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Computational Mathematics and Mathematical Physics

Dynamics of Singular Quantum States Generated by Averaging Random Shifts

Vladimir Mikhailovich Busovikov, V. Zh. Sakbaev

Published Aug 1, 2026• 0 citations
Publisher

Abstract

Abstract Random walks in a real separable Hilbert space correspond to a composition of random unitary operators in the space of functions square integrable with respect to a translation-invariant measure, as well as to a composition of random unitary channels in the space of quantum states. Conditions on the distribution of random walks that are necessary and sufficient for the continuity of the mathematical expectations of their representations in the Hilbert space and the space of quantum states are obtained. The dynamics of the set of quantum states obtained by averaging random shifts over the semigroup (with respect to the convolution operation) of centered Gaussian measures on the Hilbert space is studied. It is shown that, depending on the property of the covariance operator of the Gaussian measure to have a nuclear square root, such dynamics is either a strongly continuous quantum dynamical semigroup or instantaneously destroys the set of normal states. In the latter case, the dynamics of singular quantum states is described in terms of the evolution of mean values on an operator algebra that implements a representation of canonical commutation relations.

DOI
10.1134/s0965542526700508

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Abstract Random walks in a real separable Hilbert space correspond to a composition of random unitary operators in the space of functions square integrable with respect to a translation-invariant measure, as well as to a composition of random unitary channels in the space of quantum states. Conditions on the distribution of random walks that are necessary and sufficient for the continuity of the mathematical expectations of their representations in the Hilbert space and the space of quantum states are obtained. The dynamics of the set of quantum states obtained by averaging random shifts over the semigroup (with respect to the convolution operation) of centered Gaussian measures on the Hilbert space is studied. It is shown that, depending on the property of the covariance operator of the Gaussian measure to have a nuclear square root, such dynamics is either a strongly continuous quantum dynamical semigroup or instantaneously destroys the set of normal states. In the latter case, the dynamics of singular quantum states is described in terms of the evolution of mean values on an operator algebra that implements a 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.
We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators in the space of functions on a Hilbert space which are square integrable with respect to a shift-invariant measure. We study unitary groups of shift operators in the phase space and averaging of such shifts by Gaussian vectors, which form semigroups of self-adjoint contractions: we find conditions for their strong continuity and establish properties of their generators. Significant differences in their properties allow us to show the absence of the Fourier transform as a unitary transformation that implements the unitary equivalence of these compressive semigroups. Next, we prove the Taylor formula for a certain special subset of smooth functions for shifting to a non-finite vector. It allows us to prove convergence of quantum random walks in the coordinate representation to the evolution of a diffusion process, as well as convergence of quantum random walks in both coordinate and momentum representations to the evolution semigroup of a quantum oscillator in an infinite-dimensional phase space. We find the special essential common domain of generators of semigroups arising in averaging of random shift operators both in position and momentum representations. The invariance of this common domain with respect to both semigroups allows to establish properties of a convex combination of both generators. That convex combination are Hamiltonians of infinite-dimentional quantum oscillators. Thus, we obtain that a Weyl representation of a random walk in an infinite dimensional phase space describes the semigroup of self-adjoint contractions whose generator is the Hamiltonian of an infinite dimensional harmonic oscillator.
We investigate a model that associates random walks in a finite-dimensional Euclidean coordinate space of a classical system with random quantum walks, i.e. random transformations of the set of states of a quantum system arising from quantization of a classical system. As is known, the convolution semigroup of Gaussian measures on a coordinate space admits a representation by a semigroup of self-adjoint contractions in the space of square-integrable functions described by the heat equation. We obtain a representation of the convolution semigroup of Gaussian measures on a coordinate space by a quantum dynamic semigroup in the space of nuclear operators. We give a description of the quantum dynamic semigroup by solutions of the Cauchy problem for a degenerate diffusion equation. We establish the generalized convergence in distribution of a sequence of quantum random walks to an operator-valued random process with values in the Abelian algebra of shift operators by a vector with a normal distribution.
Abstract We investigate asymptotic properties of a sequence whose elements are probability distributions which correspond to compositions of independent random operators. We prove the following statements: 1) a sequence whose elements are the expected values which corresponds to compositions of independent identically distributed random semigroups converges to a semigroup which is generated by the expected value of the random generator; 2) an analog of the law of large numbers for a sequence whose elements are compositions of independent identically distributed random semigroups; 3) a sequence whose elements are compositions of random shift operators on position space converges to a semigroup generated by the Laplace operator; 4) a sequence whose elements are compositions of random shift operators on position space and random shift operators in momentum space converges to a semigroup generated by the Hamiltonian of the quantum oscillator; 5) a sequence whose elements are compositions of random shift operators on phase space converges to a semigroup generated by the Hamiltonian of the quantum oscillator.
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 φ(v)=Eexp⁡(iRe⁡⟨v∣ψ⟩).\varphi(v)=\mathbb{E}\exp(i\operatorname{Re}\langle v|\psi\rangle ).φ(v)=Eexp(iRe⟨v∣ψ⟩). 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)F'(0)F′(0) is an unbounded operator. However, we restrict the consideration to the case when the Markov operators of random walks with the operators GtG_tGt​ 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.
Background: Operator algebras and quantum dynamical semigroups are the mathematical tools of choice for understanding open quantum systems, especially in infinite-dimensional spaces. Such tools are extremely important for the time development of quantum systems out of equilibium under the influence of environment, where conventional finite dimensional methods do not include all possibilities for interaction. Objective: The objects of the present research are the r-properties and the decompositions of generators of quantum dynamical semigroups on infinite-dimensional spaces including their Hamiltonian and dissipative part, semigroup continuity, stationary states, and positivity improvement. The aim is to set a fundament in order to think about long-time dynamics in open quantum systems. Methods: The theorem Hille–Yosida is used to get conditions for the semigroup strongly continuous generated by unbounded operators. The time-evolution operator of the system is approximated with both Gaussian quadrature and Monte Carlo integration during infinitesimal time slices, by using Inf-LCHS in conjunction. Generators were studied under decomposition of their Hamiltonian and dissipative parts, while commutant algebras were investigated for irreducibility and positivity enhancement. Results: Most systems studied for these properties presented a singular decomposition, strong and weak semigroup continuity, existence of stationary states and positivity improvement. The counterexamples were the systems with in a lack of verification for dissipative parts, and they lost smoothing property, uniqueness of decomposition, partial semigroup continuity or large errors in approximation when we perform numerically. Conclusions: The combination of the methods of operator algebra with state-of-the-art simulation techniques yields a powerful approach to study infinite-dimensional quantum dynamics. Although dissipative components are not yet completely characterized, the results improve our understanding of generator structures and long-term system behavior which will help future work in quantum statistical mechanics and quantum information theory.