We consider the ground state of the XY model on an infinite chain at zero temperature. Following Bennett, Bernstein, Popescu, and Schumacher we use entropy of a sub-system as a measure of entanglement. Vidal, Latorre, Rico and Kitaev conjectured that von Neumann entropy of a large block of neighboring spins approaches a constant as the size of the block increases. We evaluated this limiting entropy as a function of anisotropy and transverse magnetic field. We used the methods based on integrable Fredholm operators and Riemann-Hilbert problem. The entropy is singular at phase transitions.
We investigate the presence of multipartite entanglement in macroscopic spin chains. We discuss the Heisenberg and the XY model and derive bounds on the internal energy for systems without multipartite entanglement. Based on this we show that in thermal equilibrium the above mentioned spin systems contain genuine multipartite entanglement, even at finite modest temperatures.
The thermal entanglement of a two-qubit anisotropic Heisenberg $XYZ$ chain under an inhomogeneous magnetic field b is studied. It is shown that when inhomogeneity is increased to certain value, the entanglement can exhibit a larger revival than that of less values of b. The property is both true for zero temperature and a finite temperature. The results also show that the entanglement and critical temperature can be increased by increasing inhomogeneous exteral magnetic field.
We consider two prototypical quantum models, the spin-1/2 XY chain and the quantum Ising chain and study their entanglement entropy, S(l,L), of blocks of l spins in homogeneous or inhomogeneous systems of length L. By using two different approaches, free-fermion techniques and perturbational expansion, an exact relationship between the entropies is revealed. Using this relation we translate known results between the two models and obtain, among others, the additive constant of the entropy of the critical homogeneous quantum Ising chain and the effective central charge of the random XY chain.
We consider the ground state of the XX chain that is constrained to carry a current of energy. The von Neumann entropy of a block of $L$ neighboring spins, describing entanglement of the block with the rest of the chain, is computed. Recent calculations have revealed that the entropy in the XX model diverges logarithmically with the size of the subsystem. We show that the presence of the energy current increases the prefactor of the logarithmic growth. This result indicates that the emergence of the energy current gives rise to an increase of entanglement.
We examine the entanglement of cyclic spin 1/2 chains with anisotropic XY Z Heisenberg couplings of arbitrary range at transverse factorizing magnetic fields. At these fields the system exhibits a degenerate symmetry-breaking separable ground state (GS). It is shown, however, that the side limits of the GS pairwise entanglement at these fields are actually non-zero in finite chains, corresponding such fields to a GS spin-parity transition. These limits exhibit universal properties like being independent of the pair separation and interaction range, and are directly related to the magnetization…
We describe an algorithm for studying the entanglement entropy and spectrum of 2D systems, as a coupled array of $N$ one dimensional chains in their continuum limit. Using the algorithm to study the quantum Ising model in 2D, (both in its disordered phase and near criticality) we confirm the existence of an area law for the entanglement entropy and show that near criticality there is an additive piece scaling as $c_{eff}\log (N)/6$ with $c_{eff} \approx 1$. \textcolor{black}{Studying the entanglement spectrum, we show that entanglement gap scaling can be used to detect the critical point of th…
We consider the problem of the validity of a statistical mechanical description of two-site entanglement in an infinite spin chain described by the XY model Hamiltonian. We show that the two-site entanglement of the state, evolved from the initial equilibrium state, after a change of the magnetic field, does not approach its equilibrium value. This suggests that two-site entanglement, like (single-site) magnetization, is a nonergodic quantity in this model. Moreover we show that these two nonergodic quantities behave in a complementary way.
We investigate the possibility of chaining qubits by letting pairs of nearest neighbours qubits dissipating into common environments. We then study entanglement dynamics within the chain and show that steady state entanglement can be achieved.
We address the problem of spring-like coupling between bosons in an open chain configuration where the counter-rotating terms are explicitly included. We show that fruitful insight can be gained by decomposing the time-evolution operator of this problem into a pattern of linear-optics elements. This allows us to provide a clear picture of the effects of the counter-rotating terms in the important problem of long-haul entanglement distribution. The analytic control over the variance matrix of the state of the bosonic register allows us to track the dynamics of the entanglement. This helps in de…
We study scrambling in connection to multipartite entanglement dynamics in regular and chaotic long-range spin chains, characterized by a well defined semi-classical limit. For regular dynamics, scrambling and entanglement dynamics are found to be very different: up to the Ehrenfest time they rise side by side departing only afterwards. Entanglement saturates and becomes extensively multipartite, while scrambling continues its growth up to the recurrence time. Remarkably, the exponential behaviour of scrambling emerges not only in the chaotic case, but also in the regular one, when the dynamic…
We study the entanglement in a chain of harmonic oscillators driven out of equilibrium by preparing the two sides of the system at different temperatures, and subsequently joining them together. The steady state is constructed explicitly and the logarithmic negativity is calculated between two adjacent segments of the chain. We find that, for low temperatures, the steady-state entanglement is a sum of contributions pertaining to left- and right-moving excitations emitted from the two reservoirs. In turn, the steady-state entanglement is a simple average of the Gibbs-state values and thus its s…
We calculate exactly the entanglement content of magnon excited states in the integrable spin-1/2 XXX and XXZ chains in the scaling limit. In particular, we show that as far as the number of excited magnons with respect to the size of the system is small one can decompose the entanglement content, in the scaling limit, to the sum of the entanglement of particular excited states of free fermionic or bosonic theories. In addition we conjecture that the entanglement content of the generic translational invariant free fermionic and bosonic Hamiltonians can be also classified, in the scaling limit,…
We demonstrate that a translation invariant chain of interacting quantum systems can be used for high efficiency transfer of quantum entanglement and the generation of multi-particle entanglement over large distances and between arbitrary sites without the requirement of precise spatial or temporal control. The scheme is largely insensitive to disorder and random coupling strengths in the chain. We discuss harmonic oscillator systems both in the case of arbitrary Gaussian states and in situations when at most one excitation is in the system. The latter case which we prove to be equivalent to a…
The entanglement dynamics of spin chains is investigated using Heisenberg-XY spin Hamiltonian dynamics. The various measures of two-qubit entanglement are calculated analytically in the time-evolved state starting from initial states with no entanglement and exactly one pair of maximally-entangled qubits. The localizable entanglement between a pair of qubits at the end of chain captures the essential features of entanglement transport across the chain, and it displays the difference between an initial state with no entanglement and an initial state with one pair of maximally-entangled qubits.
We study the entanglement entropy of the quantum trajectories of a free fermion chain under continuous monitoring of local occupation numbers. We propose a simple theory for entanglement entropy evolution from disentangled and highly excited initial states. It is based on generalized hydrodynamics and the quasi-particle pair approach to entanglement in integrable systems. We test several quantitative predictions of the theory against extensive numerics and find good agreement. In particular, the volume law entanglement is destroyed by the presence of arbitrarily weak measurement.
We derive some entanglement properties of the ground states of two classes of quantum spin chains described by the Fredkin model, for half-integer spins, and the Motzkin model, for integer ones. Since the ground states of the two models are known analytically, we can calculate the entanglement entropy, the negativity and the quantum mutual information exactly. We show, in particular, that these systems exhibit long-distance entanglement, namely two disjoint regions of the chains remain entangled even when the separation is sent to infinity, i.e. these systems are not affected by decoherence. T…
We investigate the dynamics of the XXZ spin chain after a geometric quench, which is realized by connecting two half-chains prepared in their ground states with zero and maximum magnetizations, respectively. The profiles of magnetization after the subsequent time evolution are studied numerically by density-matrix renormalization group methods, and a comparison to the predictions of generalized hydrodynamics yields a very good agreement. We also calculate the profiles of entanglement entropy and propose an ansatz for the noninteracting XX case, based on arguments from conformal field theory. I…
A subsystem of an entangled ground state is in a mixed state. Thus, if we isolate this subsystem from its surroundings we may be able to extract work applying unitary transformations, up to a maximal amount which is called ergotropy. Once this work has been extracted, the subsystem will still contain some bound energy above its local ground state, which can provide valuable information about the entanglement structure. We show that the bound energy for half a free fermionic chain decays as the square of the entanglement entropy divided by the chain length, thus approaching zero for large syste…
Fisher-Hartwig formula has been successful applied to describe the von Neumann and Rényi entropies of a block of spins in the ground state of XX spin chain. It was based on a determinant representation. In this paper, we generalize the free fermion method to obtain an exact formulation for the entropy of any finite subsystem in XX spin chain. Based on this, we derive a determinant representation of the entropy of multiple disjoint intervals in the ground state of $XX$ model.
We show how multi-partite entanglement, such as of W-state form, can be created in branched spin chain systems. We also discuss the preservation of such entanglement, once created. The technique could be applied to actual spin chain systems, or to other physical systems such as strings of coupled quantum dots, molecules or atoms.
A relation between entanglement and criticality of spin chains is established. The entanglement we exploit is shared between auxiliary particles, which are isolated from each other, but are coupled to the same critical spin-1/2 chain. We analytically evaluate the reduced density matrix, and numerically show the entanglement of the auxiliary particles in the proximity of the critical points of the spin chain. We find that the entanglement induced by the spin-chain may reach one, and it can signal very well the critical points of the chain. A physical understanding and experimental realization w…
Thermal entanglement of a two-qubit Heisenberg chain in presence of the Dzyaloshinski-Moriya (DM) anisotropic antisymmetric interaction and entanglement teleportation when using two independent Heisenberg chains as quantum channel are investigated. It is found that the DM interaction can excite the entanglement and teleportation fidelity. The output entanglement increases linearly with increasing value of input one, its dependences on the temperature, DM interaction and spin coupling constant are given in detail. Entanglement teleportation will be better realized via antiferromagnetic spin cha…
Entanglement entropy obeys area law scaling for typical physical quantum systems. This may naively be argued to follow from locality of interactions. We show that this is not the case by constructing an explicit simple spin chain Hamiltonian with nearest neighbor interactions that presents an entanglement volume scaling law. This non-translational model is contrived to have couplings that force the accumulation of singlet bonds across the half chain. Our result is complementary to the known relation between non-translational invariant, nearest neighbor interacting Hamiltonians and QMA complete…
We study pairwise entanglements in spin-half and spin-one Heisenberg chains with an open boundary condition, respectively. We find out that the ground-state and the first-excited-state entanglements are equal for the three-site spin-one chain. When the number of sites L>3, the concurrences and negativities display oscillatory behaviors, and the oscillations of the ground-state and the first-excited-state entanglements are out of phase or in phase.
We study the dynamics of a single excitation in an infinite XXZ spin chain, which is launched from the origin. We study the time evolution of the spread of entanglement in the spin chain and obtain an expression for the second order spatial moment of concurrence, about the origin, for both ordered and disordered chains. In this way, we show that a finite central disordered region can lead to sustained superballistic growth in the second order spatial moment of entanglement within the chain.
We exploit the inherent entanglement of the ground state of a spin chain with dimerized XX or XXZ Hamiltonian to investigate the entanglement generation between the ends of the chain. We follow the strategy has been introduced in Ref. [14] to encode the information in the entangled ground state of the system by local rotation. The amount of achieved entanglement in this scheme is higher than the attaching a pair of maximally entanglement scenarios. Also, our proposal can be implemented by using the optical lattices.
The entanglement properties in an antiferromagnetic dimerized Heisenberg spin-1/2 chain are investigated. The entanglement gap, which is the difference between the ground-state energy and the minimal energy that any separable state can attain, is calculated to detect the entanglement. It is found that the entanglement gap can be increased by varying the alternation parameter. Through thermal energy, the witness of the entanglement can determine a characteristic temperature below that an entangled state can be obtained. The entanglement detected by the energy can provide a lower bound for that …
In a ferromagnetic spin chain, the control of the local effective magnetic field allows to manipulate the static and dynamical properties of entanglement. In particular, the propagation of quantum correlations can be driven to a great extent so as to achieve an entanglement transfer on demand toward a selected site.
Recently, entanglement teleportation has been investigated in Phys. Rev. Lett. 84, 4236 (2000). In this paper we study entanglement teleportation via two separate thermally entangled states of two-qubit Heisenberg XX chain. We established the condition under which the parameters of the model have to satisfy in order to teleport entanglement. The necessary minimum amount of thermal entanglement for some fixed strength of exchange coupling is a function of the magnetic field and temperature.
We introduce a connection between entanglement induced by interaction and geometric phases acquired by a composite quantum spin system. We begin by analyzing the evaluation of cyclic (Aharonov-Anandan) and non-cyclic (Mukunda-Simon) geometric phases for general spin chains evolving in the presence of time-independent magnetic fields. Then, by considering Heisenberg chains, we show that the interaction geometric phase, namely, the total geometric phase with subtraction of free spin contributions, is directly related to the global (Meyer-Wallach) entanglement exhibited by an initially separable …
We discuss the effect of a single diagonal defect on both the static and dynamical properties of entanglement in a spin chain. We show that entanglement localizes at the defect and discuss its localization length, arguing that this can be used as a mean to store entanglement. We also show that the impurity site can behave as an entanglement mirror and characterize the bouncing process in terms of reflection and transmission coefficients.
Most of the work involving entanglement measurement focuses on systems that can be modeled by two interacting qubits. This is due to the fact that there are few studies presenting entanglement analytical calculations in systems with spins $s > 1/2$. In this paper we present for the first time an analytical way of calculating thermal entanglement in a dimension $2\otimes3$ Heisenberg chain through the distance between states. We use the Hilbert-Schmidt norm to obtain entanglement. The result obtained can be used to calculate entanglement in chains with spin-$1/2$ coupling with spin-$1$, such as…
We investigate entanglement in a linear chain of $N$ polar molecules coupled by dipole interaction. In our model, nearest neighbour interaction predominate, and we compute entanglement with the help of a two-party correlation entanglement measure. We find that, in this system, only excited states are entangled. Moreover, when an electrostatic field is applied, energy levels crossings occur, changing significantly the system's entanglement properties. We make a systematic study of the entanglement dependency on the inter molecular distance separating pairs of molecules, different partitions of …
The evolution of entanglement in a one-dimensional Ising chain is numerically studied under various initial conditions. We analyze two problems concerning the dynamics of the entanglement: (i) generation of the entanglement from the pseudopure separable state and (ii) transportation of the entanglement from one end of the chain to the other. The investigated model is a one-dimensional Ising spin-1/2 chain with nearest-neighbor interactions placed in an external magnetic field and irradiated by a weak resonant transverse field. The possibility of selective initialization of partially entangled …
In this paper we propose an expression for the entanglement entropy of several intervals in a stationary state of a free, translational invariant Hamiltonian in a fermionic chain. We check numerically the accuracy of our proposal and conjecture a new formula for the asymptotic behaviour of principal sub-matrices of a Toeplitz matrix.
We show how branching spin chains can be used to both generate and distribute entanglement from their natural dynamics. Such entanglement provides a useful resource, for example for teleportation or distributed quantum processing. Once distributed, this resource can be isolated through mapping or swapping the entanglement into specific qubits at the ends of branches. Alternatively, as we demonstrate for distributed bipartite entanglement, applying simple single-qubit operations to the end spin of one or more branches can dynamically freeze the entanglement at the branch ends.
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.
We derive temporally shaped control pulses for the creation of long-distance entanglement in disordered spin chains. Our approach is based on a time-dependent target functional and a time-local control strategy that permits to ensure that the description of the chain in terms of matrix product states is always valid. With this approach, we demonstrate that long-distance entanglement can be created even for substantially disordered interaction landscapes.
We examine the maximum entanglement reached by an initially fully aligned state evolving in an XY Heisenberg spin chain placed in a uniform transverse magnetic field. Both the global entanglement between one qubit and the rest of the chain and the pairwise entanglement between adjacent qubits is analyzed. It is shown that in both cases the maximum is not a monotonous decreasing function of the aligning field, exhibiting instead a resonant behavior for low anisotropies, with pronounced peaks (a total of [n/2] peaks in the global entanglement for an $n$-spin chain), whose width is proportional t…
Quantifying entanglement of multiple subsystems is a challenging open problem in interacting quantum systems. Here, we focus on two subsystems of length $\ell$ separated by a distance $r=α\ell$ and quantify their entanglement negativity (${\cal E}$) and mutual information (${\cal I}$) in critical random Ising chains. Both the disorder averaged ${\cal E}$ and ${\cal I}$ are found to be scale-invariant and universal, i.e. independent of the form of disorder. We find a constant ${\cal E}(α)$ and ${\cal I}(α)$ over any distances, using the asymptotically exact strong disorder renormalization group…
We study the ground state as well as the dynamics of chains of bosons with local repulsive interactions and nearest-neighbour exchange using numerical techniques based on density matrix decimation. We explore the development of entanglement between the terminal sites of such chains as mechanisms are invoked to concentrate population in these sites. We find that long-range entanglement in the ground state emerges as a result of hopping taking place at the whole chain length in systems with appropriate hopping coefficients. Additionally, we find appropriate perturbations to increase the entangle…
A strongly coupled Heisenberg chain provides an important channel for quantum communication through its many-body ground state. Yet, the nature of the effective interactions and the ability to mediate long-range entanglement differs significantly for chains of opposite parity. Here, we contrast the characters of even and odd-size chains when they are coupled to external qubits. Additional parity effects emerge in both cases, depending on the positions of the attached qubits. Some striking results include (i) the emergence of maximal entanglement and (ii) Ruderman-Kittel-Kasuya-Yosida (RKKY) in…
A formula to evaluate the entanglement in an one-dimensional ferrimagnetic system is derived. Based on the formula, we find that the thermal entanglement in a small size spin-1/2 and spin-s ferrimagnetic chain is rather robust against temperature, and the threshold temperature may be arbitrarily high when s is sufficiently large. This intriguing result answers unambiguously a fundamental question: ``can entanglement and quantum behavior in physical systems survive at arbitrary high temperatures?"
We propose to use ferromagnetic systems for entanglement generation and distribution together with perfect state transfer between distant parties in a qubit chain. The scheme relies on an effective 2-qubit dynamics, realized by leaving two empty sites in a uniformly filled chain. This allows long-range interacting qubit chains to serve as quantum channels for both tasks with optimal performances. Remarkably, the entanglement between sender and receiver sites is independent of both the transmission distance and of system size. This property opens new perspectives for short and mid-range quantum…
The limitations on the production and profitability of entanglement in a harmonic chain under strong driving are considered. We report on the limits of attainable entanglement for a given set of squeezings of the eigenmodes, showing that the higher the entanglement the more oscillatory and thus less easy to profit from. We also comment on propagation properties of entanglement, discussing the role of fast rotating terms and illustrating several issues with the example of a sudden switch of the coupling.
An exponential deformation of 1D critical Hamiltonians gives rise to ground states whose entanglement entropy satisfies a volume-law. This effect is exemplified in the XX and Heisenberg models. In the XX case we characterize the crossover between the critical and the maximally entangled ground state in terms of the entanglement entropy and the entanglement spectrum.
By using the concept of negativity, we study entanglement in spin-one Heisenberg chains. Both the bilinear chain and the bilinear-biquadratic chain are considered. Due to the SU(2) symmetry, the negativity can be determined by two correlators, which greatly facilitate the study of entanglement properties. Analytical results of negativity are obtained in the bilinear model up to four spins and the two-spin bilinear-biquadratic model, and numerical results of negativity are presented. We determine the threshold temperature before which the thermal state is doomed to be entangled.
We show that in one-dimensional isotropic Heisenberg model two-qubit thermal entanglement and maximal violation of Bell inequalities are directly related with a thermodynamical state function, i.e., the internal energy. Therefore they are completely determined by the partition function, the central object of thermodynamics. For ferromagnetic ring we prove that there is no thermal entanglement at any temperature. Explicit relations between the concurrence and the measure of maximal Bell inequality violation are given.
The entanglement in a general mixed spin chain with arbitrary spin $S$ and 1/2 is investigated in the thermodynamical limit. The entanglement is witnessed by the magnetic susceptibility which decides a characteristic temperature for an entangled thermal state. The characteristic temperature is nearly proportional to the interaction $J$ and the mixed spin $S$. The bound of negativity is obtained on the basis of the magnetic susceptibility. It is found that the macroscopic magnetic properties are affected by the quantum entanglement in the real solids. Meanwhile, the entanglement can be quantita…
Entanglement entropy (EE) in critical quantum spin chains described by 1+1D conformal field theories contains signatures of the universal characteristics of the field theory. Boundaries and defects in the spin chain give rise to universal contributions in the EE. In this work, we analyze these universal contributions for the critical Ising and XXZ spin chains for different conformal boundary conditions and defects. For the spin chains with boundaries, we use the boundary states for the corresponding continuum theories to compute the subleading contribution to the EE analytically and provide su…
We study the occurrence of multipartite entanglement in spin chains. We show that certain genuine multipartite entangled states, namely W states, can be obtained as ground states of simple XX type ferromagnetic spin chains in a transverse magnetic field, for any number of sites. Moreover, multipartite entanglement is proven to exist even at finite temperatures. A transition from a product state to a multipartite entangled state occurs when decreasing the magnetic field to a critical value. Adiabatic passage through this point can thus lead to the generation of multipartite entanglement.
The entanglement in a general Heisenberg antiferromagnetic chain of arbitrary spin-$s$ is investigated. The entanglement is witnessed by the thermal energy which equals to the minimum energy of any separable state. There is a characteristic temperature below that an entangled thermal state exists. The characteristic temperature for thermal entanglement is increased with spin $s$. When the total number of lattice is increased, the characteristic temperature decreases and then approaches a constant. This effect shows that the thermal entanglement can be detected in a real solid state system of l…
The effects of boundary impurities on the entanglement entropy in an antiferromagnetic Heisenberg opened spin-$1/2$ chain are investigated. The method of density-matrix renormalization-group is used to obtain the bipartite entanglement. The entropy increases when the length of the subsystem increases. It will approach to a constant when system length is very large. With the same impurity interaction, qutrit impurities of spin-$1$ can increase the entanglement entropy.
Using the adaptive time-dependent density-matrix renormalization group method, the dynamics of entanglement and quantum discord of a one-dimensional spin-1/2 XXZ chain is studied when anisotropic interaction quenches are applied at different temperatures. The dynamics of the quantum discord and pairwise entanglement between the nearest qubits shows that the entanglement and quantum discord will first oscillate and then approach to a constant value. The quantum discord can be used to predict the quantum phase transition, while the entanglement cannot.
We study the scaling of the traces of the integer powers of the partially transposed reduced density matrix and of the entanglement negativity for two spin blocks as function of their length and separation in the critical Ising chain. For two adjacent blocks, we show that tensor network calculations agree with universal conformal field theory (CFT) predictions. In the case of two disjoint blocks the CFT predictions are recovered only after taking into account the finite size corrections induced by the finite length of the blocks.
We study the effect of a phase shift on the amount of transferrable two-spin entanglement in a spin chain. We consider a ferromagnetic Heisenberg/XY spin chain, both numerically and analytically, and two mechanisms to generate a phase shift, the Aharonov-Casher effect and the Dzyaloshinskii-Moriya interaction. In both cases, the maximum attainable entanglement is shown to be significantly enhanced, suggesting its potential usefulness in quantum information processing.
