Quantum discord, as a measure of all quantum correlations, has been proposed as the key resource in certain quantum communication tasks and quantum computational models without containing much entanglement. Dakić, Vedral, and Brukner [Phys. Rev. Lett. 105, 190502 (2010)] introduced a geometric measure of quantum discord (GMQD) and derived an explicit formula for any two-qubit state. Luo and Fu [Phys. Rev. A 82, 034302 (2010)] introduced another form of GMQD and derived an explicit formula for arbitrary state in a bipartite quantum system. However, the explicit analytical expression for any bipartite system was not given. In this work, we give out the explicit analytical expressions of the GMQD for a two-parameter class of states in a qubit-qutrit system and study its dynamics for the states under various dissipative channels in the first time. Our results show that all these dynamic evolutions do not lead to a sudden vanishing of GMQD. Quantum correlations vanish at an asymptotic time for local or multi-local dephasing, phase-flip, and depolarizing noise channels. However, it does not disappear even though t ---> $\infty$ for local trit-flip and local trit-phase-flip channels. Our results maybe provide some important information for the application of GMQD in hybrid qubit-qutrit systems in quantum information.
In the paper we study the notion of trace norm geometric discord in the system of two qutrits. We compute this quantity for some class of states including qutrit Bell states, Werner states and bound entangled states.
We investigate the dynamics of quantum discord and entanglement for a class of mixed qubit-qutrit states assuming that only the qutrit is under the action of a dephasing channel. We demonstrate that even though the entanglement in the qubit-qutrit state disappears in a finite time interval, partial coherence left in the system enables quantum discord to remain invariant throughout the whole time evolution.
In this paper, by exploiting the weak measurement and quantum measurement re- versal (WMQMR) procedure, we propose a scheme to show how one can protect the geometric quantum discord (GQD) of a two-qutrit V-type atomic system each of which interacts with a dissipative reservoir independently. We examine the scheme for the GQD of the initial two-qutrit Werner and Horodecki states for different classes of weak mea- surement strengthes. It is found out that the presented protocol enables us to suppress decoherence due to the amplitude damping (AD) channel and preserve the quantum dis- cord of the …
We study the correlation dynamics of a system composed of arbitrary numbers of qutrits interacting with a common environment. Initially, the system is assumed to be in a low dimensional subspace of the Hamiltonian called "decoherence-free subspace". Environment induced quantum entanglement and discord is calculated between pair of qutrits of the system as measures of non-classical correlations. Finally the steady state distribution of entanglement and discord is determined with respect to the total number of qutrits of the system.
The dynamics of entanglement and quantum discord for qubit-qutrit systems are studied in the presence of phase damping and amplitude damping noises. Both one way and two couplings of the marginal systems with the environments are considered. Entanglement sudden death is unavoidable under any setup, however, the required time span depends on the way of coupling. On the other hand, the dynamics of quantum discord strongly depends both on the nature of environment and on the number of dimensions of the Hilbert space of the coupled marginal system. We show that freezing and invariance of quantum d…
We study the time evolution of classical and quantum correlations for hybrid qubit-qutrit systems in independent and common classical dephasing environments. Our discussion involves a comparative analysis of the Markovian dynamics of negativity, quantum discord, geometric measure of quantum discord and classical correlation. For the case of independent environments, we have demonstrated the phenomenon of sudden transition between classical and quantum decoherence for qubit-qutrit states. In the common environment case, we have shown that dynamics of quantum and geometric discords might be comp…
We study the measurement-induced geometric discord based on the trace norm and generalize some properties known for qutrits to qudits. Previous preliminary results for bipartite qutrit systems (i.e. d=3 systems) are here strictly proved for arbitrary d. Present study supports observations, coming also from other approaches, that systems with $d \geq 3$ show similar behaviour when quantum correlations are concerned, but there is pronounced difference between d=2 and d=3. Qubit systems are exceptionally simple. Underlying geometry of state spaces and related Lie groups are responsible for that.
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.
We investigate the dissipative dynamics of quantum discord in a decoherence model with two initially entangled qubits in addition to a quantum kicked top. The two qubits are uncoupled during the period of our study and one of them interacts with the quantum kicked top. We find that the long time behavior of quantum discord could be well described by the fidelity decay of the quantum kicked top; for short time behavior, however, the phase of the amplitude of the fidelity decay is necessary to provide more specific information about the system. We have made comparison between the quantum kicked …
We study the dynamics of geometric measure of quantum discord (GMQD) under the influences of two local phase damping noises. Consider the two qubits initially in arbitrary X-states, we find the necessary and sufficient conditions for which GMQD is unaffected for a finite period. It is further shown that such results also hold for the non-Markovian dephasing process.
We investigate the phenomenon of sudden transition between classical and quantum decoherence in the study of quantum discord for a dissipative cavity QED system, which consists of two noninteracting two-level atoms, each trapped in a dissipative cavity. It is found that the quantum discord between two atoms, which are prepared initially in the X-type quantum states, is not destroyed by the dissipation of the cavities for a finite time interval and the stationary quantum discord can arise in the interaction of atoms with cavities as the time approach to infinite. The transition time is sensitiv…
Due to decoherence, realistic quantum systems inevitably interact with the environment when quantum information is processed, which causes the loss of quantum properties. As a fundamental issue of quantum properties, quantum correlations have attracted a lot of interests in recent years. Because of the importance of high dimensional systems in quantum information, in this work, we study the quantum correlations affected by the Markovian environment by considering the quantum correlations of qutrit-qutrit quantum systems measured by the negativity and the geometric discord. The local noise chan…
The quantum dynamics of two entangled two-level atoms initially in extended Werner-like (EWL) states is studied. Each of the two atoms is located within an isolated and dissipative cavity. If the interaction time of atoms and cavities is not very long, the amount of quantum discord and entanglement between two atoms decreases as the system evolves. The sudden death of quantum discord and entanglement of two atoms occurs within a short interaction time. However, after a long interaction time, quantum discord and entanglement of two atoms could be partially preserved due to the long-lived nature…
This is a thesis submitted to university of Pune, India, for the Ph.D. degree. This work deals with entanglement production in two qubit, two qutrit and three qubit systems, entanglement in indistinguishable fermionic systems, quantum discord in a Heisenberg chain and geometric measure of quantum discord in an arbitrary state of a bipartite quantum system.
Quantum discord witnesses the nonclassicality of quantum states even when there is no entanglement in these quantum states. This type of quantum correlation also has some interesting and significant applications in quantum information processing. Quantum discord has been evaluated explicitly only for certain class of two-qubit states. We extend the previous studies to $2 \otimes d$ quantum systems and derive an analytical expression for quantum discord for a two-parameter class of states for $d \geq 3$. We compare quantum discord, classical correlation, and entanglement for qubit-qutrit system…
The study of quantum correlations in High-dimensional bipartite systems is crucial for the development of quantum computing. We propose relative entropy as a distance measure of correlations may be measured by means of the distance from the quantum state to the closest classical-classical state. In particular, We establish relations between relative entropy and quantum discord quantifiers obtained by means of orthogonal projective measurements. We show that for symmetrical X-states density matrices the quantum discord is equal to relative entropy. At the end of paper, various examples of X-sta…
The discovery of the intriguing phenomenon that certain kinds of quantum correlations remain impervious to noise up to a specific point in time and then suddenly decay, has generated immense recent interest. We exploit dynamical decoupling sequences to prolong the persistence of time-invariant quantum discord in a system of two NMR qubits decohering in independent dephasing environments. We experimentally prepare two-qubit Bell-diagonal quantum states that interact with individual dephasing channels and demonstrate the effect of dynamical decoupling on the preservation of both quantum and clas…
Quantum discord, a kind of quantum correlation based on entropic measures, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. Procedures are available for analytical calculation of discord when one of the parties is a qubit with dimension two and measurements made on it to get that one-way discord. We extend now to systems when both parties are of larger dimension, of interest to qudit-quDit with d, D > 2 or spin chains of spins > 1/2. While recognizing that no universal scheme is feasible, applicable to all density matrices, nevert…
We have investigated the effect of counter-rotating terms on the dynamics of entanglement and quantum discord between two identical atoms interacting with a lossy single mode cavity field for a system initially in a vacuum state. The counter-rotating terms are found to lead to steady states in the long time limit which can have high quantum discord, but have no entanglement. The effect of cavity decay rate on this steady state quantum discord has been also investigated, surprisingly, the increase in cavity decay rate is found to both enhance and maximize the steady quantum discord for separabl…
We present analytical and numerical simulation results for squeezing, entanglement, and quantum discord in a dissipatively coupled coherent Ising machine (CIM). Both analytical solutions and numerical simulation results, which are obtained with positive-P, truncated-Wigner and truncated-Husimi representations for the density operator, predict the presence of entanglement and quantum discord, below and above the threshold of CIM. The entanglement criteria and quantum discord are evaluated as a function of the dissipative coupling strength relative to the background loss. For coupled two DOPOs, …
We calculate the dissipative dynamics of two-qubit quantum discord under Markovian environments. We analyze various dissipative channels such as dephasing, depolarizing, and generalized amplitude damping, assuming independent perturbation, in which each qubit is coupled to its own channel. Choosing initial conditions that manifest the so-called sudden death of entanglement, we compare the dynamics of entanglement with that of quantum discord. We show that in all cases where entanglement suddenly disappears, quantum discord vanishes only in the asymptotic limit, behaving similarly to individual…
Recently it has been discovered that certain two-qutrit entangled states interacting with global and/or multi-local decoherence undergo distillability sudden death (DSD). We investigate this phenomenon for qutrit-qutrit systems interacting with statistically independent zero-temperature reservoirs. We show that certain initially prepared free-entangled states become bound-entangled in a finite time due to the action of Markovian dissipative environment. Moreover, in contrast with local dephasing, simple local unitary transformations can completely avoid distillability sudden death under amplit…
We have studied the dynamics of quantum correlations such as entanglement, Bell-nonlocality and quantum discord between identical as well as unidentical atoms interacting with a single-mode cavity field and subject to cavity decay. The effect of single atom detuning, cavity decay rate and initial preparation of the atoms on the corresponding correlation measures have been investigated. It is found that even under strong dissipation, time evolution can create high quantum discord while entanglement and Bell nonlocality stay zero for an initially separable state. Quantum discord increases while …
We investigate the time evolution of negativity and quantum discord for a pair of non-interacting qubits with one being weakly coupled to a decohering Davies--type Markovian environment. At initial time of preparation, the qubits are prepared in one of the maximally entangled pure Bell states. In the limiting case of pure decoherence (i.e. pure dephasing), both, the quantum discord and negativity decay to zero in the long time limit. In presence of a manifest dissipative dynamics, the entanglement negativity undergoes a sudden death at finite time while the quantum discord relaxes continuously…
Hilbert - Schmidt and trace norm geometric quantum discord are compared with regard to their behavior during local time evolution. We consider the system of independent two - level atoms with time evolution given by the dissipative process of spontaneous emission. It is explicitly shown that the Hilbert - Schmidt norm discord has nonphysical properties with respect to such local evolution and cannot serve as a reasonable measure of quantum correlations and the better choice is to use trace norm discord as such a measure.
Engineered dynamical maps combining coherent and dissipative transformations of quantum states with quantum measurements, have demonstrated a number of technological applications, and promise to be a crucial tool in quantum thermodynamic processes. Here, we exploit the control on the effective open spin qutrit dynamics of an NV center, to experimentally realize an autonomous feedback process (Maxwell demon) with tunable dissipative strength. The feedback is enabled by random measurement events that condition the subsequent dissipative evolution of the qutrit. The efficacy of the autonomous Max…
We analyze the roles of initial correlations between the two-qubit system and a dissipative cavity on quantum discord dynamics of two qubits. Considering two initial system-cavity states, we show that the initial system-cavity correlations not only can initially increase the two-qubit quantum discord but also would lead to a larger long-time quantum discord asymptotic value. Moreover, quantum discord due to initial correlations is more robust than the case of the initial factorized state. Finally, we show the initial correlations' importance for dynamics behaviors of mutual information and cla…
The inevitable dissipative interaction of an entangled quantum system with its environment causes degradation in quantum correlations present in the system. This can lead to a finite-time disappearance of entanglement, which is known as entanglement sudden death (ESD). Here, we consider an initially entangled qubit-qutrit system and a dissipative noise which leads to ESD, and propose a set of local unitary operations, which when applied on the qubit, qutrit, or both subsystems during the decoherence process, cause ESD to be hastened, delayed, or avoided altogether, depending on its time of app…
In this work, a mechanism for long time protection of entanglement of a two-qutrit V-type atomic system each of which interacts independently with a dissipative reservoir is investigated. It is shown that the protection process is related exclusively on the formation of bound states in the system-reservoir energy spectrum. It is demonstrated that the degree of boundedness for the system-reservoir bound states is well-controlled by entering other V-type qutrits into the reservoirs, and also through the manipulation of the spontaneously generated interference related to decay channels of the V-t…
A symmetric measure of quantum correlation based on the Hilbert-Schmidt distance is presented in this paper. For two-qubit states, we simplify considerably the optimization procedure so that numerical evaluation can be performed efficiently. Analytical expressions for the quantum correlation are attained for some special states. We further investigate the dynamics of quantum correlation of the system qubits in the presence of independent dissipative environments. Several nontrivial aspects are demonstrated. We find that the quantum correlation can increase even if the system state is suffering…
We give the explicit expressions of the pairwise quantum correlations present in superpositions of multipartite coherent states. A special attention is devoted to the evaluation of the geometric quantum discord. The dynamics of quantum correlations under a dephasing channel is analyzed. A comparison of geometric measure of quantum discord with that of concurrence shows that quantum discord in multipartite coherent states is more resilient to dissipative environments than is quantum entanglement. To illustrate our results, we consider some special superpositions of Weyl-Heisenberg, SU(2) and SU…
In this work, we investigate the dynamics of quantum correlations captured by entropic and geometric measures of discord under the influence of dissipative channels for widely used two qubit X state with maximally mixed marginals. Identifying the phenomena of the preservation, sudden change and revival of quantum correlation quantifiers, we determine their hierarchy of robustness under memory-less decoherence channels. We find the analytical expressions of decoherence probabilities at which sudden change occur when the first qubit is subjected to the decoherence channels for multiple times. We…
We investigate the dynamics of entanglement for a two-parameter class of states in a hybrid qubit-qutrit system under the influence of various dissipative channels. Our results show that entanglement sudden death (ESD) is a general phenomenon and it usually takes place in a qubit-qutrit system interacting with various noisy channels, not only the case with dephasing and depolarizing channels observed by others. ESD can only be avoided for some initially entangled states under some particular noisy channels. Moreover, the environment affects the entanglement and the coherence of the system in v…
Quantum discord is a prominent measure of quantum correlations, playing an important role in expanding its horizon beyond entanglement. Here we provide an operational meaning of (geometric) discord, which quantifies the amount of non-classical correlation of an arbitrary quantum system in terms of its minimal distance from the set of classical states, in terms of teleportation fidelity for general two qubit and $d \otimes d$ dimensional isotropic and Werner states. A critical value of the discord is found beyond which the two qubit state must violate the Bell inequality. This is illustrated by…
The dynamics of entanglement and quantum discord between two identical qubits strongly interacting with a common single mode leaky cavity field have been investigated beyond the rotating wave approximation (RWA) by using recently derived Lindblad type quantum optical master equation [F. Beaudoin, J.M. Gambetta, A. Blais, Phys. Rev. A {\bf 84}, 043832 (2011)] that can describe the losses of the cavity field in a strong atom-field coupling regime. Contrary to previous investigations of the same model in the dissipative regime by using the standard Lindblad quantum optical master equation in a st…
We investigated the dissipative dynamics of quantum discord for correlated qubits under Markovian environments. The basic idea in the present scheme is that quantum discord is more general, and possibly more robust and fundamental, than entanglement. We provide three initially correlated qubits in pure Greenberger-Horne-Zeilinger (GHZ) or W state and analyse the time evolution of the quantum discord under various dissipative channels such as: Pauli channels $σ_{x}$, $σ_{y}$, and $σ_{z}$, as well as depolarising channels. Surprisingly, we find that under the action of Pauli channel $σ_{x}$, the…
Heralded near-deterministic multi-qubit controlled phase gates with integrated error detection have recently been proposed by Borregaard et al. [Phys. Rev. Lett. 114, 110502 (2015)]. This protocol is based on a single four-level atom (a heralding quartit) and $N$ three-level atoms (operational qutrits) coupled to a single-resonator mode acting as a cavity bus. Here we generalize this method for two distant resonators without the cavity bus between the heralding and operational atoms. Specifically, we analyze the two-qubit controlled-Z gate and its multi-qubit-controlled generalization (i.e., a…
Quantum reservoir engineering is a powerful framework for autonomous quantum state preparation and error correction. However, traditional approaches to reservoir engineering are hindered by unavoidable coherent leakage out of the target state, which imposes an inherent trade off between achievable steady-state state fidelity and stabilization rate. In this work we demonstrate a protocol that achieves trade off-free Bell state stabilization in a qutrit-qubit system realized on a circuit-QED platform. We accomplish this by creating a purely dissipative channel for population transfer into the ta…
Entanglement can be considered as a special quantum correlation, but not the only kind. Even for a separable quantum system, it is allowed to exist non-classical correlations. Here we propose two dissipative schemes for generating a maximally correlated state of two qubits in the absence of quantum entanglement, which was raised by [F. Galve, G. L. Giorgi, and R. Zambrini, {\color{blue}Phys. Rev. A {\bf 83}, 012102 (2011)}]. These protocols take full advantages of the interaction between four-level atoms and strongly lossy optical cavities. In the first scenario, we alternatively change the ph…
We propose local strategies to protect global quantum information. The protocols, which are quantum error correcting codes for dissipative systems, are based on environment measurements, direct feedback control and simple encoding of the logical qubits into physical qutrits whose decaying transitions are indistinguishable and equally probable. The simple addition of one extra level in the description of the subsystems allows for local actions to fully and deterministically protect global resources, such as entanglement. We present codes for both quantum jump and quantum state diffusion measure…
We investigate the quantum Zeno and anti-Zeno effects in quantum dissipative systems by employing a hierarchical equations of motion approach which is beyond the usual Markovian approximation, the rotating wave approximation, and the perturbative approximation. The quantum Zeno and anti-Zeno dynamics of a biased qubit-boson model and a biased qutrit-boson model are provided as illustrative examples. It is found that (i) there exists multiple Zeno-anti-Zeno crossover phenomena, (ii) the non-Markovian characteristic of the bath may be favorable for the accessibility of the Zeno dynamics, and (ii…
We investigate the quantum-to-classical crossover of a dissipative optical cavity mode based on measurement of the correlations between two atoms which do not interact with each other, but interact with the cavity mode. Firstly, we note that there is a time window where the mode has a classical behavior, which depends on the cavity decay rate, the atom-field coupling strength and the number of atoms. Then, considering only two atoms inside the cavity and working in the steady state of the system, we note that the entanglement between the atoms disappears with the increasing the mean number of …
Quantum entanglement is widely recognized as one of the key resources for the advantages of quantum information processing, including universal quantum computation, reduction of communication complexity or secret key distribution. However, computational models have been discovered, which consume very little or no entanglement and still can efficiently solve certain problems thought to be classically intractable. The existence of these models suggests that separable or weakly entangled states could be extremely useful tools for quantum information processing as they are much easier to prepare a…
In this paper we use the Fano representation of two-qubit states from which we can identify a correlation matrix containing the information about the classical and quantum correlations present in the bipartite quantum state. To illustrate the use of this matrix, we analyze the behavior of the correlations under non-dissipative decoherence in two-qubit states with maximally mixed marginals. From the behavior of the elements of the correlation matrix before and after making measurements on one of the subsystems, we identify the classical and quantum correlations present in the Bell-diagonal stat…
Single and coupled quantum van der Pol oscillators: Wigner-function dynamics, exact qutrit solutions, and synchronization
Tata Institute of Fundamental Research（塔塔基础研究所）；Indian Institute of Science Education and Research（印度科学教育与研究学院）；Universität des Saarlandes（萨尔兰大学）；Center for Quantum Technologies (QuTe), Saarland University（萨尔兰大学量子技术中心）
It was recently discovered that the quantum correlations of a pair of disentangled qubits, as measured by the quantum discord, can increase solely because of their interaction with a local dissipative bath. Here, we show that a similar phenomenon can occur in continuous-variable bipartite systems. To this aim, we consider a class of two-mode squeezed thermal states and study the behavior of Gaussian quantum discord under various local Markovian non-unitary channels. While these in general cause a monotonic drop of quantum correlations, an initial rise can take place with a thermal-noise channe…
Invariant-based master equation applied to driven qutrit coupled to a bath and a leaky cavity
Homer L. Dodge Department of Physics and Astronomy, The University of Oklahoma（俄克拉荷马大学霍默·L·道奇物理与天文系）；Center for Quantum Research and Technology, The University of Oklahoma（俄克拉荷马大学量子研究与技术中心）；School of Electrical and Computer Engineering, The University of Oklahoma（俄克拉荷马大学电气与计算机工程学院）
By employing the spin-boson model in a dense limit of environmental modes, quantum entanglement and correlation of sub-Ohmic and Ohmic baths for dissipative quantum phase transitions are numerically investigated based on the variational principle. With several measures borrowed from quantum information theory, three different types of singularities are found for the first-order, second-order, and Kosterlitz-Thouless phase transitions, respectively, and the values of transition points and critical exponents are accurately determined. Besides, the scaling form of the quantum discord in the Ohmic…
