[1][2][3] 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
Note that this noise has dependent increments with respect to the output probability measure
is a Poisson process corresponding to counting observation, and the Brownian (or diffusion) type
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 evolution of which is described by the posterior Belavkin equation, which is nonlinear, because operators
in the posterior equation has independent increments with respect to the output probability 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.
[11] The Belavkin equations describe continuous-time decoherence of initially pure state
giving a rigorous description of the dynamics of the wavefunction collapse due to an observation or measurement.
[12] [13] [14] 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.
(white noise in the diffusive case) of a noisy observation
, 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
gives the linear Belavkin equation for the unnormalized random wavefunction
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
becomes standard Wiener process with respect to the input probability measure.
gives the linear Belavkin equation for the unnormalized random wavefunction
The nonlinear Belavkin equation of the diffusion type for the posterior state vector
The position and momentum observables correspond respectively to operators
these solutions correspond to optimal quantum linear filter.
the wavefunction has finite dispersion,[20] therefore resolving the quantum Zeno effect.