Pairwise entanglement properties of a symmetric multi-qubit system are analyzed through a complete set of two-qubit local invariants. Collective features of entanglement, such as spin squeezing, are expressed in terms of invariants and a classifcation scheme for pairwise entanglement is proposed. The invariant criteria given here are shown to be related to the recently proposed (Phys. Rev. Lett. 95, 120502 (2005)) generalized spin squeezing inequalities for pairwise entanglement in symmetric multi-qubit states.
Basing on the calculation of all the pairwise entanglement in the $n$ ($n \leq 6$)-qubit Heisenberg XX open chain with system impurity, we find an important result: pairwise entanglement can only be transferred through entangled pair. The non-nearest pairwise entanglement will has the possibility to exist as long as there has even number qubit in their middle. This point means that we can get longer distance entanglement in solid system.
Consider the dynamics of a two-qubit entangled system in the decoherence environment, we investigate the stability of pairwise entanglement under decoherence. We find that for different decoherence models, there exist some special class of entangled states of which the pairwise entanglement is the most stable. The lifetime of the entanglement in these states is larger than other states with the same initial entanglement. In addition, we also investigate the dynamics of pairwise entanglement in the ground state of spin models such as Heisenberg and XXY models.
The Hilbert-Schmidt distance between a mixed three-qubit state and its closest state is used to quantify the amount of pairwise quantum correlations in a tripartite system. Analytical expressions of geometric quantum discord are derived. A particular attention is devoted to two special classes of three-qubit $X$ states. They include three-qubit states of ${\rm W}$, ${\rm GHZ}$ and Bell type. We also discuss the monogamy property of geometric quantum discord in some mixed three-qubit systems.
Inspired by the discrete-variable pairwise entanglement, in this work, we in theory analyze the continuous-variable pairwise entanglement between microwave modes based on a hybrid optoelectromechanical system, where the multi-pair microwave superconducting circuits simutaneously interact with each other via a mechanical resonator, which forms a Fabry-Pérot cavity along with a standing mirror. With experimentally reachable parameter settings, wanted entanglement can be acheived when the pair number up to 10, and more is also available, which has the potential to be useful in quantum technologie…
Kempe-Invariant Ordering of Three-Qubit Pairwise Entanglement under Local Depolarization
School of Physics and Materials Engineering, Hefei Normal University（合肥师范学院物理与材料工程学院）；School of Physics, Anhui University（安徽大学物理学院）；Leibniz International Joint Research Center of Materials Sciences of Anhui Province, Anhui University（安徽大学安徽省材料科学国际联合研究中心）
The pairwise entanglement and local polarization of the ground state are discussed by studying the Heisenberg XX model in finite qubit case. The results show that: the ground state is composed by the micro state with the minimal total spin 0 (for even qubit) or 1/2 (for odd qubit), local polarization (LP) has intimate relation with the probability of the micro state in the ground state, the stronger the LP the smaller the probability, the same LP corresponding to the same probability; the pairwise entanglement of the ground state is the biggest in all the eigenvectors. We find when the qubit i…
Based on the idea of measuring the factorizability of a given density matrix, we propose a pairwise analysis strategy for quantifying and understanding multipartite entanglement. The methodology proves very effective as it immediately guarantees, in addition to the usual entanglement properties, additivity and strong super additivity. We give a specific set of quantities that fulfill the protocol and which, according to our numerical calculations, make the entanglement measure an LOCC non-increasing function. The strategy allows a redefinition of the structural concept of global entanglement.
We study the pairwise quantum discord (QD) for a symmetric multi-qubit system in different types of noisy channels, such as phase-flip, amplitude damping, phase-damping, and depolarizing channels. Using the QD and geometric measure of quantum discord (GMQD) to quantify quantum correlations, some analytical or numerical results are presented. The results show that, the dynamics of the pairwise QD is related to the number of spin particles $N$ as well as initial parameter $θ$ of the one-axis twisting collective state. With the number of spin particles $N$ increasing, the amount of the pairwise Q…
We investigate the nonlocality distributions among multiqubit systems based on the maximal violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality of reduced pairwise qubit systems. We present a trade-off relation satisfied by these maximal violations, which gives rise to restrictions on the distribution of nonlocality among the subqubit systems. For a three-qubit system, it is impossible that all pairs of qubits violate the CHSH inequality, and once a pair of qubits violates the CHSH inequality maximally, the other two pairs of qubits must both obey the CHSH inequality. Detailed exampl…
The pairwise correlations in a multi-qubit state are quantified through a linear variant of relative entropy. In particular, we derive the explicit expressions of total, quantum and classical bipartite correlations. Two different bi-partioning schemes are considered. We discuss the derivation of closest product, quantum-classical and quantum-classical product states. We also investigate the additivity relation between the various pairwise correlations existing in pure and mixed states. As illustration, some special cases are examined.
For a 3-qubit Heisenberg model in a uniform magnetic field, the pairwise thermal entanglement of any two sites is identical due to the exchange symmetry of sites. In this paper we consider the effect of a non-uniform magnetic field on the Heisenberg model, modeling a magnetic impurity on one site. Since pairwise entanglement is calculated by tracing out one of the three sites, the entanglement clearly depends on which site the impurity is located. When the impurity is located on the site which is traced out, that is, when it acts as an external field of the pair, the entanglement can be enhanc…
The concurrence, a quantitative measure of the entanglement between a pair of particles, is determined for the case where the pair is extracted from a symmetric state of N two-level systems. Examples are given for both pure and mixed states of the N-particle system, and for a pair extracted from two ensembles with correlated collective spins.
The effects of multi-impurity on the entanglement of anisotropic Heisenberg ring XXZ under a homogeneous magnetic field have been studied. The impurities make the equal pairwise entanglement in a ring compete with each other so that the pairwise entanglement exhibits oscillation. If the impurities are of larger couplings, both the critical temperature and pairwise entanglement can be improved
The pairwise entanglement, measured by concurrence and geometric phase in $d=2,3$-dimensional free-fermion lattice systems have been studied for the ground state in this paper. Their derivation with respect to the external parameter show the singularity closed to the phase transition points, and can be used to detect the phase transition in this model. Furthermore our studies show for the free-fermion model that both concurrence and geometric phase shows the intimate connection with the correlation functions. The possible relation between concurrence and geometric phase has been also discussed…
We show that spin squeezing implies pairwise entanglement for arbitrary symmetric multiqubit states. If the squeezing parameter is less than or equal to 1, we demonstrate a quantitative relation between the squeezing parameter and the concurrence for the even and odd states. We prove that the even states generated from the initial state with all qubits being spin down, via the one-axis twisting Hamiltonian, are spin squeezed if and only if they are pairwise entangled. For the states generated via the one-axis twisting Hamiltonian with an external transverse field for any number of qubits great…
We investigate the teleportation of an entangled two-qubit state using three-qubit GHZ and W channels. The effects of white noise on the average teleportation fidelity and amount of entanglement transmitted are also studied.
In previous work, we have developed a dynamic learning paradigm for "programming" a general quantum computer. A learning algorithm is used to find a set of parameters for a coupled qubit system such that the system at an initial time evolves to a state in which a given measurement results in the desired calculation value. This can be thought of as a quantum neural network (QNN). Here, we apply our method to a system of three qubits, and demonstrate training the quantum computer to estimate both pairwise and three-way entanglement.
In this Brief Report we show the relation between three-qubit entanglement invariants and two-qubit concurrence with the help of projective measurements. How to use these invariants to represent the entanglement property of three-qubit pure states is also investigated.
It is shown that the entanglement-structure of 3- and 4-qubit states can be characterized by optimized operators of the Mermin-Klyshko type. It is possible to discriminate between pure 2-qubit entanglements and higher entanglements. A comparison with a global entanglement measure and the i-concurrence is made.
We study the entanglement creation between two flux qubits interacting with electromagnetic field modes. No direct interaction between the qubits exists. Entanglement is reached using entanglement swapping method by an interference measurement performed on photons. We discuss the influence of off-resonance and multi-photon initial states on the qubit-qubit entanglement. The presented scheme is able to drive an initially separable state of two qubits into an highly entangled state suitable for quantum information processing.
Entanglement of four qubit pure states defined by the $η$-trigonometric functions is studied. We analyze the behavior of two recently proposed symmetric entanglement monotones on the chosen $n=4$ qubit states .
The concurrence of two alternate qubits in a four-qubit Heisenberg XX chain is investigated when a uniform magnetic field B is included. It is found that there is no thermal entanglement between alternate qubits if B is close to zero. Magnetic field can induce entanglement in a certain range both for the antiferromagnetic and ferromagnetic cases. Near zero temperature, the entanglement undergoes two sudden changes with increasing value of the magnetic field B. This is due to the changes in the ground state. This novel property may be used as quantum entanglement switch. The anisotropy in the s…
Universal ion trap computation on Decoherence Free Subspaces (DFS) using only two qubit operations is presented. The DFS is constructed for the collective dephasing model. Encoded single and two-qubit logical operations are implemented via the Sorensen-Molmer interaction. Alternation of the effective Hamiltonians for two particular phase configurations of control fields approximates an anisotropic exchange interaction. This is universal over suitable encodings of one logical qubit into three physical qubits which are also DFS under collective decoherence.
Two qubits in pure entangled states going through separate paths and interacting with their own individual environments will gradually lose their entanglement. Here we show that the entanglement change of a two-qubit state due to amplitude damping noises can be recovered by entanglement swapping. Some initial states can be asymptotically purified into maximally entangled states by iteratively using our protocol.
We investigate the thermal entanglement in two superconducting qubits for arbitrary interaction strength and ground state frequencies. We calculate the concurrence of the system to quantify the thermal entanglement. We suggest a scheme, where an external tunable coupler qubit sandwich between two superconducting qubits generates entanglement. The behavior of concurrence is analyzed for three different cases, in which we consider the effects of the temperature, the qubit-qubit effective coupling strength, and the qubit frequencies on the thermal entanglement. What deserves mentioning here is th…
In this work, we propose a scheme to design the time evolution of the entropy of entanglement between two qubits. It is shown an explicit accurate solution for the inverse problem of determining the time dependence of the coupling constant from a user-defined dynamical entanglement function. Such an active control of entanglement can be implemented in many different physical implementations of coupled qubits, and we briefly comment on the use of interacting flux qubits.
Recently, Coffman, Kundu, and Wootters introduced the residual entanglement for three qubits to quantify the three-qubit entanglement in Phys. Rev. A 61, 052306 (2000). In Phys. Rev. A 65, 032304 (2007), we defined the residual entanglement for $n$ qubits, whose values are between 0 and 1. In this paper, we want to show that the residual entanglement for $n$ qubits is a natural measure of entanglement by demonstrating the following properties. (1). It is SL-invariant, especially LU-invariant. (2). It is an entanglement monotone. (3). It is invariant under permutations of the qubits. (4). It va…
In this paper we consider the generation of a three-qubit GHZ-like thermal state by applying the entanglement swapping scheme of Zukowski {\it et al.} [Ann. N. Y. Acad. Sci. {\bf 755}, 91 (1995)] to three pairs of two-qubit Heisenberg XY chains. The quality of the resulting three-qubit entanglement is studied by analyzing the teleportation fidelity, when it is used as a resource in the teleportation protocol of Karlsson {\it et al.}[Phys. Rev. A {\bf 58}, 4394 (1998)]. We show that even though thermal noise in the original two-qubit states is amplified by the entanglement swapping process, we …
We provide an initial characterization of pairwise concurrence in quantum states which are invariant under cyclic permutations of party labeling. We prove that maximal entanglement can be entirely described by adjacent pairs, then give explicit descriptions of those states in specific subsets of 4 and 5 qubit states - X states. We also construct a monogamy bound on shared concurrences in the same subsets in 4 and 5 qubits, finding that above non-maximal entanglement thresholds, no other entanglements are possible.
We study the degree to which quantum entanglement survives when a three-qubit entangled state is copied by using local and non-local processes, respectively, and investigate iterating quantum copying for the three-qubit system. There may exist inter-three-qubit entanglement and inter-two-qubit entanglement for the three-qubit system. We show that both local and non-local copying processes degrade quantum entanglement in the three-particle system due to a residual correlation between the copied output and the copying machine. We also show that the inter-two-qubit entanglement is preserved bette…
We construct entanglement monotones for multi-qubit states based on Plücker coordinate equations of Grassmann variety, which are central notion in geometric invariant theory. As an illustrative example, we in details investigate entanglement monotones of a three-qubit state.
Based on the geometry of entangled three and two qubit states, we present the connection between the entanglement measure of the three-qubit state defined using the last Hopf fibration and the entanglement measures known as two- and three-tangle. Moreover, the generalization of the geometric representation of four qubit state and a potential entanglement measure is studied using sedenions for the simplification of the Hilbert space S^31 of the four qubit system. An entanglement measure is proposed and the degree of entanglement is calculated for specific states. The difficulties of a possible …
In this paper, we find the invariant for $n$-qubits and propose the residual entanglement for $n$-qubits by means of the invariant. Thus, we establish a relation between SLOCC entanglement and the residual entanglement. The invariant and the residual entanglement can be used for SLOCC entanglement classification for $n$-qubits.
In this work we study the entanglement of pure fourpartite of qubit states. The analysis is realized through the comparison between two different entanglement measures: the Groverian entanglement measure and the residual entanglement measured with negativities. Furthemore, we discuss some applications of four-way entangled fourpartite states.
In this paper, an unextendible product basis and exact-entanglement bases of three qubit is given, and the properties of entanglement for exact-entanglement bases are also discussed. In addition, the bound entangled mixed state is obtained from the exact-entanglement bases.
Using an exactly solvable pure dephasing model, we show how entanglement between qubits can be generated via the interaction with a common environment and concurrent application of suitable control pulses. The control pulses are able to effectively remove the detrimental effect of the environment while preserving the indirect interaction between the qubits, thereby leading to the generation of near-perfect entanglement. Furthermore, we also investigate the entanglement dynamics if the qubits are directly interacting; this interaction may even contain a noise term. The present of this additiona…
We analytically prove the necessary and sufficient criterion for the full separability of three-qubit Greenberger-Horne-Zeilinger (GHZ) diagonal states. The corresponding entanglement is exactly calculable for some GHZ diagonal states and is tractable for the others using the relative entropy of entanglement. We show that the biseparable criterion and the genuine entanglement are determined only by the biggest GHZ diagonal element regardless of all the other smaller diagonal elements. We have completely solved the entanglement problems of three-qubit GHZ diagonal states.
Pairwise correlation is really an important property for multi-qubit states. For the two-qubit X states extracted from Dicke states and their superposition states, we obtain a compact expression of the quantum discord by numerical check. We then apply the expression to discuss the quantum correlation of the reduced two-qubit states of Dicke states and their superpositions, and the results are compared with those obtained by entanglement of formation, which is a quantum entanglement measure.
In this paper we consider a system consist of a qubit and a qutrit, and find a formula to evaluate the concurrence for it. We show that entanglement of formation for this system obeys the same relation as for two-qubits.
We study the asymptotic entanglement of three identical qubits under the action of a Markovian open system dynamics that does not distinguish them. We show that by adding a completely depolarized qubit to a special class of two qubit states, by letting them reach the asymptotic state and by finally eliminating the added qubit, can provide more entanglement than by direct immersion of the two qubits within the same environment.
We establish a relation between concurrence and entanglement witnesses. In particular, we construct entanglement witnesses for three-qubit W and GHZ states in terms of concurrence and different set of operators that generate it. We also generalize our construction for multi-qubit states.
Recently, the theoretical work of Fröwis and W. Dür (Phys. Rev. Lett. \textbf{106}, 110402 (2011)) and the experiment of Lu \emph{et al.} (Nat. Photon. \textbf{8}, 364 (2014)) both showed that the logic-qubit entanglement has its potential application in future quantum communication and quantum network. However, the entanglement will suffer from the noise and decoherence. In this paper, we will investigate the entanglement purification for logic-qubit entanglement. We show that both the bit-flip error and phase-flip error in logic-qubit entanglement can be well purified. Moreover, the bit-flip…
We establish a characterization of multi-qubit entanglement constraints in terms of non-negative power of entanglement measures based on unified-$(q,s)$ entropy. Using the Hamming weight of the binary vector related with the distribution of subsystems, we establish a class of tight monogamy inequalities of multi-qubit entanglement based on the $α$th-power of unified-$(q,s)$ entanglement for $α\geq 1$. For $0 \leq β\leq 1$, we establish a class of tight polygamy inequalities of multi-qubit entanglement in terms of the $β$th-power of unified-$(q,s)$ entanglement of assistance. Thus our results c…
The qubit entanglement induced by quasiparticle excitations in the Heisenberg spin chain and its relationship to the Bethe Ansatz structure of the eigenmodes is studied. A phenomenon called entanglement quenching, which suppresses eigenstate entanglement, is described and shown to be mediated by Goldstone magnons. Scattering states are characterized by short-range entanglement, and never exhibit entanglement at the longest range. In contrast, bound states have complex long-range entanglement structures.
We reduce the necessary and sufficient biseparable conditions of the four qubit cluster diagonal state to concise forms. Only 4 out of the 15 parameters are proved to be relevant in specifying the genuine entanglement of the state. Using the relative entropy of entanglement as the entanglement measure, we analytically find the genuine entanglement of all the four qubit cluster diagonal states. The formulas of the genuine entanglement are of five kinds, for seven different parameter regions of entanglement.
We discuss entanglement evolution of a multi-qubit system when one of its qubits is subjected to a general noisy channel. For such a system, an evolution equation of entanglement for a lower bound for multi-qubit concurrence is derived. Using this evolution equation, the entanglement dynamics of an initially mixed three-qubit state composed of a GHZ and a W state is analyzed if one of the qubits is affected by a phase, an amplitude or a generalized amplitude damping channel.
We consider a general unitary operator acting on two qubits in a product state. We find the conditions such that the state of the qubits after the action is as entangled as possible. We also consider the possibility of using ancilla qubits to increase the entanglement.
The three-qubit space of entanglement types is the orbit space of the local unitary action on the space of three-qubit pure states, and hence describes the types of entanglement that a system of three qubits can achieve. We show that this orbit space is homeomorphic to a certain subspace of R^6, which we describe completely. We give a topologically based classification of three-qubit entanglement types, and we argue that the nontrivial topology of the three-qubit space of entanglement types forbids the existence of standard states with the convenient properties of two-qubit standard states.
Operator entanglement of two-qubit joint unitary operations is revisited. Schmidt number is an important attribute of a two-qubit unitary operation, and may have connection with the entanglement measure of the unitary operator. We found the entanglement measure of two-qubit unitary operators is classified by the Schmidt number of the unitary operators. The exact relation between the operator entanglement and the parameters of the unitary operator is clarified too.
We show that the restricted sharability and distribution of multi-qubit entanglement can be characterized by Tsallis-$q$ entropy. We first provide a class of bipartite entanglement measures named Tsallis-$q$ entanglement, and provide its analytic formula in two-qubit systems for $1 \leq q \leq 4$. For $2 \leq q \leq 3$, we show a monogamy inequality of multi-qubit entanglement in terms of Tsallis-$q$ entanglement, and we also provide a polygamy inequality using Tsallis-$q$ entropy for $1 \leq q \leq 2$ and $3 \leq q \leq 4$.
Entanglement and Berry phase are investigated in two interacting qubit systems. The XXZ spin interaction model with a slowly rotating magnetic field is employed for the interaction between the two qubits. We show how the anisotropy of interaction reveals unique relations between the Berry phases and the entanglements for the eigenstates of the system.
Using Rényi-$α$ entropy to quantify bipartite entanglement, we prove monogamy of entanglement in multi-qubit systems for $α\geq 2$. We also conjecture a polygamy inequality of multi-qubit entanglement with strong numerical evidence for $0.83-ε\leq α\leq 1.43+ε$ with $0<ε<0.01$.
In Phys. Rev. Lett. 80, 2245 (1998) an explicit expression for entanglement of formation for any two qubit state was given. Based on this result we present an expression for the Bures measure of entanglement for two qubit states. This measure was proposed in Phys. Rev. A 57, 1619 (1998), where the authors showed that it satisfies all properties every entanglement measure must have.
We define an entanglement measure, called the partial tangle, which represents the residual two-qubit entanglement of a three-qubit pure state. By its explicit calculations for three-qubit pure states, we show that the partial tangle is closely related to the faithfulness of a teleportation scheme over a three-qubit pure state.
Making use of the master equation and effective Hamiltonian approach, we investigate the steady state entanglement in a three-qubit $XX$ model. Both symmetric and nonsymmetric qubit-qubit couplings are considered. The system (the three qubits) is coupled to two bosonic baths at different temperatures. We calculate the steady state by the effective Hamiltonian approach and discuss the dependence of the steady state entanglement on the temperatures and couplings. The results show that for symmetric qubit-qubit couplings, the entanglements between the nearest neighbor are equal, independent of th…
