This paper is devoted to the aspects of derivation of dynamics equation of quantum system under a stochastic dynamics. We study the conditions, under which a sequence of random variations of wave function can approximate a random diffusion process in a Hilbert space. A random variation $|\psi_0\rangle \mapsto G_{t_N}\ldots G_{t_1}|\psi_0\rangle =|\psi_{t_N}\rangle $ is associated with a transform of distribution of vector $|\psi\rangle $, as well as with the variation of its characteristic functional $\varphi(v)=\mathbb{E}\exp(i\operatorname{Re}\langle v|\psi\rangle ).$ For a continuous random walk we study the approximation of a Markov semigroup by the Markov operators of discrete random walk. We pay a special attention to the cases, when the derivative of random operator $F'(0)$ is an unbounded operator. However, we restrict the consideration to the case when the Markov operators of random walks with the operators $G_t$ mutually commute. The characteristic functional is transformed by the Markov operator of adjoint process, and in contrast to the dynamics of wave function, it has a deterministic nature that allows us to rely on the developed theory of semigroups in Banach spaces. The most illustrative examples are the process of continuous measurement, that is, the measurement of trajectories of some observable, and the random control.This paper is devoted to the aspects of derivation of dynamics equation of quantum system under a stochastic dynamics. We study the conditions, under which a sequence of random variations of wave function can approximate a random diffusion process in a Hilbert space. A random variation $|\psi_0\rangle \mapsto G_{t_N}\ldots G_{t_1}|\psi_0\rangle =|\psi_{t_N}\rangle $ is associated with a transform of distribution of vector $|\psi\rangle $, as well as with the variation of its characteristic functional $\varphi(v)=\mathbb{E}\exp(i\operatorname{Re}\langle v|\psi\rangle ).$ For a continuous random walk we study the approximation of a Markov semigroup by the Markov operators of discrete random walk. We pay a special attention to the cases, when the derivative of random operator $F'(0)$ is an unbounded operator. However, we restrict the consideration to the case when the Markov operators of random walks with the operators $G_t$ mutually commute. The characteristic functional is transformed by the Markov operator of adjoint process, and in contrast to the dynamics of wave function, it has a deterministic nature that allows us to rely on the developed theory of semigroups in Banach spaces. The most illustrative examples are the process of continuous measurement, that is, the measurement of trajectories of some observable, and the random control.

Ufimskii Matematicheskii ZhurnalJan 1, 20255 citations

Banach and counting measures, and dynamics of singular quantum states generated by averaging of operator random walks

E. A. Dzhenzher, S. V. Dzhenzher, V. Zh. Sakbaev

In this paper the random channels and their compositions in the space of quantum states are studied. For compositions of i.i.d. random unitary channels, the limit behaviour of probability distributions is described. The sufficient condition for convergence in probability is obtained. The generalized convergence in distribution w.r.t. weak operator topology is obtained. The analysis of transmission of pure and normal states to the set of singular states is done. The dynamics of quantum states is described in terms of the evolution of the values of quadratic forms of operators from the algebra that implements the representation of canonical commutation relations.In this paper the random channels and their compositions in the space of quantum states are studied. For compositions of i.i.d. random unitary channels, the limit behaviour of probability distributions is described. The sufficient condition for convergence in probability is obtained. The generalized convergence in distribution w.r.t. weak operator topology is obtained. The analysis of transmission of pure and normal states to the set of singular states is done. The dynamics of quantum states is described in terms of the evolution of the values of quadratic forms of operators from the algebra that implements the representation of canonical commutation relations.

quant-pharXiv:2603.25151Lobachevskii Journal of MathematicsMar 27, 20260 citations

Quantum random walk polynomial and quantum random walk measure

Yuanbao Kang, Caishi Wang

Quantum Information ProcessingJan 4, 20142 citations

Quantum random walks

Luigi Accardi, G. S. Watson

Lecture notes in mathematicsJan 1, 198923 citations

The convergence of unitary quantum random walks

Alexander C. R. Belton, Michał Gnacik, J. Martin Lindsay

We give a simple and direct treatment of the convergence of quantum random walks to quantum stochastic operator cocycles, using the semigroup method. The pointwise product of two such quantum random walks is shown to converge to the quantum stochastic Trotter product of the respective limit cocycles. Since such Trotter products themselves reduce to pointwise products when the cocycles inhabit commuting subspaces of the system algebra, this yields an elementary approach to the quantum random walk approximation of the 'tensorisation' of cocycles with common noise dimension space. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.We give a simple and direct treatment of the convergence of quantum random walks to quantum stochastic operator cocycles, using the semigroup method. The pointwise product of two such quantum random walks is shown to converge to the quantum stochastic Trotter product of the respective limit cocycles. Since such Trotter products themselves reduce to pointwise products when the cocycles inhabit commuting subspaces of the system algebra, this yields an elementary approach to the quantum random walk approximation of the 'tensorisation' of cocycles with common noise dimension space. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.

Apr 25, 20141 citation

\varphi(v)=\mathbb{E}\exp(i\operatorname{Re}\langle v|\psi\rangle ).

For a continuous random walk we study the approximation of a Markov semigroup by the Markov operators of discrete random walk. We pay a special attention to the cases, when the derivative of random operator

F'(0)

is an unbounded operator. However, we restrict the consideration to the case when the Markov operators of random walks with the operators

G_t

mutually commute. The characteristic functional is transformed by the Markov operator of adjoint process, and in contrast to the dynamics of wave function, it has a deterministic nature that allows us to rely on the developed theory of semigroups in Banach spaces. The most illustrative examples are the process of continuous measurement, that is, the measurement of trajectories of some observable, and the random control.