Thermal Entanglement Research Articles

Magnetic shielding of quantum entanglement states

The measure of quantum entanglement is determined for any dimer, either ferromagnetic or antiferromagnetic, spin-1/2 Heisenberg systems in the presence of external magnetic field. The physical quantity proposed as a measure of thermal quantum entanglement is the distance between states defined through the Hilbert-Schmidt norm. It has been shown that for ferromagnetic systems there is no entanglement at all. However, although under applied magnetic field, antiferromagnetic spin-1/2 dimers exhibit entanglement for temperatures below the decoherence temperature -- the one above which the entanglement vanishes. In addition to that, the decoherence temperature shows to be proportional to the exchange coupling constant and independent on the applied magnetic field, consequently, the entanglement may not be destroyed by external magnetic fields -- the phenomenon of {\it magnetic shielding effect of quantum entanglement states}. This effect is discussed for the binuclear nitrosyl iron complex [Fe$_2$(SC$_3$H$_5$N$_2$)$_2$(NO)$_4$] and it is foreseen that the quantum entanglement survives even under high magnetic fields of Tesla orders of magnitude.

Thermal Entanglement Between Two Two-Level Atoms in a Two-Photon Jaynes-Cummings Model with an Added Kerr Medium

In this paper, we consider a Hamiltonian model that includes interaction of two coupled two-level atoms with a single-mode quantized electromagnetic field in a cavity via the degenerate two-photon transition. The cavity is filled with a Kerr-like medium and is held at a temperature T. The free field Hamiltonian possesses the su(1,1) symmetry which realized by either even or odd photon-number states. The total number of excitation as a constant of motion, provides a decomposition of the Hilbert space of system into direct sums of invariant subspaces. As a results, the representation of the Hamiltonian becomes block-diagonal matrix with three blocks. After diagonalizing each block, we obtain thermal state of system in the whole Hilbert space and within its excitation subspaces. Finally, the effect of temperature, atom-atom and Kerr-type couplings on the degree of thermal entanglement between the atoms are investigated. Our results show that within the single-excitation subspace spanned with odd photon-number states, the entanglement between the atoms is thermally robust.

Thermal Entanglement Between Atoms in the Four-Cavity Linear Chain Coupled by Single-Mode Fibers

Natural thermal entanglement between atoms of a linear arranged four coupled cavities system is studied. The results show that there is no thermal pairwise entanglement between atoms if atom-field interaction strength f or fiber-cavity coupling constant J equals to zero, both f and J can induce thermal pairwise entanglement in a certain range. Numerical simulations show that the nearest neighbor concurrence CAB is always greater than alternate concurrence CAC in the same condition. In addition, the effect of temperature T on the entanglement of alternate qubits is much stronger than the nearest neighbor qubits.

Thermal entanglement of a V-type three-level atom and bimodal photons

In this paper the properties of thermal entanglement between a V-type three-level atom and bimodal photons in an optical cavity is studied. The cavity is assumed to be in thermal equilibrium with a heat reservoir thus all atom–photon states with probabilities defined by Boltzmann factor are present. Introducing total atom–photon excitation operator (as conserved operator), it is shown that the Hamiltonian representation is block-diagonal each of dimensions 3N+1, where N is the eigenvalues of the total excitation operator. Each of these blocks consists of two 2 × 2 and N-1 3 × 3 irreducible subblocks. The thermal (Gibbs) density operator for the present system is then analytically calculated by diagonalizing the block matrices. The partially transposed density matrix with respect to the atomic states and consequently, the negativity, as a measure of entanglement, are determined as functions of temperature. The negativity shows that the system of the three-level atom and photons is separable at zero temperature, exhibits a maximum at a certain temperature and disentangles at a threshold finite temperature. Effect of atom–photon couplings and detunings (as controllable parameters) on thermal entanglement is also discussed in detail.

Entropic uncertainty relations in the Heisenberg XXZ model and its controlling via filtering operations

The uncertainty principle is recognized as an elementary ingredient of quantum theory and sets up a significant bound to predict outcome of measurement for a couple of incompatible observables. In this work, we develop dynamical features of quantum memory-assisted entropic uncertainty relations (QMA-EUR) in a two-qubit Heisenberg XXZ spin chain with an inhomogeneous magnetic field. We specifically derive the dynamical evolutions of the entropic uncertainty with respect to the measurement in the Heisenberg XXZ model when spin A is initially correlated with quantum memory B. It has been found that the larger coupling strength \( J \) of the ferromagnetism (\( J 0 \)) chains can effectively degrade the measuring uncertainty. Besides, it turns out that the higher temperature can induce the inflation of the uncertainty because the thermal entanglement becomes relatively weak in this scenario, and there exists a distinct dynamical behavior of the uncertainty when an inhomogeneous magnetic field emerges. With the growing magnetic field \( \left| B \right| \), the variation of the entropic uncertainty will be non-monotonic. Meanwhile, we compare several different optimized bounds existing with the initial bound proposed by Berta et al. and consequently conclude Adabi et al.’s result is optimal. Moreover, we also investigate the mixedness of the system of interest, dramatically associated with the uncertainty. Remarkably, we put forward a possible physical interpretation to explain the evolutionary phenomenon of the uncertainty. Finally, we take advantage of a local filtering operation to steer the magnitude of the uncertainty. Therefore, our explorations may shed light on the entropic uncertainty under the Heisenberg XXZ model and hence be of importance to quantum precision measurement over solid state-based quantum information processing.

Competition of direct and indirect sources of thermal entanglement in a spin-star network

A spin-star system consisting of three peripheral two-state systems and a central one is considered, with the peripheral spins assumed to interact with each other, as well as with the central one. It is shown that such two couplings, each one being a thermal entanglement source, can significantly compete in the formation of quantum correlations in the thermal state to the point that they can destroy any thermal entanglement of the peripheral spins.

Thermal entanglement in a five-qubit XXZ Heisenberg spin chain with the next nearest neighboring interaction

In the study of thermal entanglement of the Heisenberg spin chain model, one usually considers only the spin interaction between the nearest neighboring qubits. Actually, a generalized Heisenberg model, so-called J1-J2 Heisenberg model, which is constructed by considering the fact that not only the nearest neighboring but also the next nearest neighboring spin interaction also plays an important role. In J1-J2 Heisenberg model, due to the next nearest neighboring spin interaction, the frustration effect can occur and has an important influence on the magnetic properties of the model. In this paper we investigate the thermal entanglement of a five-qubit XXZ Heisenberg spin chain with the next nearest neighboring interaction in a magnetic field. Using the numerical method, we calculate the pairwise concurrences of the nearest neighbouring qubits and the next nearest neighboring qubits, abbreviated as C12 and C13 respectively. The numerical results show that the frustration parameter α has an important effect on the pairwise thermal entanglement. Moreover, C12 and C13 have different variations with the change of the frustration parameter α. Meanwhile, it is found that the temperature, magnetic field, Dzyaloshinkii-Moriya (DM) interaction and anisotropic parameter also have great effects on the thermal entanglement. The increasing of temperature can reduce the thermal entanglement. The magnetic field can enhance the thermal entanglement between both two nearest and next nearest neighboring qubits, but when the magnetic field becomes strong enough, only the thermal entanglement between the two nearest neighboring qubits is suppressed. A certain extent of DM interaction can enhance the thermal entanglement between the two nearest neighboring qubits. But for the next nearest neighboring qubits, without the magnetic field, the increasing of DM interaction mainly enlarge the entanglement vanishing area of frustration parameter α. When the system changes from anisotropic to isotropic state, the entanglement vanishing area also changes obviously for C12 and C13. Thus, we can choose appropriate magnetic field strength, temperature, frustration parameter, DM interaction parameter and anisotropic parameter to effectively control and enhance the thermal entanglement of the system.

Generation of Quantum Correlations in Bipartite Gaussian Open Quantum Systems

We describe the generation of quantum correlations (entanglement, discord and steering) in a system composed of two coupled non-resonant bosonic modes immersed in a common thermal reservoir, in the framework of the theory of open systems. We show that for separable initial squeezed thermal states entanglement generation may take place, for definite values of squeezing parameter, average photon numbers, temperature of the thermal bath, dissipation constant and strength of interaction between the two bosonic modes. We also show that for initial uni-modal squeezed states Gaussian discord can be generated for all non-zero values of the strength of interaction between the modes. Likewise, for an initial separable state, a generation of Gaussian steering may take place temporarily, for definite values of the parameters characterizing the initial state and the thermal environment, and the strength of coupling between the two modes.

Phase transitions and thermal entanglement of the distorted Ising–Heisenberg spin chain: topology of multiple-spin exchange interactions in spin ladders

We consider a symmetric spin-1/2 Ising-XXZ double sawtooth spin ladder obtained from distorting a spin chain, with the XXZ interaction between the interstitial Heisenberg dimers (which are connected to the spins based on the legs via an Ising-type interaction), the Ising coupling between nearest-neighbor spins of the legs and rungs spins, respectively, and additional cyclic four-spin exchange (ring exchange) in the square plaquette of each block. The presented analysis supplemented by results of the exact solution of the model with infinite periodic boundary implies a rich ground state phase diagram. As well as the quantum phase transitions, the characteristics of some of the thermodynamic parameters such as heat capacity, magnetization and magnetic susceptibility are investigated. We prove here that among the considered thermodynamic and thermal parameters, solely heat capacity is sensitive versus the changes of the cyclic four-spin exchange interaction. By using the heat capacity function, we obtain a singularity relation between the cyclic four-spin exchange interaction and the exchange coupling between pair spins on each rung of the spin ladder. All thermal and thermodynamic quantities under consideration should be investigated by regarding those points which satisfy the singularity relation. The thermal entanglement within the Heisenberg spin dimers is investigated by using the concurrence, which is calculated from a relevant reduced density operator in the thermodynamic limit.

INFLUENCE OF ANISOTROPY AND MAGNETIC FIELD ON THERMAL ENTANGLEMENT IN HEISENBERG MODEL AND THERMODYNAMIC ANALYSIS OF MODEL

The thermal entanglement in a two-qubit anisotropic Heisenberg XXZ system, also XYZ system, with Dzyaloshinskii-Moriya (DM) couplings in an inhomogenous magnetic field, was studied. The effects of these two kinds of anisotropies on the thermal entanglement have been studied in detail in the concept for concurrence, the measure of entanglement. The analytical expressions of concurrence are obtained for this model. It is found that the DM interaction can enhance thermal entanglement and can be efficiently controlled by the DM interaction parameter and ehchange interaction Jx, Jy and Jz. When D is large enough, the entanglement can exist for larger temperatures and strong magnetic field. We also analysed thermodynamic properties of Heisenberg model and the most important results were shown in the paper.

Effect of three-spin interaction on thermal entanglement in Heisenberg XXZ model

The effect of three-spin interaction k on thermal entanglement between alternate qubits is studied using pairwise concurrence C and energy level diagram. It is found that k breaks the symmetry about the effect of magnetic field h on C. It shifts a dip structure and gradually effaces a boot structure when |k|<|J| (J is spin exchange coupling). A peak with C maintains 1 appears and expands, elbowing the dip backwards when |k|>|J|. A sudden change of the concurrence occurs around |k|=|J|, h=-k. Similar conclusions about nearest-neighbor qubits are directly given.

Thermal Entanglement in XXZ Heisenberg Model for Coupled Spin-Half and Spin-One Triangular Cell

In this paper, we investigate the thermal entanglement of two-spin subsystems in an ensemble of coupled spin-half and spin-one triangular cells, (1/2, 1/2, 1/2), (1/2, 1, 1/2), (1, 1/2, 1) and (1, 1, 1) with the XXZ anisotropic Heisenberg model subjected to an external homogeneous magnetic field. We adopt the generalized concurrence as the measure of entanglement which is a good indicator of the thermal entanglement and the critical points in the mixed higher dimensional spin systems. We observe that in the near vicinity of the absolute zero, the concurrence measure is symmetric with respect to zero magnetic field and changes abruptly from a non-null to null value for a critical magnetic field that can be signature of a quantum phase transition at finite temperature. The analysis of concurrence versus temperature shows that there exists a critical temperature, that depends on the type of the interaction, i.e. ferromagnetic or antiferromagnetic, the anisotropy parameter and the strength of the magnetic field. Results show that the pairwise thermal entanglement depends on the third spin which affects the maximum value of the concurrence at absolute zero and at quantum critical points.

Entropic uncertainty for spin-1/2 XXX chains in the presence of inhomogeneous magnetic fields and its steering via weak measurement reversals

The uncertainty principle configures a low bound to the measuring precision for a pair of non-commuting observables, and hence is considerably nontrivial to quantum precision measurement in the field of quantum information theory. In this letter, we consider the entropic uncertainty relation (EUR) in the context of quantum memory in a two-qubit isotropic Heisenberg spin chain. Specifically, we explore the dynamics of EUR in a practical scenario, where two associated nodes of a one-dimensional XXX-spin chain, under an inhomogeneous magnetic field, are connected to a thermal entanglement. We show that the temperature and magnetic field effect can lead to the inflation of the measuring uncertainty, stemming from the reduction of systematic quantum correlation. Notably, we reveal that, firstly, the uncertainty is not fully dependent on the observed quantum correlation of the system; secondly, the dynamical behaviors of the measuring uncertainty are relatively distinct with respect to ferromagnetism and antiferromagnetism chains. Meanwhile, we deduce that the measuring uncertainty is dramatically correlated with the mixedness of the system, implying that smaller mixedness tends to reduce the uncertainty. Furthermore, we propose an effective strategy to control the uncertainty of interest by means of quantum weak measurement reversal. Therefore, our work may shed light on the dynamics of the measuring uncertainty in the Heisenberg spin chain, and thus be important to quantum precision measurement in various solid-state systems.

Probabilistic quantum teleportation via thermal entanglement

We study the probabilistic (conditional) teleportation protocol when the entanglement needed to its implementation is given by thermal entanglement, i.e., when the entangled resource connecting Alice and Bob is an entangled mixed state described by the canonical ensemble density matrix. Specifically, the entangled resource we employ here is given by two interacting spin-1/2 systems (two qubits) in equilibrium with a thermal reservoir at temperature T. The interaction between the qubits is described by a Heisenberg-like Hamiltonian, encompassing the Ising, the XX, the XY, the XXX, and XXZ models, with or without external fields. For all those models we show analytically that the probabilistic protocol is exactly equal to the deterministic one whenever we have no external field. However, when we turn on the field the probabilistic protocol outperforms the deterministic one in several interesting ways. Under certain scenarios, for example, the efficiency (average fidelity) of the probabilistic protocol is greater than the deterministic one and increases with increasing temperature, a counterintuitive behavior. We also show regimes in which the probabilistic protocol operates with relatively high success rates and, at the same time, with efficiency greater than the classical limit 2/3, a threshold that cannot be surpassed by any protocol using only classical resources (no entanglement shared between Alice and Bob). The deterministic protocol's efficiency under the same conditions is below 2/3, highlighting that the probabilistic protocol is the only one yielding a genuine quantum teleportation. We also show that near the quantum critical points for almost all those models the qualitative and quantitative behaviors of the efficiency change considerably, even at finite T.

Thermal entanglement in a spin-1/2 Ising-XYZ distorted diamond chain with the second-neighbor interaction between nodal Ising spins

We consider a spin-1/2 Ising-XYZ distorted diamond chain with the XYZ interaction between the interstitial Heisenberg dimers, the nearest-neighbor Ising coupling between the nodal and interstitial spins, respectively, and the second-neighbor Ising coupling between the nodal spins. The ground-state phase diagram of the spin-1/2 Ising-XYZ distorted diamond chain exhibits several intriguing phases due to the XY anisotropy and the second-neighbor interaction, whereas the model can be exactly solved using the transfer-matrix technique. The quantum entanglement within the Heisenberg spin dimers is studied through a bipartite measure concurrence, which is calculated from a relevant reduced density operator. The concurrence may either show a standard thermal dependence with a monotonous decline with increasing temperature or a more peculiar thermal dependence accompanied with reentrant behavior of the concurrence. Based in the present model, it is conjectured that the bipartite entanglement between the interstitial Heisenberg spin pairs in the natural mineral azurite is quite insensitive to the applied magnetic field and it persists up to approximately 30 K.

Thermal Entanglement and Quantum Non-Locality along the Stepwise Magnetization Curve of the Spin-1/2 Ising-Heisenberg Trimerized Chain

Thermal Entanglement and Quantum Non-Locality along the Stepwise Magnetization Curve of the Spin-1/2 Ising-Heisenberg Trimerized Chain

Thermal entanglement of the spin-1 Ising–Heisenberg diamond chain with biquadratic interaction**Project supported by the National Natural Science Foundation of China (Grant No. 11274102), the New Century Excellent Talents in University of Ministry of Education of China (Grant No. NCET-11-0960), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20134208110001).

We investigate the thermal entanglement of the spin-1 Ising–Heisenberg diamond chain, which can be regarded as a theoretical model for the homometallic molecular ferrimagnet . Two cases, i.e., the isotropic Heisenberg (Ising–XXX) coupling model and anisotropic Heisenberg (Ising–XXZ) coupling model, are discussed respectively. The negativity is chosen as the measurement of the thermal entanglement. By means of the transfer-matrix approach, we focus on the effects of biquadratic interaction parameters on the negativity of the infinite spin-1 Ising–Heisenberg diamond chain. In the Ising–XXX coupling model, it is shown that for the case with ferromagnetic coupling the thermal entanglement can be induced by the biquadratic interaction, but the external magnetic field will suppress the occurrence of the entanglement induced by the biquadratic interaction. In the Ising–XXZ coupling model, for the case with antiferromagnetic coupling, due to the biquadratic interaction the effect of the anisotropy parameter on the entanglement will be suppressed at near-zero temperature. Moreover, the biquadratic interaction makes the threshold temperature increase. The effects of the external magnetic field on the thermal entanglement are also discussed, and it is observed that the entanglement revival phenomena exist in both models considered.

Pairwise thermal entanglement and quantum discord in a three-ligand spin-star structure

In this work, we perform a comparative study between the pairwise thermal entanglement (PWTE) and thermal quantum discord (TQD) to detect quantum phase transitions (QPT)s in a three-ligand spin-star structure whose magnetic interactions are described by different model Hamiltonians such as pure Dzyaloshinskii---Moriya (DM) interaction, anisotropic Heisenberg model (XXZ), and XXZ model with the different components of the DM interaction. Representing the system's energy spectrum, we also focus on the critical points of QPTs where the ground-state level crossing happens in such models. Taking advantage of the concurrence as a measure of the PWTE, we found that while the ligand---ligand concurrence in all models is sensitive to the ground-state level crossing, the concurrence between the central qubit and a ligand cannot exhibit a QPT. In contrast, the TQD between any two arbitrary qubits can be a signature of a QPT in a large range of temperature. However, depending on the model studied, the behavior of the TQD at the critical point will be different. In addition, the TQD behaves quite differently than the concurrence. Moreover, in order to confirm the numerical results, we analytically study the entanglement behavior at the low-temperature limit as well as the high-temperature regime. We realized that, at the low-temperature limit, the maximum value of the concurrence is approximately equal to 0.33, independent of the model studied. On the other hand, at high-temperature regime, the concurrence is suppressed down to zero rapidly beyond a critical value of temperature. The dependence of the critical temperature on the DM interaction and the anisotropy parameter is obtained explicitly. Finally we show that there is a perfect agreement between the analytical results and the numerical predictions.

Quantum-memory-assisted entropic uncertainty relation in a Heisenberg XYZ chain with an inhomogeneous magnetic field

The uncertainty principle provides a nontrivial bound to expose the precision for the outcome of the measurement on a pair of incompatible observables in a quantum system. Therefore, it is of essential importance for quantum precision measurement in the area of quantum information processing. Herein, we investigate quantum-memory-assisted entropic uncertainty relation (QMA-EUR) in a two-qubit Heisenberg spin chain. Specifically, we observe the dynamics of QMA-EUR in a realistic model there are two correlated sites linked by a thermal entanglement in the spin chain with an inhomogeneous magnetic field. It turns out that the temperature, the external inhomogeneous magnetic field and the field inhomogeneity can lift the uncertainty of the measurement due to the reduction of the thermal entanglement, and explicitly higher temperature, stronger magnetic field or larger inhomogeneity of the field can result in inflation of the uncertainty. Besides, it is found that there exists distinct dynamical behaviors of the uncertainty for ferromagnetism and antiferromagnetism chains. Moreover, we also verify that the measuring uncertainty is dramatically anti-correlated with the purity of the bipartite spin system, the greater purity can result in the reduction of the measuring uncertainty, vice versa. Therefore, our observations might provide a better understanding of the dynamics of the entropic uncertainty in the Heisenberg spin chain, and thus shed light on quantum precision measurement in the framework of versatile systems, particularly solid states.

Entanglement and coherence in a spin-s XXZ system under non-uniform fields

We investigate entanglement and coherence in an XXZ spin s pair immersed in a non-uniform transverse magnetic field. The ground state and thermal entanglement phase diagrams are analyzed in detail in both the ferromagnetic and antiferromagnetic cases. It is shown that a non-uniform field can control the energy levels and the entanglement of the corresponding eigenstates, making it possible to entangle the system for any value of the exchange couplings, both at zero and finite temperatures. Moreover, the limit temperature for entanglement is shown to depend only on the difference between the fields applied at each spin, leading for to a separability stripe in the field plane such that the system becomes entangled above a threshold value of . These results are demonstrated to be rigorously valid for any spin s. On the other hand, the relative entropy of coherence in the standard basis, which coincides with the ground state entanglement entropy at T = 0 for any s, becomes non-zero for any value of the fields at , decreasing uniformly for sufficiently high T. A special critical point arising at T = 0 for non-uniform fields in the ferromagnetic case is also discussed.