We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway to faithfully represent quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model-parameter and initial-state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov's stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular, explicit results for the synthesis of stabilizing semigroups and noiseless subspaces in finite-dimensional Markovian systems are obtained.
Quantum non-Markovianity modifies the environmental decoherence of a system. This situation is enriched in complex systems owing to interactions among subsystems. We consider the problem of distinguishing the multiple sources of non-Markovianity using a simple power spectrum technique, applied to a qubit interacting with another qubit via a Jaynes-Cummings type Hamiltonian and simultaneously subjected to some well known noise channels, such as, the random telegraph noise and non-Markovian amplitude damping, which exhibit both Markovian as well as non-Markovian dynamics under different paramete…
We analyze the dynamics of a spin 1/2 subsystem coupled to a spin chain. We simulate numerically the full quantum many-body system for various sets of parameters and initial states of the chain, and characterize the divisibility of the subsystem dynamics, i.e. whether it is Markovian and can be described by a (time dependent) master equation. We identify regimes in which the subsystem admits such Markovian description, despite the many-body setting, and provide insight about why the same is not possible in other regimes. Interestingly, coupling the subsystem at the edge, instead of the center,…
Analytical solution and entanglement swapping of a double Jaynes-Cummings model in non-Markovian environments are investigated by the timeconvolutionless master equation method. We obtain the analytical solution of this model and discuss in detail the influence of atom-cavity coupling, non-Markovian effect and initial state purity on entanglement dynamics. The results show that, in the non-Markovian environments, the entanglement between two cavities can be swapped to other bipartite subsystems by interaction between an atom and its own cavity. Due to the dissipation of environment, the entang…
By using the effective non-Markovian measure [H.P. Breuer, E.M. Laine, J. Piilo, Phys. Rev. Lett. 103, 210401 (2009)], we investigate non-Markovian dynamics of a pair of two-level atoms (TLAs) system, each of which interacting with a local reservoir. We show that subsystem dynamics can be controlled by manipulating the coupling between TLAs, temperature and relaxation rate of the atoms. Moreover, the correlation between non-Markovianity of subsystem and entanglement between the subsystem and the structured bath is investigated, the results show that the emergence of non-Markovianity has a nega…
We propose a theory of adiabaticity in quantum Markovian dynamics based on a decomposition of the Hilbert space induced by the asymptotic behavior of the Lindblad semigroup. A central idea of our approach is that the natural generalization of the concept of eigenspace of the Hamiltonian in the case of Markovian dynamics is a noiseless subsystem with a minimal noisy cofactor. Unlike previous attempts to define adiabaticity for open systems, our approach deals exclusively with physical entities and provides a simple, intuitive picture at the underlying Hilbert-space level, linking the notion of …
In this article, we investigate the problem of engineering synchronization in non-Markovian quantum systems. First, a time-convoluted linear quantum stochastic differential equation is derived which describes the Heisenberg evolution of a localized quantum system driven by multiple colored noise inputs. Then, we define quantum expectation synchronization in an augmented system consisting of two subsystems. We prove that, for two homogenous subsystems, synchronization can always be synthesized without designing direct Hamiltonian coupling given that the degree of non-Markovianity is below a cer…
In this Letter, we mainly investigate the dynamic behavior of quantum steering and how to effectively recover the lost steerability of quantum states within non-Markovian environments. We consider two different cases (one-subsystem or all-subsystem interacts with the dissipative environments), and obtain that the dynamical interaction between system initialized by a Werner state and the non-Markovian environments can induce the quasi-periodic quantum entanglement (concurrence) resurgence, however, quantum steering cannot retrieve in such a condition. And we can obtain that the resurgent quantu…
We examine the non-Markovian dynamics in a multipartite system of two initially correlated atomic qubits, each located in a single-mode leaky cavity and interacting with its own bosonic reservoir. We show the dominance of non-Markovian features, as quantified by the difference in fidelity of the evolved system with its density matrix at an earlier time, in three specific two-qubit partitions associated with the cavity-cavity and atom-reservoir density matrices within the same subsystem, and the cavity-reservoir reduced matrix across the two subsystems. The non-Markovianity in the cavity-cavity…
Non-Markovian quantum evolution of the electronic subsystem in a laser-driven molecule is characterized through the appearance of negative decoherence rates in the canonical form of the electronic master equation. For a driven molecular system described in a bipartite Hilbert space H=Hel x Hvib of dimension 2 x Nv, we derive the canonical form of the electronic master equation, deducing the canonical measures of non-Markovianity and the Bloch volume of accessible states. We find that one of the decoherence rates is always negative, accounting for the inherent non-Markovian character of the ele…
We study the curves of coherence for the Bell-diagonal states including $l_{1}$-norm of coherence and relative entropy of coherence under the Markovian channels in the first subsystem once. For a special Bell-diagonal state under bit-phase flip channel, we find frozen coherence under $l_{1}$ norm occurs, but relative entropy of coherence decrease. It illustrates that the occurrence of frozen coherence depends on the type of the measure of coherence. We study the coherence evolution of Bell-diagonal states under Markovian channels in the first subsystem $n$ times and find coherence under depola…
It is common, when dealing with quantum processes involving a subsystem of a much larger composite closed system, to treat them as effectively memory-less (Markovian). While open systems theory tells us that non-Markovian processes should be the norm, the ubiquity of Markovian processes is undeniable. Here, without resorting to the Born-Markov assumption of weak coupling or making any approximations, we formally prove that processes are close to Markovian ones, when the subsystem is sufficiently small compared to the remainder of the composite, with a probability that tends to unity exponentia…
We study open quantum systems whose evolution is governed by a master equation of Kossakowski-Gorini-Sudarshan-Lindblad type and give a characterization of the convex set of steady states of such systems based on the generalized Bloch representation. It is shown that an isolated steady state of the Bloch equation cannot be a center, i.e., that the existence of a unique steady state implies attractivity and global asymptotic stability. Necessary and sufficient conditions for the existence of a unique steady state are derived and applied to different physical models including two- and four-level…
The developing of (non-Markovian) memory effects strongly depends on the underlying system-environment dynamics. Here we study this problem in multipartite arrangements where all subsystems are coupled to each other by non-diagonal Markovian (Lindblad) dephasing mechanisms. Taking as system and environment arbitrary sets of complementary subsystems it is shown that both operational and non-operational approaches to quantum non-Markovianity can be characterized in an exact analytical way. Similarly to previous studies about dissipative-entanglement-generation in this kind of dynamics [Seif, Wan…
We study the unitary time evolution of photons interacting with a dielectric resonator using coherent control pulses. We show that non-Markovianity of transient photon dynamics in the resonator subsystem may be controlled to within a photon-resonator transit time. In general, appropriate use of coherent pulses and choice of spatial subregion may be used to create and control a wide range of non-Markovian transient dynamics in photon resonator systems.
Full information about a many-body quantum system is usually out-of-reach due to the exponential growth -- with the size of the system -- of the number of parameters needed to encode its state. Nonetheless, in order to understand the complex phenomenology that can be observed in these systems, it is often sufficient to consider dynamical or stationary properties of local observables or, at most, of few-body correlation functions. These quantities are typically studied by singling out a specific subsystem of interest and regarding the remainder of the many-body system as an effective bath. In t…
We investigate a total thermodynamic entropy production rate of an isolated quantum system. In particular, we consider a quantum model of coupled harmonic oscillators in a star configuration, where a central harmonic oscillator (system) is coupled to a finite number of surrounding harmonic oscillators (bath). In this model, when the initial state of the total system is given by the tensor product of the Gibbs states of the system and the bath, every harmonic oscillator is always in a Gibbs state with a time-dependent temperature. This enables us to define time-dependent thermodynamic entropy f…
P-divisibility conditions on subsystems in quantum dynamics
TCG Centres for Research and Education in Science and Technology (TCG CREST)（科学技术研究与教育中心）；Academy of Scientific and Innovative Research (AcSIR)（科学与创新研究学院）；Ramakrishna Mission Vivekananda Educational and Research Institute（罗摩克里希那使命维韦卡南达教育与研究所）
We analyze the asymptotic behavior of discrete-time, Markovian quantum systems with respect to a subspace of interest. Global asymptotic stability of subspaces is relevant to quantum information processing, in particular for initializing the system in pure states or subspace codes. We provide a linear-algebraic characterization of the dynamical properties leading to invariance and attractivity of a given quantum subspace. We then construct a design algorithm for discrete-time feedback control that allows to stabilize a target subspace, proving that if the control problem is feasible, then the …
We present an algorithmic method for the digital quantum simulation of many-body locally indivisible non-Markovian open quantum systems. It consists of two parts: firstly, a Suzuki-Lie-Trotter decomposition of the global system propagator into the product of subsystem propagators, which may not be quantum channels, and secondly, an algorithmic procedure for the implementation of the subsystem propagators through unitary operations and measurements on a dilated space. By providing rigorous error bounds for the relevant Suzuki-Lie-Trotter decomposition, we are able to analyze the efficiency of t…
We introduce the multipartite collision model, defined in terms of elementary interactions between subsystems and ancillae, and show that it can simulate the Markovian dynamics of any multipartite open quantum system. We develop a method to estimate an analytical error bound for any repeated interactions model, and we use it to prove that the error of our scheme displays an optimal scaling. Finally, we provide a simple decomposition of the multipartite collision model into elementary quantum gates, and show that it is efficiently simulable on a quantum computer according to the dissipative qua…
Quantum non-Markovianity of a quantum noisy channel manifests typically as information backflow, characterized by the departure of the intermediate map from complete positivity, though we indicate certain noisy channels that don't exhibit this behavior. In complex systems, non-Markovianity becomes more involved on account of subsystem dynamics. Here we study various facets of non-Markovian evolution, in the context of coined quantum walks, with particular stress on disambiguating the internal vs. environmental contributions to non-Markovian backflow. For the above problem of disambiguation, we…
We study the entanglement dynamics of two qubits, each of which is embedded into its local amplitude-damping reservoir, and the entanglement distribution among all the bipartite subsystems including qubit-qubit, qubit-reservoir, and reservoir-reservoir. It is found that the entanglement can be stably distributed among all components, which is much different to the result obtained under the Born-Markovian approximation by C. E. López {\it et al.} [Phys. Rev. Lett. \textbf{101}, 080503 (2008)], and particularly it also satisfies an identity. Our unified treatment includes the previous results as…
Steady-state manifolds of open quantum systems, such as decoherence-free subspaces and noiseless subsystems, are of great practical importance to the end of quantum information processing. Yet, it is a difficult problem to find steady-state manifolds of open quantum systems, especially of non-Markovian systems. In this paper, we propose an approach to find the steady-state manifolds, which is generally applicable to both Markovian and non-Markovian systems. Our approach is based on an arbitrarily given steady state, and by following the standard steps of the approach, the steady-state manifold…
Continuous-time quantum error correction (CTQEC) is an approach to protecting quantum information from noise in which both the noise and the error correcting operations are treated as processes that are continuous in time. This chapter investigates CTQEC based on continuous weak measurements and feedback from the point of view of the subsystem principle, which states that protected quantum information is contained in a subsystem of the Hilbert space. We study how to approach the problem of constructing CTQEC protocols by looking at the evolution of the state of the system in an encoded basis i…
It is widely spread in the literature that non-Markovianity (NM) may be regarded as a resource in quantum mechanics. However, it is still unclear how and when this alleged resource may be exploited. Here, we study the relationship between NM and quantum optimal control under the objective of generating entanglement within M non-interacting subsystems, each one coupled to the same non-Markovian environment. Thus, we design a variety of entangling protocols that are only achievable due to the existence of the environment. We show that NM plays a crucial role in all the entangling protocols consi…
The question, whether an open system dynamics is Markovian or non-Markovian can be answered by studying the direction of the information flow in the dynamics. In Markovian dynamics, information must always flow from the system to the environment. If the environment is interacting with only one of the subsystems of a bipartite system, the dynamics of the entanglement in the bipartite system can be used to identify the direction of information flow. Here we study the dynamics of a two-level system interacting with an environment, which is also a heat bath, and consists of a large number of two-l…
Recent observation that almost all quantum states bear non-classical correlations [A. Ferraro et al, Phys. Rev. A 81, 052328 (2010)] may seem to imply that the Markovian bipartite systems are practically deprived of zero discord states. Nevertheless, complementary to the result of Ferraro et al, we construct a model of a Markovian bipartite system providing zero discord for arbitrary long time interval, that we term 'Markovian classicality'. Our model represents a matter-of-principle formal proof, i.e. a sufficient condition for the, otherwise not obvious, existence of Markovian classicality. …
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) quant…
Quantum correlation lies at the very heart of almost all the non-classical phenomena exhibited by quantum systems composed of more than one subsystem. In the recent days it has been pointed out that there exists quantum correlation, namely discord which is more general than entanglement. Some authors have investigated that for certain initial states the quantum correlations as well as classical correlation exhibit sudden change under simple Markovian noise. We show that, this dy- namical behavior of the both types of correlations can be explained using the idea of complementary correlations in…
Decoherence-free state (DFS) encoding supplies a useful way to avoid the detrimental influence of the environment on quantum information processing. The DFS was previously well established in either the two subsystems locating at the same spatial position or the dynamics under the Born--Markovian approximation. Here, we investigate the exact DFS of two spatially separated quantum systems consisting of two-level systems or harmonic oscillators coupled to a common non-Markovian zero-temperature bosonic environment. The exact distance-dependent DFS and the explicit criterion for forming the DFS a…
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, …
We introduce the Markovian matrix product density operator, which is a special subclass of the matrix product density operator. We show that the von Neumann entropy of such ansatz can be computed efficiently on a classical computer. This is possible because one can efficiently certify that the global state forms an approximate quantum Markov chain by verifying a set of inequalities. Each of these inequalities can be verified in time that scales polynomially with the bond dimension and the local Hilbert space dimension. The total number of inequalities scale linearly with the system size. We us…
The coarse-graining approach to deriving the quantum Markovian master equation is revisited, with close attention given to the underlying approximations. It is further argued that the time interval over which the coarse-graining is performed is a free parameter that can be given a physical measurement-based interpretation. In the case of the damping of composite systems to reservoirs of different temperatures, currently of much interest in the study of quantum thermal machines with regard to the validity of `local' and `global' forms of these equations, the coupling of the subsystems leads to …
Correlations between different regions of a quantum many-body system can be quantified through measures based on entropies of (reduced) subsystem states. For closed systems, several analytical and numerical tools, e.g., hydrodynamic theories or tensor networks, can accurately capture the time-evolution of subsystem entropies, thus allowing for a profound understanding of the unitary dynamics of quantum correlations. However, so far, these methods either cannot be applied to open quantum systems or do not permit an efficient computation of quantum entropies for mixed states. Here, we make progr…
We obtain new types of exponential decay laws for solutions of density-matrix master equations in the weak-coupling limit: after comparing with results already present in the literature and developing the necessary techniques, we study the crucial aspect of complete positivity under fairly general conditions. We propose a new type of time average that guarantees complete positivity and approximates, in markovian fashion, the exact dynamics for a plethora of physical applications, no matter which are the spectral properties of the subsystem, or its dimensions. We shall comment on some interesti…
We investigate nonlocal quantum correlations arising between multiple two-level impurity atoms coupled to an ultracold bosonic gas. We find that the environment-induced dynamics of the impurity subsystem can generate nonlocal states that are robust against noise and violate a multipartite Bell inequality when projective spin measurements are made. Genuine multipartite nonlocality is also observed in a system of three impurities. We show that non-Markovian effects, and the persistence of coherences in the impurity subsystem, are crucial for preventing complete loss of nonlocality and allow for …
