support for the stochastic process driving the random evolution.
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References
d’Espagnat B. (1976). Conceptual Foundations of Quantum Mechanics, 2nd comp. rev. and enl. edn. Addison-Wesley, New York
Santos E. (1991). “Does quantum mechanics violate the Bell inequalities”. Phys. Rev. Lett. 66: 1388
Wheeler J.A., Zurek W.H. ed. (1983). Quantum Theory and Measurement. Princeton University Press, Princeton
Cini M., Lévy-Leblond J.-M. ed. (1990). Quantum Theory without Reduction. Adam Hilger, New York
Peres A. (1993). Quantum Theory: Concepts and Methods. Kluwer Academic, Dordrecht
Bub J. (1999). Interpreting the Quantum World. Cambridge University Press, Cambridge
de la Peña L., Cetto A.M. (1996). The Quantum Dice. An Introduction to Stochastic Electrodynamics. Kluwer Academic, Dordrecht
Gisin N. (1984). “Quantum measurements and stochastic processes”. Phys. Rev. Lett. 52: 1657
Ghirardi G.C., Rimini A., Weber T. (1986). “Unified dynamics for microscopic and macroscopic systems”. Phys. Rev. D 34, 470
Pearle P. (1989). “Combining stochastic dynamical state-vector reduction with spontaneous localization”. Phys. Rev. A 39: 2277
Ghirardi G.C., Pearle P., Rimini A. (1990). “Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles”. Phys. Rev. A 42, 78
Diosi L. (1989). “Models for universal reduction of macroscopic quantum fluctuations”. Phys. Rev. A 40: 1165
Gisin N., Percival I.C. (1992). “The quantum state diffusion model applied to open quantum systems”. J. Phys. A: Math. Gen. 25: 5677
Pearle P. “Collapse models,” in Open Systems and Measurement in Relativistic Quantum Theory, F. Petruccione and H.P. Breuer, eds. (Springer, Berlin, 2000).
Percival I.C. (1998). Quantum State Diffusion. Cambridge University Press, Cambridge
Bassi A., Ghirardi G.-C. (2003). “Dynamical reduction models”. Phys. Rep. 379: 257
Dalibard J., Castin Y., Molmer K. (1992). “Wave-function approach to dissipative processes in quantum optics”. Phys. Rev. Lett. 68, 580
Gardiner C.W., Parkins A.S., Zoller P. (1992). “Wave-function quantum stochastic differential equations and quantum-jump simulation methods”. Phys. Rev. A 46: 4363
Dum R., Parkins A.S., Zoller P., Gardiner C.W. (1992). “Monte Carlo simulation of master equations in quantum optics for vacuum, thermal, and squeezed reservoirs”. Phys. Rev. A 46: 4382
Adler S.L., Brody D.C., Brun T.A., Hughston L.P. (2001). “Martingale models for quantum state reduction”. J. Phys. A 34: 8795
Dirac P.A.M. (1958). The Principles of Quantum Mechanics, 4th rev. edn. Clarendon Press, Oxford
Øksendal B. (1998). Stochastic Differential Equations, 5th edn. Springer, Berlin
Hudson R.L., Parthasarathy K.R. (1984). “Quantum Ito’s formula and stochastic evolutions”. Commun. Math. Phys. 93: 301
Belavkin V.P. (1989). “A continuous counting observation and posterior quantum dynamics”. J. Phys. A: Math. Gen. 22: L1109
Parthasarathy K.R. (1992). An Introduction to Quantum Stochastic Calculus. Birkhäuser, Berlin
G. W. Johnson and G. Kallianpur, “Stochastic Dyson series and the solution to associated stochastic evolution equations,” in Stochastic Analysis and Mathematical Physics, R. Rebolledo, ed., Proceedings of the 2nd Intl. Workshop (World Scientific, Singapore, 1996), p. 82.
Karatzas I., Shrève S.E. (1991). Brownian Motion and Stochastic Calculus, Graduate Text in Mathematics, 2nd edn. Springer, Berlin
Bassi A., Ghirardi G.-C. (2002). “Dynamical reduction models with general gaussian noises”. Phys. Rev. A 65: 042114
Gardiner C.W. (1995). Handbook of Stochastic Methods. Springer, Berlin
Lindblad G. (1976). “On the generators of quantum dynamical semigroups”. Commun. Math. Phys. 48: 119
Alicki R., Lendi K. (1987). Quantum Dynamical Semigroups and Applications, Lecture Notes in Physics 286 Springer, Berlin
Diosi L., Wiseman H.M. (2001). “Complete parameterization and invariance of diffusive quantum trajectories for Markovian open systems”. J. Chem. Phys. 268: 91
Gisin N. (1989). “Stochastic quantum dynamics and relativity”. Helv. Phys. Acta 62: 363
Hughston L.P. (1996). “Geometry of stochastic state vector reduction”. Proc. R. Soc. Lond. A 452: 953
Adler S.L., Horwitz L.P. (2000). “Structure and properties of Hughston’s stochastic extension of the Schrödinger equation”. J. Math. Phys. 41: 2485
Gisin N. (1981). “A simple nonlinear dissipative quantum evolution equation”. J. Phys. A: Math. Gen. 14: 2259
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This paper is dedicated to Prof. Emilio Santos on the occasion of his 70th birthday.
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Salgado, D., Sánchez-Gómez, J.L. & Ferrero, M. A Mathematical Characterization of Quantum Gaussian Stochastic Evolution Schemes. Found Phys 36, 526–540 (2006). https://doi.org/10.1007/s10701-005-9034-8
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DOI: https://doi.org/10.1007/s10701-005-9034-8