A large class of non-Markovian quantum processes in open systems can be formulated through time-local master equations which are not in Lindblad form. It is shown that such processes can be embedded in a Markovian dynamics which involves a time dependent Lindblad generator on an extended state space. If the state space of the open system is given by some Hilbert space ${\mathcal{H}}$, the extended state space is the triple Hilbert space ${\mathcal{H}}\otimes{\mathbb C}^3$ which is obtained by combining the open system with a three state system. This embedding is used to derive an unraveling for non-Markovian time evolution by means of a stochastic process in the extended state space. The process is defined through a stochastic Schrödinger equation which generates genuine quantum trajectories for the state vector conditioned on a continuous monitoring of an environment. The construction leads to a continuous measurement interpretation for non-Markovian dynamics within the framework of the theory of quantum measurement.
We devise a novel protocol to detect genuinely multipartite entangled states by harnessing quantum non-Markovian operations. We utilize a particular type of non-Markovianity known as the eternal non-Markovianity to construct a non-complete positive map to filter out the bi-separable states and detect genuine multipartite entanglement. We further propose a witness operator to detect genuinely multipartite entangled states experimentally based on this theory. Our study sheds light on a hitherto unexplored connection between entanglement theory and quantum non-Markovianity.
We study dynamics of genuine entanglement for quantum states of three and four qubits under non-Markovian dephasing. Using a computable entanglement monotone for multipartite systems, we find that GHZ state is quite resilient state whereas the W state is the most fragile. We compare dynamics of chosen quantum states with dynamics of random pure states and weighted graph states.
Decoherence-free subspaces and Markovian revival of genuine multipartite entanglement in a dissipative system
Poornaprajna Institute of Scientific Research (PPISR)（普尔纳普拉吉纳科学研究学院）；Manipal Academy of Higher Education（马帕尔高等教育学院）；Amrita Vishwa Vidyapeetham（阿弥塔大学）；National Institute for Theoretical and Computational Sciences (NITheCS)（理论与计算科学国家研究所）；Chennai Institute of Technology（金奈理工学院）
We present an algorithm to simulate genuine, measurement-conditioned quantum trajectories for a class of non-Markovian systems, using a collision model for the environment. We derive two versions of the algorithm, the first corresponding to photodetection and the second to homodyne detection with a finite local oscillator amplitude. We use the algorithm to simulate trajectories for a system with delayed coherent feedback, as well as a system with a continuous memory.
We develop a notion of stochastic quantum trajectories. First, we construct a basis set of trajectories, called elementary trajectories, and go on to show that any quantum dynamical process, including those that are non-Markovian, can be expressed as a linear combination of this set. We then show that the set of processes divide into two natural classes: those that can be expressed as convex mixture of elementary trajectories and those that cannot be. The former are shown to be entanglement breaking processes (in each step), while the latter are dubbed coherent processes. This division of proc…
In this work we generalize the quantum trajectory (QT) theory from Markovian to non-Markovian environments. We model the non-Markovian environment by using a Lorentzian spectral density function with bandwidth ($Λ$), and find perfect "scaling" property with the measurement frequency ($τ^{-1}$) in terms of the scaling variable $x=Λτ$. Our result bridges the gap between the existing QT theory and the Zeno effect, by rendering them as two extremes corresponding to $x\to\infty$ and $x\to 0$, respectively. This $x$-dependent criterion improves the idea of using $τ$ alone, and quantitatively identif…
Semiclassical path integral expression for a quantum system coupled to a harmonic bath is derived based on the stationary phase condition. It is discovered that the system path is non-Markovian. Most strikingly, the system path not only couples to its past (as in the Langevin equation), but also to its future, i.e. the equation of motion for the system is an integro-differential equation that involves all times. Numerical tests are performed to confirm that the future-involved term is indeed necessary. Because of the future-non-Markovian nature of the equation, the numerical solution cannot be…
We analyze the non-Markovian stochastic Schroedinger equation describing a particle subject to spontaneous collapses in space (in the language of collapse models), or subject to a continuous measurement of its position (in the language of continuous quantum measurement). For the first time, we give the explicit general solution for the free particle case (H = p^2/2m), and discuss the main properties. We analyze the case of an exponential correlation function for the noise, giving a quantitative description of the dynamics and of its dependence on the correlation time.
We study the evolution of the entanglement of noninteracting qubits coupled to reservoirs under monitoring of the reservoirs by means of continuous measurements. We calculate the average of the concurrence of the qubits wavefunction over all quantum trajectories. For two qubits coupled to independent baths subjected to local measurements, this average decays exponentially with a rate depending on the measurement scheme only. This contrasts with the known disappearance of entanglement after a finite time for the density matrix in the absence of measurements. For two qubits coupled to a common b…
In this paper, we investigate the dynamics behaviors of genuine multipartite Einstein-Podolsky-Rosen steering (GMS) and genuine multipartite nonlocality (GMN), and explore how to recover the lost GMS and GMN under a mixed decoherence system. Explicitly, the decoherence system can be modeled by that a tripartite Werner-type state suffers from the non-Markovian regimes and one subsystem of the tripartite is under a non-inertial frame. The conditions for steerable and nonlocal states can be obtained with respect to the tripartite Werner-type state established initially. GMS and GMN are very fragi…
We investigate the dynamics of information among the parties of tripartite systems. We start by proving two results concerning the monogamy of mutual information. The first one states that mutual information is monogamous for generic tripartite pure states. The second shows that in general mutual information is monogamous only if the amount of genuine tripartite correlations is large enough. Then, we analyze the internal dynamics of tripartite systems whose parties do not exchange energy. In particular, we allow for one of the subsystems to play the role of a finite thermal bath. As a result, …
The non-Markovia dynamics of quantum evolution plays an important role in open quantum sytem. However, how to quantify non-Markovian behavior and what can be obtained from non- Markovianity are still open questions, especially in complex solid systems. Here we address the problem of quantifying non-Markovianity with entanglement in a genuine noisy solid state system at room temperature. We observed the non-Markovianity of quantum evolution with entanglement. By prolonging entanglement with dynamical decoupling, we can reveal the non-Markovianity usually concealed in the environment and obtain …
In this paper, the non-Markovian quantum dynamics of a coupled $N$-cavity model is studied based on the quantum state diffusion (QSD) approach. The time-local Diósi-Gisin-Strunz equation and the corresponding exact master equation are derived for the model consisting of a coupled cavity array. The resulting QSD equation serves as a stochastic solution to a genuine $N$-partite continuous-variable (CV) system. Several non-Markovian effects are studied in two interesting examples -- two-cavity and three-cavity, under different boundary conditions. We have shown that the environment-memory can fac…
We determine the total state dynamics of a dephasing open quantum system using the standard environment of harmonic oscillators. Of particular interest are random unitary approaches to the same reduced dynamics and system-environment correlations in the full model. Concentrating on a model with an at times negative dephasing rate, the issue of "non-Markovianity" will also be addressed. Crucially, given the quantum environment, the appearance of non-Markovian dynamics turns out to be accompanied by a loss of system-environment correlations. Depending on the initial purity of the qubit state, th…
Starting from a generalization of the quantum trajectory theory (based on the stochastic Schrödinger equation - SSE), non-Markovian models of quantum dynamics are derived. In order to describe non-Markovian effects, the approach used in this article is based on the introduction of random coefficients in the usual linear SSE. A major interest is that this allows to develop a consistent theory of quantum measurement in continuous time for these non-Markovian quantum trajectory models. In this context, the notions of instrument, a priori and a posteriori states are rigorously described. The key p…
We develop a systematic and efficient approach for numerically solving the non-Markovian quantum state diffusion equations for open quantum systems coupled to an environment up to arbitrary orders of noises or coupling strengths. As an important application, we consider a real-time simulation of a spin-boson model in a strong coupling regime that is difficult to deal with using conventional methods. We show that the non-Markovian stochastic Schrödinger equation can be efficiently implemented as a real--time simulation for this model, so as to give an accurate description of spin-boson dynamics…
We characterize a class of Markovian dynamics using the concept of divisible dynamical map. Moreover we provide a family of criteria which can distinguish Markovian and non-Markovian dynamics. These Markovianity criteria are based on a simple observation that Markovian dynamics implies monotonic behavior of several well known quantities like distinguishability of states, fidelity, relative entropy and genuine entanglement measures.
The linear and the nonlinear non-Markovian quantum state diffusion equation (NMQSD) are well known tools for the description of certain non-Markovian open quantum systems. In this work, we systematically investigate whether the normalized linear NMQSD or the nonlinear NMQSD solutions can be generated by means of a time-continuous measurement. By considering any conceivable measurement scheme in the framework of instruments, we derive a necessary criterion for a measurement interpretation of both equations. Concrete examples show that the normalized linear NMQSD solutions are realizable only in…
The fully quantized model of double qubits coupled to a common bath is solved using the quantum state diffusion (QSD) approach in the non-Markovian regime. We have established the explicit time-local non-Markovian QSD equations for the two-qubit dissipative and dephasing models. Diffusive quantum trajectories are applied to the entanglement estimation of two-qubit systems in a non-Markovian regime. In both cases, non-Markovian features of entanglement evolution are revealed through quantum diffusive unravellings in the system state space.
The initial stages of the evolution of an open quantum system encode the key information of its underlying dynamical correlations, which in turn can predict the trajectory at later stages. We propose a general approach based on non-Markovian dynamical maps to extract this information from the initial trajectories and compress it into non-Markovian transfer tensors. Assuming time-translational invariance, the tensors can be used to accurately and efficiently propagate the state of the system to arbitrarily long timescales. The non-Markovian Transfer Tensor Method (TTM) demonstrates the coherent…
We unveil a novel source of non-Markovianity for the dynamics of quantum systems, which appears when the system does not explore the full set of dynamical trajectories in the interaction with its environment. We term this effect non-Markovianity by undersampling and demonstrate its appearance in the operation of an all-optical quantum simulator involving a polarization qubit interacting with a dephasing fluctuating environment.
The quantum jump approach allows to characterize the stochastic dynamics associated to an open quantum system submitted to a continuous measurement action. In this paper we show that this formalism can consistently be extended to non-Markovian system dynamics. The results rely in studying a measurement process performed on a bipartite arrangement characterized by a Markovian Lindblad evolution. Both a renewal and non-renewal extensions are found. The general structure of non-local master equations that admit an unravelling in terms of the corresponding non-Markovian trajectories are also found…
Markovian dynamics for a quantum/classical system and quantum trajectories
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Milano（意大利国家核物理研究所（INFN）米兰分部）；Istituto Nazionale di Alta Matematica (INDAM-GNAMPA)（意大利高等数学研究所（INDAM-GNAMPA））
We introduce jumptime unraveling as a distinct description of open quantum systems. As our starting point, we consider quantum jump trajectories, which emerge, physically, from continuous quantum measurements, or, formally, from the unraveling of Markovian quantum master equations. If the stochastically evolving quantum trajectories are ensemble-averaged at specific times, the resulting quantum states are solutions to the associated quantum master equation. We demonstrate that quantum trajectories can also be ensemble-averaged at specific jump counts. The resulting jumptime-averaged quantum st…
A long-standing open problem in non-Markovian quantum state diffusion (QSD) approach to open quantum systems is to establish the non-Markovian QSD equations for multiple qubit systems. In this paper, we settle this important question by explicitly constructing a set of exact time-local QSD equations for $N$-qubit systems. Our exact time-local (convolutionless) QSD equations have paved the way towards simulating quantum dynamics of many-body open systems interacting with a common bosonic environment. The applicability of this multiple-qubit stochastic equation is exemplified by numerically solv…
We investigate the non-Markovian quantum dynamics of a hybrid open system consisting of one qubit and one qutrit by employing a stochastic Schrödinger equation to generate diffusive quantum trajectories. We have established an exact quantum state diffusion (QSD) equation for the dissipative qubit-qutrit system coupled to a bosonic heat bath at zero temperature. As an important application, the non-Markovian QSD equation is employed to simulate the entanglement decay and generation measured by negativity. Finally, some steady state properties of the hybrid system are also discussed.
The large deviation theory has recently been applied to open quantum systems to uncover dynamical crossovers in the space of quantum trajectories associated to Markovian evolutions. Such dynamical crossovers are characterized by qualitative changes in the fluctuations of rare quantum jump trajectories, and have been observed in the statistics of coherently driven quantum systems. In this article we investigate the counting statistics of rare quantum trajectories of an undriven system undergoing a pure relaxation process, namely, Markovian exciton transport. We find that dynamical crossovers oc…
In this Letter we investigate Quantum Darwinism and the emergence of a classical world from the quantum one in connection with the spectral properties of the environment. We use a microscopic model of quantum environment in which, by changing a simple system parameter, we can modify the information back flow from environment into the system, and therefore its non-Markovian character. We show that the presence of memory effects hinders the emergence of classical objective reality. We conjecture that this phenomenon is connected to the absence of a measurement-scheme interpretation for quantum t…
A natural non-Markovian extension of the theory of white noise quantum trajectories is presented. In order to introduce memory effects in the formalism an Ornstein-Uhlenbeck coloured noise is considered as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, non-Markovian stochastic Schrödinger equations which unravel non-Markovian master equations are derived.
Solving realistic quantum systems coupled to an environment is a challenging task. Here we develop a hierarchical functional derivative (HFD) approach for efficiently solving the non-Markovian quantum trajectories of an open quantum system embedded in a bosonic bath. An explicit expression for arbitrary order HFD equation is derived systematically. Moreover, it is found that for an analytically solvable model, this hierarchical equation naturally terminates at a given order and thus becomes exactly solvable. This HFD approach provides a systematic method to study the non-Markovian quantum dyna…
More than a century after the inception of quantum theory, the question of which traits and phenomena are fundamentally quantum remains under debate. Here we give an answer to this question for temporal processes which are probed sequentially by means of projective measurements of the same observable. Defining classical processes as those that can---in principle---be simulated by means of classical resources only, we fully characterize the set of such processes. Based on this characterization, we show that for non-Markovian processes (i.e., processes with memory), the absence of coherence does…
Reconfigurable qubit states and quantum trajectories in a synthetic artificial neuron network with a process to direct information generation from co-integrated burst-mode spiking under non-Markovianity
A wide class of exact master equations for a multiple qubit system can be explicitly constructed by using the corresponding exact non-Markovian quantum state diffusion equations. These exact master equations arise naturally from the quantum decoherence dynamics of qubit system as a quantum memory coupled to a collective colored noisy source. The exact master equations are also important in optimal quantum control, quantum dissipation and quantum thermodynamics. In this paper, we show that the exact non-Markovian master equation for a dissipative N-qubit system can be derived explicitly from th…
We consider the Dicke model, describing an ensemble of $N$ quantum spins interacting with a cavity field, and study how the coupling to a non-Markovian environment with power-law spectrum changes the physics of superradiant phase transition. Quite remarkably we find that dissipation can induce, rather than suppress, the ordered phase, a result which is in striking contrast with both thermal and Markovian quantum baths. We interpret this dissipation-induced superradiance as a genuine dissipative quantum phase transition that exists even at finite $N$ due to the coupling with the bath modes and …
