Universal simulation of Markovian open quantum systems
Ryan Sweke, Ilya Sinayskiy, Denis Bernard, Francesco Petruccione
备注：
Revised version. Restricted recombination method to first order Suzuki-Lie-Trotter integrators, and added discussion concerning issues with the application of higher order integrators in the open quantum systems setting
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 $d^2-3$ 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.
Although the conditions for performing arbitrary unitary operations to simulate the dynamics of a closed quantum system are well understood, the same is not true of the more general class of quantum operations (also known as superoperators) corresponding to the dynamics of open quantum systems. We propose a framework for the generation of Markovian quantum dynamics and study the resources needed for universality. For the case of a single qubit, we show that a single nonunitary process is necessary and sufficient to generate all unital Markovian quantum dynamics, whereas a set of processes para…
We exploit the properties of chain mapping transformations of bosonic environments to identify a finite collection of modes able to capture the characteristic features, or fingerprint, of the environment. Moreover we show that the countable infinity of residual bath modes can be replaced by a universal Markovian closure, namely a small collection of damped modes undergoing a Lindblad-type dynamics whose parametrization is independent of the spectral density under consideration. We show that the Markovian closure provides a quadratic speed-up with respect to standard chain mapping techniques an…
Non-Markovian processes can often be turned Markovian by enlarging the set of variables. Here we show, by an explicit construction, how this can be done for the dynamics of a Brownian particle obeying the generalized Langevin equation. Given an arbitrary bath spectral density $J_{0}$, we introduce an orthogonal transformation of the bath variables into effective modes, leading stepwise to a semi-infinite chain with nearest-neighbor interactions. The transformation is uniquely determined by $J_{0}$ and defines a sequence $\{J_{n}\}_{n\in\mathbb{N}}$ of residual spectral densities describing the…
We address the dynamics of quantum correlations in continuous variable open systems and analyze the evolution of bipartite Gaussian states in independent noisy channels. In particular, upon introducing the notion of dynamical path through a suitable parametrization for symmetric states, we focus attention on phenomena that are common to Markovian and non-Markovian Gaussian maps under the assumptions of weak coupling and secular approximation. We found that the dynamical paths in the parameter space are universal, that is they do depend only on the initial state and on the effective temperature…
A universal characterization of non-Markovianity for any open hybrid quantum systems is presented. This formulation is based on the negativity volume of the generalized Wigner function, which serves as an indicator of the quantum correlations in any composite quantum systems. It is shown, that the proposed measure can be utilized for any single or multi-partite quantum system, containing any discrete or continuous variables. To demonstrate its power in revealing non-Markovianity in such quantum systems, we additionally consider a few illustrative examples.
Firstly, the Markovian stochastic Schrödinger equations are presented, together with their connections with the theory of measurements in continuous time. Moreover, the stochastic evolution equations are translated into a simulation algorithm, which is illustrated by two concrete examples - the damped harmonic oscillator and a two-level atom with homodyne photodetection. Then, we consider how to introduce memory effects in the stochastic Schrödinger equation via coloured noise. Specifically, the approach by using the Ornstein-Uhlenbeck process is illustrated and a simulation for the non-Markov…
Non-Markovian open quantum systems represent the most general dynamics when the quantum system is coupled with a bath environment. The quantum dynamics arising from many important applications are non-Markovian. Although for special cases, such as Hamiltonian evolution and Lindblad evolution, quantum simulation algorithms have been extensively studied, efficient quantum simulations for the dynamics of non-Markovian open quantum systems remain underexplored. The most immediate obstacle for studying such systems is the lack of a universal succinct description of their dynamics. In this work, we …
It is by now well understood that quantum dissipative processes can be harnessed and turned into a resource for quantum-information processing tasks. In this paper we demonstrate yet another way in which this is true by providing a dissipation-assisted protocol for the simulation of general Markovian dynamics. More precisely, we show how a suitable coherent coupling of a quantum system to a set of Markovian dissipating qubits allows one to enact an effective Liouvillian generator of any Lindbladian form. This effective dynamical generator arises from high-order virtual-dissipative processes an…
The system-environment interaction is simulated by light propagating in coupled photonic waveguides. The profile of the electromagnetic field provides intuitive physical insight to study the Markovian and non-Markovian dynamics of open quantum systems. The transition from non-Markovian to Markovian process is demonstrated by increasing the size of environment, as the energy evolution changes from oscillating to an exponential decay, and the revival period increases. Moreover, the dynamical decoupling with a sequence of phase modulations is introduced to such a photonic open system to form a ba…
We demonstrate a scaling method for non-Markovian Monte Carlo wave-function simulations used to study open quantum systems weakly coupled to their environments. We derive a scaling equation, from which the result for the expectation values of arbitrary operators of interest can be calculated, all the quantities in the equation being easily obtainable from the scaled Monte Carlo simulations. In the optimal case, the scaling method can be used, within the weak coupling approximation, to reduce the size of the generated Monte Carlo ensemble by several orders of magnitude. Thus, the developed meth…
Open quantum systems exhibit a range of novel out-of-equilibrium behavior due to the interplay between coherent quantum dynamics and dissipation. Of particular interest in these systems are driven, dissipative transitions, the emergence of dynamical phases with novel broken symmetries, and critical behavior that lies beyond the conventional paradigms of Landau-Ginzburg phenomenology. Here, we consider a parametrically driven two-mode system in the presence of non-Markovian system-reservoir interactions. We show that non-Markovianity modifies the phase diagram of this system resulting in the em…
We derive a new memory-assisted entropic uncertainty relation for non-degenerate Hermitian observables where both quantum correlations, in the form of conditional von Neumann entropy, and quantum discord between system and memory play an explicit role. Our relation is `universal', in the sense that it does not depend on the specific observable, but only on properties of the quantum state. We contrast such an uncertainty relation with previously known memory-assisted relations based on entanglement and correlations. Further, we present a detailed comparative study of entanglement- and discord-a…
Entanglement enhanced quantum metrology has been well investigated for beating the standard quantum limit (SQL). However, the metrological advantage of entangled states becomes much more elusive in the presence of noise. Under strictly Markovian dephasing noise, the uncorrelated and maximally entangled states achieve exactly the same measurement precision. However, it was predicted that in a non-Markovian dephasing channel, the entangled probes can recover their metrological advantage. Here, by using a highly controlled photonic system, we simulate a non-Markovian dephasing channel fulfill the…
Faster Quantum Simulation Of Markovian Open Quantum Systems Via Randomisation
University of KwaZulu-Natal（夸祖鲁-纳塔尔大学）；National Institute for Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）；Stellenbosch University（斯坦伦布什大学）
A numerical method of calculating the non-Markovian evolution of a driven atom radiating into a structured continuum is developed. The formal solution for the atomic reduced density matrix is written as a Markovian algorithm by introducing a set of additional, virtual density matrices which follow, to the level of approximation of the algorithm, all the possible trajectories of the photons in the electromagnetic field. The technique is perturbative in the sense that more virtual density matrices are required as the product of the effective memory time and the effective coupling strength become…
In this paper, we use the quantum state diffusion (QSD) equation to implement the Uhrig dynamical decoupling (UDD) to a three-level quantum system coupled to a non-Markovian reservoir comprising of infinite numbers of degrees of freedom. For this purpose, we first reformulate the non-Markovian QSD to incorporate the effect of the external control fields. With this stochastic QSD approach, we demonstrate that an unknown state of the three-level quantum system can be universally protected against both colored phase and amplitude noises when the control-pulse sequences and control operators are p…
A sequence of controlled collisions between a quantum system and its environment (composed of a set of quantum objects) naturally simulates (with arbitrary precision) any Markovian quantum dynamics of the system under consideration. In this paper we propose and study the problem of simulation of an {\it arbitrary} quantum channel via collision models. We show that a correlated environment is capable to simulate {\it non-Markovian} evolutions leading to any indivisible qubit channel. In particular, we derive the corresponding master equation generating a continuous time non-Markovian dynamics i…
We suggest a quantum simulator that allows to study the role of memory effects in the dynamics of open quantum systems. Our proposal is based on a bipartite quantum system consisting, for simplicity, of a harmonic oscillator and a few-level system; it exploits the formal analogy between dissipation and quantum measurements. The interaction between the subsystems gives rise to quantum Zeno dynamics, and the dissipation strength experienced by the harmonic oscillator can be tuned by changing the parameters of the measurement, i.e., the interaction with the few-level system. Extension of the prop…
Simulation methods based on stochastic realizations of state vector evolutions are commonly used tools to solve open quantum system dynamics, both in the Markovian and non-Markovian regime. Here, we address the question of waiting time distribution (WTD) of quantum jumps for non-Markovian systems. We generalize Markovian quantum trajectory methods in the sense of deriving an exact analytical WTD for non-Markovian quantum dynamics and show explicitly how to construct this distribution for certain commonly used quantum optical systems.
It is shown that the exact dynamics of a composite quantum system can be represented through a pair of product states which evolve according to a Markovian random jump process. This representation is used to design a general Monte Carlo wave function method that enables the stochastic treatment of the full non-Markovian behavior of open quantum systems. Numerical simulations are carried out which demonstrate that the method is applicable to open systems strongly coupled to a bosonic reservoir, as well as to the interaction with a spin bath. Full details of the simulation algorithms are given, …
We present a non-Markovian quantum trajectory method for treating atoms radiating into a reservoir with a non-flat density of states. The results of an example numerical simulation of the case where the free space modes of the reservoir are altered by the presence of a cavity are presented and compared with those of an extended system approach.
The measurement-result-conditioned evolution of a system (e.g. an atom) with spontaneous emissions of photons is well described by the quantum trajectory (QT) theory. In this work we generalize the associated QT theory from infinitely wide bandwidth Markovian environment to the case of finite bandwidth non-Markovian environment. In particular, we generalize the treatment for arbitrary spectrum, which is not restricted by the specific Lorentzian case. We rigorously prove a general existence of a perfect scaling behavior jointly defined by the bandwidth of environment and the time interval betwe…
Universal Precision Limits in General Open Quantum Systems
Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所引力物理与量子信息中心）；Department of Physics, Kyoto University（京都大学物理学系）
A conjecture for the universal constraints for relaxation rates of a quantum dynamical semigroup is proposed. It is shown that it holds for several interesting classes of semigroups, e.g. unital semigroups and semigroups derived in the weak coupling limit from the proper microscopic model. Moreover, proposed conjecture is supported by numerical analysis. This conjecture has several important implications: it allows to provide universal constraints for spectra of quantum channels and provides necessary condition to decide whether a given channel is consistent with Markovian evolution.
The standard dynamical approach to quantum thermodynamics is based on Markovian master equations describing the thermalization of a system weakly coupled to a large environment, and on tools such as entropy production relations. Here we develop a new framework overcoming the limitations that the current dynamical and information theory approaches encounter when applied to this setting. More precisely, we introduce the notion of continuous thermomajorization, and employ it to obtain necessary and sufficient conditions for the existence of a Markovian thermal process transforming between given i…
To understand typical dynamics of an open quantum system in continuous time, we introduce an ensemble of random Lindblad operators, which generate Markovian completely positive evolution in the space of density matrices. Spectral properties of these operators, including the shape of the spectrum in the complex plane, are evaluated by using methods of free probabilities and explained with non-Hermitian random matrix models. We also demonstrate universality of the spectral features. The notion of ensemble of random generators of Markovian qauntum evolution constitutes a step towards categorizati…
In this paper we present a linear quadratic Gaussian (LQG) feedback control strategy for a class of linear non-Markovian quantum systems. The feedback control law is designed based on the estimated states of a whitening quantum filter for an augmented Markovian model of the non-Markovian open quantum systems. In this augmented Markovian model, an ancillary system plays the role of internal modes of the environment converting white noise into Lorentzian noise and a principal system obeys non-Markovian dynamics due to the direct interaction with the ancillary system. The simulation results show …
Forked Physics-Informed Neural Networks for Non-Markovian Open Quantum Dynamics and Control
College of Physics and Optoelectronic Engineering, Ocean University of China（中国海洋大学物理与光电工程学院）；China Mobile (Suzhou) Software Technology Co., Ltd（中国移动（苏州）软件技术有限公司）；School of Information Management and School of Artificial Intelligence, Qingdao University of Technology（青岛理工大学信息管理学院和人工智能学院）；Engineering Research Center of Advanced Marine Physical Instruments and Equipment of Ministry of Education, Ocean University of China（教育部海洋先进物理仪器装备工程研究中心）；Qingdao Key Laboratory of Optics and Optoelectronics, Ocean University of China（青岛市光学与光电技术重点实验室）
