A brief presentation of the basic concepts in quantum probability theory is given in comparison to the classical one. The notion of quantum white noise, its explicit representation in Fock space, and necessary results of noncommutative stochastic analysis and integration are outlined. Algebraic differential equations that unify the quantum non Markovian diffusion with continuous non demolition observation are derived. A stochastic equation of quantum diffusion filtering generalising the classical Markov filtering equation to the quantum flows over arbitrary *-algebra is obtained. A Gaussian quantum diffusion with one dimensional continuous observation is considered.The a posteriori quantum state difusion in this case is reduced to a linear quantum stochastic filter equation of Kalman-Bucy type and to the operator Riccati equation for quantum correlations. An example of continuous nondemolition observation of the coordinate of a free quantum particle is considered, describing a continuous collase to the stationary solution of the linear quantum filtering problem found in the paper.
A brief presentation is given of the basic concepts in quantum probability theory in analogy to the classical ones, as well as the necessary information about noncommutative stochastic integration and its explicit representation in the Fock space. Algebraic differential relations that define quantum diffusion motion with nondemolition observation are derived. In the Markov case we obtain the stochastic equation of quantum diffusion filtering which is analogous to the Zakai equation of classical nonlinear filtering. A quantum linear stochastic model with continuous observation is given for which this equation is reduced to a linear stochastic quantum filtering equation of Kalman-Busi type and to the operator Riccati equation. We also consider an example of observing the coordinate of a free quantum particle, for which the solution of a stationary quantum filtering problem is obtained.
Stationary quantum stochastic process j is introduced as a *-homomorphism embedding an involutive graded algebra
K~=⊕i=1∞Ki into a ring of (abelian) cohomologies of the one-parameter group
α consisting of *-automorphisms of certain operator algebra in a Hilbert space such that every x from
Ki is translated into an additive
i−α-cocycle j(x). It is shown that (noncommutative) multiplicative markovian cocycle defines a perturbation of the stationary quantum stochastic process in the sense of such definition. The
E0-semigroup
β~ on the von Neumann algebra
N associated with the markovian perturbation of K-flow j posseses the restriction
β~∣N0,N0⊂N, which is conjugate to the flow of Powers shifts
β associated with j. It yields for
β~ an analogue of the Wold decomposition for classical stochastic process on completely nondeterministic and deterministic parts. The examples of quantum stationary stochastic processes on the algebras of canonical commutation, anticommutation and square of white noise relations are considered. In the model situation of the space
L2(R) all markovian cocycles of the group of shifts are described up to unitary equivalence of perturbations.
The mathematical model describing a light beam prepared in an arbitrary quantum optical state is a quasifree quantum stochastic process on the C* algebra of the canonical commutatation relations. For such quantum stochastic processes the concept of stochastic states is introduced. Stochastic quantum states have a classical analog in the following sense: If the light beam is prepared in a stochastic state, one can construct a generalized classical stochastic process, such that the distributions of the quantum observables and the classical random variables coincide. A sufficient algebraic condition for the stochasticity of a quantum state is formulated. The introduced formalism generalizes the Wigner representation from a single field mode to a continuum of modes. For the special case of a single field mode the stochasticity condition provides a new criterion for the positivity of the Wigner function related to the given state. As an example the quantized eletromagnetic field in empty space at temperature T=0 is discussed. It turns out that the corresponding classical stochastic process is not a white noise but a colored noise with a linearly increasing spectrum.
We define a stationary Gaussian quantum stochastic process (GQSP) on the C*-algebra of the canonical commutation relations over a real symplectic Hilbert space. Physically GQSPs can describe diffusion with non-Markovian memory effects in quantum harmonic oscillators of arbitrary dimension. We find that in analogy with the commutative case a GQSP is completely specified by a operator-valued autocovariance function satisfying certain positive definiteness and reality conditions. The autocovariance function also determines the response of the system to a class of time-dependent generalized forces, and it has a spectral representation in terms of a positive operator-valued measure.
In this chapter, we define and explore basic properties of a quantum Markov semigroup { I t , t ≥ 0} of linear maps on the C* -algebra or von Neumann algebra A that characterizes the quantum system. The quantum Markov semigroup (QMS) plays a key role in describing quantum Markov processes that are to be explored in the subsequent chapters. The concept of QMS extends the semigroup of probability transition operators { T t , t ≥ 0} for a classical Markov process to its noncommutative counterpart. Suppose the classical Markov process { X t , t ≥ 0} is defined on the complete filtered classical probability space (Ω, Ƒ, P ; {Ƒ t , t ≥ 0}) and with values in a measurable space (핏, B (X)). Recall that for each t ≥ 0, the probability transition operator T t : L ∞ (X, B (X)) → L ∞ (X, B (X)) by ( T t f )( X s ) = E x [ f ( X s + t ) | Ƒ s ] ∀ s , t ≥ 0. A semigroup of linear maps on the C* -algebra or von Neumann algebra A is said to be a quantum dynamical semigroup (QDS) if (i) I 0 = J (the identity operator on A ); (ii) I t I s = I t + s for all t , s ≥ 0; (iii) I t is completely positive for each t ≥ 0; and (iv) I t is σ weakly continuous on A , i.e., a ↦ tr ( ρI t ( a )) is continuous from A to ℂ for each ρ ∈ S ( A ) (the space of quantum states) and for each t ≥ 0. If in addition I t ( I ) = I , (respectively, I t ( I ) ≤ I ) for all t ≥ 0, then the QDS is said to be a quantum Makrov semigroup (QMS) (respectively, quantum sub-Markov semigroup).