In quantum probability, the Belavkin equation, also known as Belavkin-Schrödinger equation, quantum filtering equation, stochastic master equation, is a quantum stochastic differential equation describing the dynamics of a quantum system undergoing observation in continuous time. It was derived and henceforth studied by Viacheslav Belavkin in 1988.[1][2][3]

Overview

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Unlike the Schrödinger equation, which describes the deterministic evolution of the wavefunction   of a closed system (without interaction), the Belavkin equation describes the stochastic evolution of a random wavefunction   of an open quantum system interacting with an observer:

 

Here,   is a self-adjoint operator (or a column vector of operators) of the system coupled to the external field,   is the Hamiltonian,   is the imaginary unit,   is the Planck constant, and   is a stochastic process representing the measurement noise that is a martingale with independent increments with respect to the input probability measure  . Note that this noise has dependent increments with respect to the output probability measure   representing the output innovation process (the observation). For  , the equation becomes the standard Schrödinger equation.

The stochastic process   can be a mixture of two basic types: the Poisson (or jump) type  , where   is a Poisson process corresponding to counting observation, and the Brownian (or diffusion) type  , where   is the standard Wiener process corresponding to continuous observation. The equations of the diffusion type can be derived as the central limit of the jump type equations with the expected rate of the jumps increasing to infinity.

The random wavefunction   is normalized only in the mean-squared sense  , but generally   fails to be normalized for each  . The normalization of   for each   gives the random posterior state vector  , the evolution of which is described by the posterior Belavkin equation, which is nonlinear, because operators   and   depend on   due to normalization. The stochastic process   in the posterior equation has independent increments with respect to the output probability measure  , but not with respect to the input measure. Belavkin also derived linear equation for unnormalized density operator   and the corresponding nonlinear equation for the normalized random posterior density operator  . For two types of measurement noise, this gives eight basic quantum stochastic differential equations. The general forms of the equations include all types of noise and their representations in Fock space.[4] [5]

The nonlinear equation describing observation of position of a free particle, which is a special case of the posterior Belavkin equation of the diffusion type, was also obtained by Diosi[6] and appeared in the works of Gisin,[7] Ghirardi, Pearle and Rimini,[8] although with a rather different motivation or interpretation. Similar nonlinear equations for posterior density operators were postulated (although without derivation) in quantum optics and the quantum trajectories theory,[9] where they are called stochastic master equations. The averaging of the equations for the random density operators   over all random trajectories   leads to the Lindblad equation,[10] which is deterministic.

The nonlinear Belavkin equations for posterior states play the same role as the Stratonovich–Kushner equation in classical probability, while the linear equations correspond to the Zakai equation.[11] The Belavkin equations describe continuous-time decoherence of initially pure state   into a mixed posterior state   giving a rigorous description of the dynamics of the wavefunction collapse due to an observation or measurement.[12] [13] [14]

Non-demolition measurement and quantum filtering

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Noncommutativity presents a major challenge for probabilistic interpretation of quantum stochastic differential equations due to non-existence of conditional expectations for general pairs of quantum observables. Belavkin resolved this issue by discovering the error-perturbation uncertainty relation and formulating the non-demolition principle of quantum measurement.[13][15] In particular, if the stochastic process   corresponds to the error   (white noise in the diffusive case) of a noisy observation   of operator   with the accuracy coefficient  , then the indirect observation perturbs the dynamics of the system by a stochastic force  , called the Langevin force, which is another white noise of intensity   that does not commute with the error  . The result of such a perturbation is that the output process   is commutative  , and hence   corresponds to a classical observation, while the system operators   satisfy the non-demolition condition: all future observables must commute with the past observations (but not with the future observations):   for all   (but not  ). Note that commutation of   with   and another operator   with   does not imply commutation of   with  , so that the algebra of future observables is still non-commutative. The non-demolition condition is necessary and sufficient for the existence of conditional expectations  , which makes the quantum filtering possible.[16]

Posterior state equations

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Counting observation

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Let   be a Poisson process with forward increments   almost everywhere and   otherwise and having the property  . The expected number of events is  , where   is the expected rate of jumps. Then substituting   for the stochastic process   gives the linear Belavkin equation for the unnormalized random wavefunction   undergoing counting observation. Substituting  , where   is the collapse operator, and  , where   is the energy operator, this equation can be written in the following form

 

Normalized wavefunction   is called the posterior state vector, the evolution of which is described by the following nonlinear equation

 

where   has expectation  . The posterior equation can be written in the standard form

 

with  ,  , and  . The corresponding equations for the unnormalized random density operator   and for the normalized random posterior density operator   are as follows

 

where  . Note that the latter equation is nonlinear.

Continuous observation

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Stochastic process  , defined in the previous section, has forward increments  , which tend to   as  . Therefore,   becomes standard Wiener process with respect to the input probability measure. Substituting   for   gives the linear Belavkin equation for the unnormalized random wavefunction   undergoing continuous observation. The output process   becomes the diffusion innovation process   with increments  . The nonlinear Belavkin equation of the diffusion type for the posterior state vector   is

 

with   and  . The corresponding equations for the unnormalized random density operator   and for the normalized random posterior density operator   are as follows

 

where  . The second equation is nonlinear due to normalization. Because  , taking the average of these stochastic equations over all   leads to the Lindblad equation

 

Example: continuous observation of position of a free particle

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Consider a free particle of mass  . The position and momentum observables correspond respectively to operators   of multiplication by   and  . Making the following substitutions in the Belavkin equation

 

the posterior stochastic equation becomes

 

where   is the posterior expectation of  . Motivated by the spontaneous collapse theory rather than the filtering theory, this equation was also obtained by Diosi,[17] showing that the measurement noise   is the increment   of a standard Wiener process. There are closed-form solutions to this equation,[18] as well as equations for a particle in a linear or quadratic potentials.[1][3][19] For a Gaussian initial state   these solutions correspond to optimal quantum linear filter.[15] Solutions to the Belavkin equation show that in the limit   the wavefunction has finite dispersion,[20] therefore resolving the quantum Zeno effect.[11]

References

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  1. ^ a b Belavkin, V.P. (1988). "Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes". In A. Blaquiere (ed.). Proc of Bellmann Continuum Workshop 'Modelling and Control of Systems'. Lecture notes in Control and Inform Sciences. Vol. 121. Sophia-Antipolis: Springer-Verlag. pp. 245–265.
  2. ^ Belavkin, V.P. (1989). "A continuous counting observation and posterior quantum dynamics". J Phys A. 22 (23): L1109–L1114. Bibcode:1989JPhA...22L1109B. doi:10.1088/0305-4470/22/23/006.
  3. ^ a b Belavkin, V.P. (1989). "A new wave equation for a continuous nondemolition measurement". Physics Letters A. 140 (7–8): 355–358. arXiv:quant-ph/0512136. Bibcode:1989PhLA..140..355B. doi:10.1016/0375-9601(89)90066-2. S2CID 6083856.
  4. ^ Belavkin, V.P. (1995). "On stochastic generators of completely positive cocycles". Russ Journ of Math Phys. 3 (4): 523–528.
  5. ^ Belavkin, V.P. (1997). "Quantum stochastic positive evolutions: characterization, construction, dilation". Commun. Math. Phys. 184 (3): 533–566. arXiv:math-ph/0512042. Bibcode:1997CMaPh.184..533B. doi:10.1007/s002200050072. S2CID 17593922.
  6. ^ Di'osi, L. (1989). "Models for universal reduction of macroscopic quantum fluctuations". Physical Review A. 40 (3): 1165–1174. Bibcode:1989PhRvA..40.1165D. doi:10.1103/PhysRevA.40.1165. PMID 9902248.
  7. ^ Gisin, N. (1989). "Stochastic quantum dynamics and relativity". Helvetica Physica Acta. 62: 363–371.
  8. ^ 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 (1): 78–89. Bibcode:1990PhRvA..42...78G. doi:10.1103/PhysRevA.42.78. PMID 9903779.
  9. ^ Carmichael, H.J. (1993). An Open Systems Approach to Quantum Optics. Springer-Verlag.
  10. ^ Smolyanov, O.; Truman, A. (1999). "Schrödinger-Belavkin equations and associated Kolmogorov and Lindblad equations". Theoretical and Mathematical Physics. 120 (2): 973–984. Bibcode:1999TMP...120..973S. doi:10.1007/BF02557405. S2CID 121436901.
  11. ^ a b Holevo, A.S. (1991). "Quantum probability and quantum statistics". In Prokhorov, Y.V. (ed.). Probability Theory~8. Itogi Nauki i Tekhniki (in Russian). Vol. 83. VINITI. pp. 5–132.
  12. ^ Belavkin, V.P. (1990). "A quantum posterior stochastics and spontaneous collapse". In Truman, A.; Davies, I.M. (eds.). Stochastics and Quantum Mechanics. World Scientific. pp. 40–68.
  13. ^ a b Belavkin, V.P. (1992). "Quantum continual measurements and a posteriori collapse on CCR". Comm. Math. Phys. 146 (3): 611–635. arXiv:math-ph/0512070. Bibcode:1992CMaPh.146..611B. doi:10.1007/BF02097018. S2CID 17016809.
  14. ^ Belavkin, V.P.; Melsheimer, O. (1995). "A hamiltonian solution to quantum collapse, state diffusion and spontaneous localization". Quantum Communications and Measurement. Plenum Publisher. pp. 201–222. doi:10.1007/978-1-4899-1391-3_20. ISBN 978-1-4899-1393-7.
  15. ^ a b Belavkin, V.P. (1980). "Optimal filtering of Markov signals with quantum white noise". Radio Eng Electron Physics. 25: 1445–1453. arXiv:quant-ph/0512091. doi:10.1007/978-1-4899-1391-3_37. S2CID 15021588.
  16. ^ Bouten, L.; van Handel, R.; James, M.R. (2009). "A discrete invitation to quantum filtering and feedback control". SIAM Review. 51 (2): 239–316. arXiv:math/0606118. Bibcode:2009SIAMR..51..239B. doi:10.1137/060671504. S2CID 10435983.
  17. ^ Diosi, L. (1988). "Continuous quantum measurement and Itô formalism". Phys Lett A. 129 (8–9): 419–423. arXiv:1812.11591. Bibcode:1988PhLA..129..419D. doi:10.1016/0375-9601(88)90309-X. S2CID 118831121.
  18. ^ Diosi, L. (1988). "Localized solution of simple nonlinear quantum Langevin-equation". Phys Lett A. 132 (5): 233–236. Bibcode:1988PhLA..132..233D. doi:10.1016/0375-9601(88)90555-5.
  19. ^ Belavkin, V.P.; Staszewski, P. (1992). "Nondemolition observation of a free quantum particle". Physical Review A. 45 (3): 1347–1357. arXiv:quant-ph/0512138. Bibcode:1992PhRvA..45.1347B. doi:10.1103/PhysRevA.45.1347. PMID 9907114. S2CID 14637898.
  20. ^ Kolokol'tsov1, V.N. (1995). "Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate". Journal of Mathematical Physics. 36 (6): 2741–2760. Bibcode:1995JMP....36.2741K. doi:10.1063/1.531063.{{cite journal}}: CS1 maint: numeric names: authors list (link)