Belavkin equation

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.
Overview.
Unlike the Schrödinger equation, which describes the deterministic evolution of the wavefunction formula_1 of a closed system (without interaction), the Belavkin equation describes the stochastic evolution of a random wavefunction formula_2 of an open quantum system interacting with an observer:
Here, formula_3 is a self-adjoint operator (or a column vector of operators) of the system coupled to the external field, formula_4 is the Hamiltonian, formula_5 is the imaginary unit, formula_6 is the Planck constant, and formula_7 is a stochastic process representing the measurement noise that is a martingale with independent increments with respect to the input probability measure formula_8. Note that this noise has dependent increments with respect to the output probability measure formula_9 representing the output innovation process (the observation). For formula_10, the equation becomes the standard Schrödinger equation.
The stochastic process formula_11 can be a mixture of two basic types: the "Poisson" (or "jump") type formula_12, where formula_13 is a Poisson process corresponding to counting observation, and the "Brownian" (or "diffusion") type formula_14, where formula_15 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 formula_2 is normalized only in the mean-squared sense formula_17, but generally formula_2 fails to be normalized for each formula_19. The normalization of formula_2 for each formula_19 gives the random "posterior state vector" formula_22, the evolution of which is described by the posterior Belavkin equation, which is nonlinear, because operators formula_3 and formula_4 depend on formula_25 due to normalization. The stochastic process formula_11 in the posterior equation has independent increments with respect to the output probability measure formula_9, but not with respect to the input measure. Belavkin also derived linear equation for unnormalized density operator formula_28 and the corresponding nonlinear equation for the normalized random posterior density operator formula_29. 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.
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 and appeared in the works of Gisin, Ghirardi, Pearle and Rimini, 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, where they are called "stochastic master equations". The averaging of the equations for the random density operators formula_30 over all random trajectories formula_19 leads to the Lindblad equation, 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. The Belavkin equations describe continuous-time decoherence of initially pure state formula_32 into a mixed posterior state formula_33 giving a rigorous description of the dynamics of the wavefunction collapse due to an observation or measurement.
Non-demolition measurement and quantum filtering.
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. In particular, if the stochastic process formula_34 corresponds to the error formula_35 (white noise in the diffusive case) of a noisy observation formula_36 of operator formula_37 with the accuracy coefficient formula_38, then the indirect observation perturbs the dynamics of the system by a stochastic force formula_39, called the "Langevin force", which is another white noise of intensity formula_40 that does not commute with the error formula_41. The result of such a perturbation is that the output process formula_42 is commutative formula_43, and hence formula_44 corresponds to a classical observation, while the system operators formula_45 satisfy the non-demolition condition: all future observables must commute with the past observations (but not with the future observations): formula_46 for all formula_47 (but not formula_48). Note that commutation of formula_49 with formula_42 and another operator formula_51 with formula_42 does not imply cummutation of formula_49 with formula_51, 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 formula_55, which makes the quantum filtering possible.
Posterior state equations.
Counting observation.
Let formula_13 be a Poisson process with forward increments formula_57 almost everywhere and formula_58 otherwise and having the property formula_59. The expected number of events is formula_60, where formula_61 is the expected rate of jumps. Then substituting formula_62 for the stochastic process formula_11 gives the linear Belavkin equation for the unnormalized random wavefunction formula_2 undergoing counting observation. Substituting formula_65, where formula_66 is the collapse operator, and formula_67, where formula_68 is the energy operator, this equation can be written in the following form
Normalized wavefunction formula_22 is called the "posterior state vector", the evolution of which is described by the following nonlinear equation
where formula_70 has expectation formula_71. The posterior equation can be written in the standard form
with formula_73, formula_74, and formula_75. The corresponding equations for the unnormalized random density operator formula_28 and for the normalized random posterior density operator formula_29 are as follows
-\rho\right)d\tilde n
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where formula_78. Note that the latter equation is nonlinear.
Continuous observation.
Stochastic process formula_79, defined in the previous section, has forward increments formula_80, which tend to formula_81 as formula_82. Therefore, formula_79 becomes standard Wiener process with respect to the input probability measure. Substituting formula_15 for formula_11 gives the linear Belavkin equation for the unnormalized random wavefunction formula_2 undergoing continuous observation. The output process formula_87 becomes the diffusion innovation process formula_88 with increments formula_89. The nonlinear Belavkin equation of the diffusion type for the posterior state vector formula_22 is
with formula_91 and formula_92. The corresponding equations for the unnormalized random density operator formula_28 and for the normalized random posterior density operator formula_29 are as follows
where formula_95. The second equation is nonlinear due to normalization. Because formula_96, taking the average of these stochastic equations over all formula_19 leads to the Lindblad equation
Example: continuous observation of position of a free particle.
Consider a free particle of mass formula_99. The position and momentum observables correspond respectively to operators formula_100 of multiplication by formula_101 and formula_102. Making the following substitutions in the Belavkin equation
the posterior stochastic equation becomes
where formula_105 is the posterior expectation of formula_100. Motivated by the spontaneous collapse theory rather than the filtering theory, this equation was also obtained by Diosi, showing that the measurement noise formula_107 is the increment formula_108 of a standard Wiener process. There are closed-form solutions to this equation, as well as equations for a particle in a linear or quadratic potentials. For a Gaussian initial state formula_109 these solutions correspond to optimal quantum linear filter. Solutions to the Belavkin equation show that in the limit formula_110 the wavefunction has finite dispersion, therefore resolving the quantum Zeno effect.