Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Doklady Mathematics

Markov Approximations of the Evolution of Quantum Systems

John Edward Gough, Yu. N. Orlov, Vsevolod Zhanovich Sakbaev, Oleg Georgievich Smolyanov

Published Apr 1, 2022• 11 citations
Publisher

Abstract

Abstract The convergence in probability of a sequence of iterations of independent random quantum dynamical semigroups to a Markov process describing the evolution of an open quantum system is studied. The statistical properties of the dynamics of open quantum systems with random generators of Markovian evolution are described in terms of the law of large numbers for operator-valued random processes. For compositions of independent random semigroups of completely positive operators, the convergence of mean values to a semigroup described by the Gorini–Kossakowski–Sudarshan–Lindblad equation is established. Moreover, a sequence of random operator-valued functions with values in the set of operators without the infinite divisibility property is shown to converge in probability to an operator-valued function with values in the set of infinitely divisible operators.

DOI
10.1134/s1064562422020107

Related papers

Operator Random Walks and Quantum Oscillator

Yu. N. Orlov, Vsevolod Zhanovich Sakbaev, Дмитрий Викторович Завадский

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.

Lobachevskii Journal of Mathematics•Apr 1, 2020•6 citations

Dynamics of Open Quantum Systems II, Markovian Approximation

Marco Merkli

A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperature T>0 . We show that for fixed, small values of the coupling constant λ , the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics is O(|λ|1/4) for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracy O(|λ|1/4) for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as 1/t4 or faster, which is a significant improvement relative to the previously required exponential decay.

arXiv:2105.00023•Quantum•Jan 3, 2022•24 citations

Bi - continuous semigroups of stochastic quantum dynamics

Andrey Vladimirovich Utkin

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 Zhurnal•Jan 1, 2025•5 citations

Universal simulation of Markovian open quantum systems

Ryan Sweke, Ilya Sinayskiy, Denis Bernard +1 more

We consider the problem of constructing a ``universal set'' of Markovian processes, such that any Markovian open quantum system, described by a one-parameter semigroup of quantum channels, can be simulated through sequential simulations of processes from the universal set. In particular, for quantum systems of dimension $d$, we explicitly construct a universal set of semigroup generators, parametrized by ${d}^{2}\ensuremath{-}3$ continuous parameters, and prove that a necessary and sufficient condition for the dynamical simulation of a $d$-dimensional Markovian quantum system is the ability to implement (a) quantum channels from the semigroups generated by elements of the universal set of generators, and (b) unitary operations on the system. Furthermore, we provide an explicit algorithm for simulating the dynamics of a Markovian open quantum system using this universal set of generators, and show that it is efficient, with respect to this universal set, when the number of distinct Lindblad operators (representing physical dissipation processes) scales polynomially with respect to the number of subsystems.

arXiv:1503.05028•Physical Review A•Jun 8, 2015•67 citations

Random Quantization of Hamiltonian Systems

John Edward Gough, Yu. N. Orlov, Vsevolod Zhanovich Sakbaev +1 more

Abstract A quantization of a Hamiltonian system is an ambiguous procedure. Accordingly, we introduce the notion of random quantization, related random variables with values in the set of self-adjoint operators, and random processes with values in the group of unitary operators. The procedures for the averaging of random unitary groups and averaging of random self-adjoint operators are defined. The generalized weak convergence of a sequence of measures and the corresponding generalized convergence in distribution of a sequence of random variables are introduced. The generalized convergence in distribution for some sequences of compositions of random mappings is obtained. In the case of a sequence of compositions of shifts by independent random vectors of Euclidean space, the obtained convergence coincides with the statement of the central limit theorem for a sum of independent random vectors. The results are applied to the dynamics of quantum systems arising in random quantization of a Hamiltonian system.

Doklady Mathematics•May 1, 2021•19 citations

Operator Algebras and Quantum Dynamics: Semigroup Methods in Infinite Dimensions

Anam Ghafoor, Ume Roman, Sawaiba Yousaf

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.

Annual Methodological Archive Research Review•Dec 16, 2025•0 citations
Abstract The convergence in probability of a sequence of iterations of independent random quantum dynamical semigroups to a Markov process describing the evolution of an open quantum system is studied. The statistical properties of the dynamics of open quantum systems with random generators of Markovian evolution are described in terms of the law of large numbers for operator-valued random processes. For compositions of independent random semigroups of completely positive operators, the convergence of mean values to a semigroup described by the Gorini–Kossakowski–Sudarshan–Lindblad equation is established. Moreover, a sequence of random operator-valued functions with values in the set of operators without the infinite divisibility property is shown to converge in probability to an operator-valued function with values in the set of infinitely divisible operators.
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.
A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperature T>0 . We show that for fixed, small values of the coupling constant λ , the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics is O(|λ|1/4) for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracy O(|λ|1/4) for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as 1/t4 or faster, which is a significant improvement relative to the previously required exponential decay.
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.
We consider the problem of constructing a ``universal set'' of Markovian processes, such that any Markovian open quantum system, described by a one-parameter semigroup of quantum channels, can be simulated through sequential simulations of processes from the universal set. In particular, for quantum systems of dimension ddd, we explicitly construct a universal set of semigroup generators, parametrized by ${d}^{2}\ensuremath{-}3$ continuous parameters, and prove that a necessary and sufficient condition for the dynamical simulation of a ddd-dimensional Markovian quantum system is the ability to implement (a) quantum channels from the semigroups generated by elements of the universal set of generators, and (b) unitary operations on the system. Furthermore, we provide an explicit algorithm for simulating the dynamics of a Markovian open quantum system using this universal set of generators, and show that it is efficient, with respect to this universal set, when the number of distinct Lindblad operators (representing physical dissipation processes) scales polynomially with respect to the number of subsystems.
Abstract A quantization of a Hamiltonian system is an ambiguous procedure. Accordingly, we introduce the notion of random quantization, related random variables with values in the set of self-adjoint operators, and random processes with values in the group of unitary operators. The procedures for the averaging of random unitary groups and averaging of random self-adjoint operators are defined. The generalized weak convergence of a sequence of measures and the corresponding generalized convergence in distribution of a sequence of random variables are introduced. The generalized convergence in distribution for some sequences of compositions of random mappings is obtained. In the case of a sequence of compositions of shifts by independent random vectors of Euclidean space, the obtained convergence coincides with the statement of the central limit theorem for a sum of independent random vectors. The results are applied to the dynamics of quantum systems arising in random quantization of a Hamiltonian system.
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.