Geometric Discord Research Articles

Characterizing quantum correlation with the correlated part of partial coherence

One of the fundamental features of the quantum world that distinguishes it from the classical counterpart is the existence of quantum correlations. In this paper, we study the quantum correlation from the viewpoint of partial coherence. Partial coherence in a bipartite system often contains local coherence of a subsystem. By removing the local coherence, we show that the remaining part of partial coherence can be regarded as quantum correlations. In this way, we introduce two quantum correlations: partial correlated coherence (PCC) and minimal partial correlated coherence (MPCC). Furthermore, we prove that PCC and MPCC induced by contractive pseudodistances are good candidates of quantum correlations. For contractive pseudodistances, we also show that the induced geometric partial correlated coherence (GPCC) and geometric minimal partial correlated coherence are actually the upper and lower bound of corresponding geometric quantum discord. Particularly, GPCCs based on a family of distances are proved to be bona fide measures of quantum correlations. By connecting to local quantum uncertainty, we provide an operational meaning for geometric quantum discord and GPCC induced by affinity of distance.

Comparing geometric discord and negativity for bipartite states

The geometric discord D of a state is a measure of the quantumness of the state and the negativity N is a measure of the entanglement of a state. It was proved by D. Girolami and G. Adesso that for states on C2⊗C2, the geometric discord is always greater than or equal to the square of the negativity and conjectured that this holds in general. S. Rana and P. Parashar showed that this relation does not hold for all states on C2⊗Cn for n>2. We provide several analytic families of states on C2⊗C3 violating this relation. Certain upper and lower bounds for N2−D are obtained for states on Cm⊗Cn for any m,n∈N.

Geometrical quantum discord and negativity of two separable and mixed qubits**Project supported by the National Natural Science Foundation of China (Grant No. 11604090).

We studied quantum correlation and quantum entanglement of a quantum system in which a coherent state light field interacts with two qubits that are initially prepared in a separable and mixed state. The influence of mean photon number of the coherent field and distribution probability of the atom on the geometrical quantum discord and the negativity are discussed. Our results show that the mean photon number of light field and distribution function of the atom can regulate and control the quantum correlation and quantum entanglement.

Control of atom-mirror entanglement versus Gaussian geometric discord with RWA

In this study, we control the quantum correlations existing between a movable mirror and atoms in hybrid atom-optomechanical system using rotating wave approximation (RWA) in adiabatic regime. We use the Mancini criterion to measure the entanglement, the purity to quantify the degree of mixedness and the Gaussian geometric discord (GGD) to characterize the quantum correlations even beyond entanglement. We study the effect of the optomechanical cooling rate and the cooperativity atomic on the transfer of quantum correlations between the movable mirror and atoms under the thermal effect. We also investigate the robustness of the GGD with respect to entanglement by exploiting recent experimental parameters.

Machine learning study of the relationship between the geometric and entropy discord

As an important resource to realize quantum information, quantum correlation displays different behaviors, freezing phenomenon and nonlocalization, which are dissimilar to the entanglement and classical correlation, respectively. In our setup, the ordering of the value of quantum correlation is represented for different quantization methods by considering an open quantum system scenario. The machine learning method (neural network method) is then adopted to train for the construction of a bridge between the Rényi discord and the geometric discord (Bures distance) for X form states. Our results clearly demonstrate that the machine learning method is useful for studying the differences and commonalities of different quantizing methods of quantum correlation.

Steady geometric quantum discord in driven coupled quantum spin systems

A geometric measure of the quantum discord in driven coupled quantum spin systems is investigated. In the rotating wave and Born–Markov approximation, by calculating the quantum master equation, the effects of noise strength and spin-coupling constant on quantum discord have been discussed in detail. The results show that we need to reduce the noise strength to get a higher value of geometric quantum discord. We also suggest a more effective way to approach the steady geometric quantum discord by controlling the noise strength and coupling constant.

Study on Non-Commutativity Measure of Quantum Discord

In this paper, we are concerned with the non-commutativity measure of quantum discord. We first present an explicit expression of the non-commutativity measure of quantum discord in the two-qubit case. Then we compare the geometric quantum discords for two dynamic models with their non-commutativity measure of quantum discords. Furthermore, we show that the results conducted by the non-commutativity measure of quantum discord are different from those conducted by both or one of the Hilbert-Schmidt distance discord and trace distance discord. These intrinsic differences indicate that the non-commutativity measure of quantum discord is incompatible with at least one of the well-known geometric quantum discords in the quantitative and qualitative representation of quantum correlations.

Fidelity-based unitary operation-induced quantum correlation for continuous-variable systems

We propose a measure of nonclassical correlation [Formula: see text] in terms of local Gaussian unitary operations based on square of the fidelity [Formula: see text] for bipartite continuous-variable systems. This quantity is easier to be calculated or estimated and is a remedy for the local ancilla problem associated with the geometric measurement-induced nonlocality. A simple computation formula of [Formula: see text] for any [Formula: see text]-mode Gaussian states is presented and an estimation of [Formula: see text] for any [Formula: see text]-mode Gaussian states is given. For any [Formula: see text]-mode Gaussian states, [Formula: see text] does not increase after performing a local Gaussian channel on the unmeasured subsystem. Comparing [Formula: see text] in scale with other quantum correlations such as Gaussian geometric discord for two-mode symmetric squeezed thermal states reveals that [Formula: see text] is much better in detecting quantum correlations of Gaussian states.

Characterizing nonclassical correlation using affinity

Geometric discord, a measure of quantumness of bipartite system, captures minimal nonlocal effects of a quantum state due to locally invariant von Neumann projective measurements. Original version of this measure is suffered by the local ancilla problem. In this article, we propose a new version of geometric discord using affinity. This quantity satisfies all criteria of a good measure of quantum correlation of the bipartite system and resolves local ancilla problem of Hilbert-Schmidt norm based discord. We evaluate analytically the proposed quantity for both pure and mixed states. For an arbitrary pure state, it is shown that affinity based geometric discord equal to geometric measure of entanglement. Further, we obtain a lower bound of this measure for m \times n dimensional arbitrary mixed state and a closed formula of proposed version of geometric discord for 2 \times n dimensional mixed state is obtained. Finally, as an illustration, we have studied the nonlocality of Bell diagonal state, isotropic and Werner states.

Two-qubit trace-norm geometric discord: the complete solution

We present the complete solution of the problem of determination of trace-norm geometric discord for arbitrary two-qubit state. Final answer is achieved due to effective reduction of the problem to the study of critical points of certain mapping depending on projectors. Our results are illustrated on various, also new, families of two-qubit states and compared to already known special solutions.

Effect of the Dzyaloshinski Moria Interaction on Quantum Correlations for the XXZ Heisenberg Spin Chain

We have studied the effect of the Dzyaloshinskii Moria (DM) interaction on geometric quantum discord (GQD) and concurrence for two spin 1/2 particles in the XXZ Heisenberg model, we have also investigated the effects of the temperature, the strength of DM interaction (D) and the anisotropic coupling constant of the Heisenberg model Jz on them in the GQD and concurrence. Further, we have compared the behaviors of GQD and entanglement for our system under the above parameters.

Influences of Detuning on Geometrical Quantum Discord Dynamics in Double Jaynes-Cummings System

We consider the case that the two-level atom off-resonantly interacts with cavity field via a one-photon hopping in double Jaynes-Cummings system. Geometrical quantum discord (GQD) is adopted to quantify the degree of correlation between two subsystems. GQD dynamics between two atoms, between cavity fields, as well as between atom and cavity fields are studied. The influences of detuning on GQD are discussed. The results obtained using the numerical method show that GQD between two atoms, between cavity fields, and between atom and cavity fields all display periodic oscillations. Furthermore, their periodic oscillations frequencies increase with the increase of detuning. On the other hand, GQD between two cavity fields and that between atom and cavity are weakened with increase of detuning.

Asymmetric broadcasting of quantum correlations

In this work, we exhaustively investigate $1 \rightarrow 2$ local and nonlocal broadcasting of entanglement as well as correlations beyond entanglement (geometric discord) using asymmetric Pauli cloners with most general two qubit state as the resource. We exemplify asymmetric broadcasting of entanglement using Maximally Entangled Mixed States. We demonstrate the variation of broadcasting range with the amount of entanglement present in the resource state as well as with the asymmetry in the cloner. We show that it is impossible to optimally broadcast geometric discord with the help of these asymmetric Pauli cloning machines. We also study the problem of $1 \rightarrow 3$ broadcasting of entanglement using non-maximally entangled state (NME) as the resource. For this task, we introduce a method we call successive broadcasting which involves application of $1 \rightarrow 2$ optimal cloning machines multiple times. We compare and contrast the performance of this method with the application of direct $1 \rightarrow 3$ optimal cloning machines. We show that $1 \rightarrow 3$ optimal cloner does a better job at broadcasting than the successive application of $1 \rightarrow 2$ cloners and the successive method can be beneficial in the absence of $1 \rightarrow 3$ cloners. We also bring out the fundamental difference between the tasks of cloning and broadcasting in the final part of the manuscript. We create examples to show that there exist local unitaries which can be employed to give a better range for broadcasting. Such unitary operations are not only economical, but also surpass the best possible range obtained using existing cloning machines enabling broadcasting of lesser entangled states. This result opens up a new direction in exploration of methods to facilitate broadcasting which may outperform the standard strategies implemented through cloning transformations.

Pairwise nonclassical correlations for superposition of Dicke states via local quantum uncertainty and trace distance discord

The pairwise nonclassical correlations for two-qubit states, extracted from multi-qubit system with exchange symmetry and parity, are quantified by local quantum uncertainty and trace distance discord. The explicit expressions of local quantum uncertainty and geometric trace distance discord for Dicke states and their superpositions are given. A comparison between the two quantum correlations quantifiers is discussed.

Quantum Correlation Based on Uhlmann Fidelity for Gaussian States.

A quantum correlation for -mode continuous-variable systems is introduced in terms of local Gaussian unitary operations performed on Subsystem A based on Uhlmann fidelity F. This quantity is a remedy for the local ancilla problem associated with the geometric measurement-induced correlations; is local Gaussian unitary invariant; is non-increasing under any Gaussian quantum channel performed on Subsystem B;and is an entanglement monotone when restricted to pure Gaussian states in the -mode case. A concrete formula for -mode symmetric squeezed thermal states (SSTSs) is presented. We also compare with other quantum correlations in scale, such as Gaussian quantum discord and Gaussian geometric discord, for two-mode SSTSs, which reveals that has some advantage in detecting quantum correlations of Gaussian states.

Geometrical Quantum Discord in the Coupled Cavities System with Tetrahedral Structure

In this paper we consider the situation that four identical two-level atoms are separately trapped in four single-mode optical cavities, which are placed at the vertices of a tetrahedron and coupled by six fibers. Each atom resonantly interacts with cavity via a one-photon hopping. The evolution of the state vector of the system is given by solving the schrodinger equation when the total excitation number of the system equals one. The geometrical quantum discords between atoms and between cavities are investigated. The effects of cavity-fiber coupling coefficient on the geometrical quantum discords between atoms and between cavities are discussed. The results obtained using the numerical method indicate that the geometrical quantum discords between atoms and between cavities are all weakened with increase of cavity-fiber coupling coefficient.

Squeezing Effect on Entanglement Swapping: A Quantum Trajectory Approach

Using the results of our previous work (J. Korean Phys. Soc. 56, 10 (2010)), we calculate the negativity N and the geometric discord G of the final qubits of the entanglement swapping protocol through an amplitude-damping multi-qubit channel interacting with a squeezed thermal bath. Our numerical results confirm that the squeezing suppresses the decay rate of quantum coherence.

Entanglement, EPR steering and Gaussian geometric discord in a double cavity optomechanical systems

The aim of the present paper is the quantification of non-classical correlations between mechanical and optical modes in an optomechanical system with two spatially separated Fabry-Perot cavities, using resolver sideband regime. To measure the amount of stationary entanglement, we use the criterion of Reid et al, while the evolution of the Gaussian quantum steering and its asymmetry make it possible to study the steerability of the correlations transfer between the different modes. Then, a theoretical study of the robustness of the Gaussian geometric discord (GGD), with respect to the temperature of the environment, is realized, for the two-mode symmetrical squeezed thermal states, to quantify quantum correlations beyond entanglement between mechanical and optical modes. By considering recent experimental parameters, we show that GGD is more robust than entanglement for a high thermal bath temperature. Finally, we find that these three indicators of quantumness enhance with the increase of the squeezing parameter.

Thermal quantum correlation and entanglement in the Bose–Hubbard Hamiltonian

We study a two-qutrit system which is described by the Bose–Hubbard Hamiltonian with two external magnetic fields. The entanglement (through the negativity) and quantum correlation (through the geometric discord) between the qutrits are calculated as functions of the magnetic field (B), the temperature (T), the linear and nonlinear coupling constants among two spins (J and K). Then, we compare the effect of these parameters on entanglement and quantum correlation of this system. For some values of system parameters, we show that the negativity is zero while, the geometric discord is nonzero. Moreover, we investigate the effect of finite external magnetic fields direction on these measures. This study leads to some new and interesting results as well.

Non-classical correlations in two quantum dots coupled in a coherent resonator field under decoherence

An analytical description is obtained for two excitons; each exciton is in one of two distant quantum dots embedded in a nano-mechanical resonator which is initially prepared in a superposition of coherent states with intrinsic decoherence. We use a particular method based on the dressed states of the model Hamiltonian. The robustness of the generated non-classical correlations is investigated via the measurement-induced nonlocality and geometric quantum discord, compared with the log-negativity. The three measures present generated different correlations that depend on the initial coherence states and their intensities, and intrinsic decoherence. It is witnessed that the phenomena of sudden appearance and disappearance of entanglement are occurring and repeated at chosen intervals of time; they can disappear due to the intrinsic decoherence. The correlations were observed to attain highest robustness under initial coherent state, with decoherence parameter. Quantum correlation functions survive in the stationary states, and the amount of stationary correlations could be controlled by adjusting the values of the intrinsic decoherence, the initial state of the nanomechanical resonator and its coherence intensity.