Different indicators for Markovian and non-Markovian dynamics

The Markovianity/non-Markovianity of two different systems are discussed by means of the quantum speed limit time and quantum Fisher information. The first system is described by a central mass particle interacts locally with its surrounding particles, while the second and third models consist of a single qubit interacts with a non-detuning Lorentzian cavity and with a thermal reservoir, respectively. For the first model, the large distance between the central particle and the surrounding particles is guaranty for a fixed quantum speed limit, while the driving time plays the central role on the fixed behavior of the quantum speed limit time. Due to the stable behavior of the quantum speed limit time and the quantum Fisher information, the exchange information between the systems and their surrounding is limited. The distance between the central mass particle and its surrounding particle plays the main role on predicating the Markovianity/non-Markovianity. For the second system the driving time is an important parameter that control on the Markovianity/non-Markovianity behavior. Finally the third model proves that non-Markovianity dynamic may increase the speed and the sensitivity of the open system.     

Memory Effects in Quantum Dynamics Modelled by Quantum Renewal Processes

Simple, controllable models play an important role in learning how to manipulate and control quantum resources. We focus here on quantum non-Markovianity and model the evolution of open quantum systems by quantum renewal processes. This class of quantum dynamics provides us with a phenomenological approach to characterise dynamics with a variety of non-Markovian behaviours, here described in terms of the trace distance between two reduced states. By adopting a trajectory picture for the open quantum system evolution, we analyse how non-Markovianity is influenced by the constituents defining the quantum renewal process, namely the time-continuous part of the dynamics, the type of jumps and the waiting time distributions. We focus not only on the mere value of the non-Markovianity measure, but also on how different features of the trace distance evolution are altered, including times and number of revivals.     

Quantum Speed Limit and Divisibility of the Dynamical Map

The quantum speed limit (QSL) is the theoretical lower limit of the time for a quantum system to evolve from a given state to another one. Interestingly, it has been shown that non-Markovianity can be used to speed-up the dynamics and to lower the QSL time, although this behaviour is not universal. In this paper, we further carry on the investigation on the connection between QSL and non-Markovianity by looking at the effects of P- and CP-divisibility of the dynamical map to the quantum speed limit. We show that the speed-up can also be observed under P- and CP-divisible dynamics, and that the speed-up is not necessarily tied to the transition from P-divisible to non-P-divisible dynamics.     

Quantum speed limits of a qubit system interacting with a nonequilibrium environment

The speed of evolution of a qubit undergoing a nonequilibrium environment with spectral density of general ohmic form is investigated. First we reveal non-Markovianity of the model, and find that the non-Markovianity quantified by information backflow of Breuer et al. [Phys. Rev. Lett. 103 210401(2009)] displays a nonmonotonic behavior for different values of the ohmicity parameter s in fixed other parameters and the maximal non-Markovianity can be achieved at a specified value s. We also find that the non-Markovianity displays a nonmonotonic behavior with the change of a phase control parameter. Then we further discuss the relationship between quantum speed limit(QSL) time and non-Markovianity of the open-qubit system for any initial states including pure and mixed states. By investigation, we find that the QSL time of a qubit with any initial states can be expressed by a simple factorization law: the QSL time of a qubit with any qubitinitial states are equal to the product of the coherence of the initial state and the QSL time of maximally coherent states,where the QSL time of the maximally coherent states are jointly determined by the non-Markovianity, decoherence factor and a given driving time. Moreover, we also find that the speed of quantum evolution can be obviously accelerated in the wide range of the ohmicity parameter, i.e., from sub-Ohmic to Ohmic and super-Ohmic cases, which is different from the thermal equilibrium environment case.     

Non-Markovianity and Indivisible Quantum dynamics of a Apin-S System

Using a measure for the divisibility of a dynamical map,we study the non-Markovian character of a quantum evolution of a spin-S system,which is in an external field and weakly coupled to a bosonic bath with a certain temperature.The finite-temperature dynamics of the open system is obtained by the time-convolutionless master equation in the secular approximation.Besides the influence of the environmental spectral density function,the external field and low temperatures can afect the quantum non-Markovianity.It is found out that the non-Markovian feature of a dynamical map of a high-dimensional spin system is noticeable in contrast to that of a low-dimension spin system.     

Control of trapped-ion quantum states with optical pulses

We present new results on the quantum control of systems with infinitely large Hilbert spaces. A control-theoretic analysis of the control of trapped-ion quantum states via optical pulses is performed. We demonstrate how resonant bichromatic fields can be applied in two contrasting ways--one that makes the system completely uncontrollable and the other that makes the system controllable. In some interesting cases, the Hilbert space of the qubit-harmonic oscillator can be made finite, and the Schr?dinger equation controllable via bichromatic resonant pulses. Extending this analysis to the quantum states of two ions, a new scheme for producing entangled qubits is discovered.     

Effect of entanglement embedded in environment on quantum non-Markovianity based on collision model

By means of collision models(CMs) where the environment is simulated by a collection of ancillas consisting of two entangled qubits, we investigate the effects of entanglement in the environment on the non-Markovianity of an open quantum system. Two CMs are considered in this study, in the first one the open quantum system S directly collides with the environment,while in the second one the system interacts with two intermediate qubits which, in turn, are coupled to the environment. We show that it is possible to enhance the non-Markovianity by environment entanglement in both models. In particular, in the second model, we show that the initial state of the auxiliary qubits can also affect the non-Markovianity of the system and there exists the optimal combination of the initial environmental state and the initial state of auxiliary qubits. In this case, the non-Markovianity can be greatly enhanced.     

The Role of the Total Entropy Production in the Dynamics of Open Quantum Systems in Detection of Non-Markovianity

The interaction between system and environment is a fundamental concept in the theory of open quantum systems. As a result of the interaction, an amount of correlation (both classical and quantum) emerges between the system and the environment. In this work, we recall the quantity that will be very useful to describe the emergence of the correlation between the system and the environment, namely, the total entropy production. Appearance of total entropy production is due to the entanglement production between the system and the environment. In this work, we discuss about the role of the total entropy production for detecting the non-Markovianity. By utilizing the relation between the total entropy production and total correlation between subsystems, one can see a temporary decrease of total entropy production is a signature of non-Markovianity. We apply our criterion for the special case, where the composite system has initial correlation with environment.     

Detecting Non-Markovianity via Linear Entropy of Choi State

Non-Markovian dynamics detection is one of the most popular subjects in quantum information science. In this paper, a linear -entropy-based non-Markovianity witness scheme is constructed. The positive definiteness of the Choi state will be broken in non-Markovian evolution, which can be witnessed by its linear entropy. Thus, the linear entropy of the Choi state can be used to witness non-Markovian dynamics. The effectiveness of the proposed method is verified using the pure dephasing channel as an example. Finally, it is shown that this method can be extended to the one based on Rényi entropy.     

Non-Markovianity in the presence of multiple thermal environments via collision model

By means of the collision model, we study the non-Markovianity of an open quantum system S being coupled to M thermal reservoirs. In our model, each reservoir is modeled as a chain of ancillas whose intracollisions account for the occurrence of non-Markovian dynamics. We show that by incorporating M reservoir ancillas into the system, the non-Markovian dynamics of S can be embedded in the extended system that experiences a completely Markovian dynamics. The number M of involved reservoirs can thus be identified as the memory depth and determines the degree of the non-Markovianity. In the equilibrium case with identical temperatures for all the reservoirs, we show that though the non-Markovianity is proportional to M in the zero and relatively low temperature regimes, in the relatively high temperature regime such proportional relation holds only for the weak intracollisions of the reservoir ancillas. In the nonequilibrium situation, we examine the effect of temperature difference of reservoirs on the non-Markovianity. Focusing on a simple situation with two reservoirs, we observe that the nonzero temperature difference has a significant impact on the non-Markovianity.     

Dynamics of non-Markovianity in the presence of a driving field

We investigate a two-level system in a cavity QED by considering the effects of amplitude damping, phase damping and driving field. We have studied the non-Markovianity in resonance and non-resonance limits in the presence of these effects using Breuer–Laine–Piilo (BLP) non-Markovianity measure (N_BLP). The evolution of the system is derived using the time convolutionless (TCL) master equation. In some conditions, it is shown that in the presence of a driving field, the N_BLP increases in the resonance and non-resonance limits. We have also found the exact solution of the master equation in order to investigate the effect of temperature- and environment-excited states. We have shown that the behaviour of non-Markovianity is very different from what one can see from the TCL approach. We have also presented some explanation about the behaviour of non-Markovianity in the exact solution using quantum discord (QD).     

Majorana's stellar representation for the quantum geometric tensor of symmetric states

Majorana's stellar representation provides an intuitive picture in which quantum states in high-dimensional Hilbert space can be observed using the trajectory of Majorana stars. We consider the Majorana's stellar representation of the quantum geometric tensor for a spin state up to spin-3/2. The real and imaginary parts of the quantum geometric tensor, corresponding to the quantum metric tensor and Berry curvature, are therefore obtained in terms of the Majorana stars. Moreover, we work out the expressions of quantum geometric tensor for arbitrary spin in some important cases. Our results will benefit the comprehension of the quantum geometric tensor and provide interesting relations between the quantum geometric tensor and Majorana's stars.     

Distribution of non-Markovian intervals for open qubit systems

We study the non-Markovianity of open qubit systems using the measure N proposed by Breuer, Laine and Piilo [Phys. Rev. Lett. 103 210401 (2009)]. We find that for the three types of quantum noises, amplitude-damping, dephasing and depolarizing noises, there exist some non-Markovian time intervals whose distribution is independent of the selection of the pair of initial states. Therefore, the maximization in the definition of measure N can be actually removed without influencing the detection of non-Markovianity.     

Testing the dimension of Hilbert spaces

Given a set of correlations originating from measurements on a quantum state of unknown Hilbert space dimension, what is the minimal dimension d necessary to describe such correlations? We introduce the concept of dimension witness to put lower bounds on d. This work represents a first step in a broader research program aiming to characterize Hilbert space dimension in various contexts related to fundamental questions and quantum information applications.     

Classical Systems, Standard Quantum Systems, and Mixed Quantum Systems in Hilbert Space

Traditionally, there has been a clear distinction between classical systems and quantum systems, particularly in the mathematical theories used to describe them. In our recent work on macroscopic quantum systems, this distinction has become blurred, making a unified mathematical formulation desirable, so as to show up both the similarities and the fundamental differences between quantum and classical systems. This paper serves this purpose, with explicit formulations and a number of examples in the form of superconducting circuit systems. We introduce three classes of physical systems with finite degrees of freedom: classical, standard quantum, and mixed quantum, and present a unified Hilbert space treatment of all three types of system. We consider the classical/quantum divide and the relationship between standard quantum and mixed quantum systems, illustrating the latter with a derivation of a superselection rule in superconducting systems.     

Time-dependent Hilbert spaces, geometric phases, and general covariance in quantum mechanics

We investigate consequences of allowing the Hilbert space of a quantum system to have a time-dependent metric. For a given possibly nonstationary quantum system, we show that the requirement of having a unitary Schrödinger time-evolution identifies the metric with a positive-definite (Ermakov–Lewis) dynamical invariant of the system. Therefore the geometric phases are determined by the metric. We construct a unitary map relating a given time-independent Hilbert space to the time-dependent Hilbert space defined by a positive-definite dynamical invariant. This map defines a transformation that changes the metric of the Hilbert space but leaves the Hamiltonian of the system invariant. We propose to identify this phenomenon with a quantum mechanical analogue of the principle of general covariance of general relativity. We comment on the implications of this principle for geometrically equivalent quantum systems and investigate the underlying symmetry group.     

On a tilted Liouville-master equation of open quantum systems

A tilted Liouville-master equation in Hilbert space is presented for Markovian open quantum systems. We demonstrate that it is the unraveling of the tilted quantum master equation. The latter is widely used in the analysis and calculations of stochastic thermodynamic quantities in quantum stochastic thermodynamics.     

Finite quantum environments as thermostats: an analysis based on the Hilbert space average method

We consider discrete quantum systems coupled to finite environments which may possibly consist of only one particle in contrast to the standard baths which usually consist of continua of oscillators, spins, etc. We find that such finite environments may, nevertheless, act as thermostats, i.e., equilibrate the system though not necessarily in the way predicted by standard open system techniques. Thus, we apply a novel technique called the Hilbert space Average Method (HAM) and verify its results numerically.