Speaker
Description
The talk is devoted to one method to derive the equation of stochastic dynamics of a quantum system for a wave function and its applications in quantum information theory. The main example of stochastic dynamics is the process of continuous-time weak measurements of an observable $L=L^*$ described
by the stochastic differential Belavkin equation
$ d|\psi_t\rangle =-iH|\psi_t\rangle dt-\frac{1}{2}L^2|\psi_t\rangle dt+L|\psi_t\rangle dW_t.$
Here $|\psi_t\rangle =|\psi_t(\omega)\rangle \in \mathcal{H}$ is a wave function, as a random vector process on the probability space $\Omega$, $H$ is a Hamiltonian, $W_t$ is a real Wiener process.
In my talk we obtain a generalization of the Belavkin equation by writing the equation not for a random vector $|\psi_t\rangle$, but for its distribution. Thus, we get rid of stochasticity and deal with Markov semigroups and their generators. The conditions of approximation of semigroups of diffusion processes corresponding to a continuous random process in Hilbert space by discrete ones are investigated. Formally, the semigroup operators are considered on the space of bounded weakly continuous functions $C_{BW}(\mathcal{H})$ on the Hilbert space, which carry out the evolution of the characteristic functions $\varphi_t(v)=\mathbb{E} e^{i\langle v|\psi_t \rangle_{\mathbb{R}}}$ of the process. The proof uses the developed powerful theory of continuous and bi-continuous semigroups on a Banach space.
In quantum physics, stochastic processes continuous in time find interesting applications: from models of collisions of particles in a gas to random circuits.
