Tetrapartite entanglement measures of generalized GHZ state in the noninertial frames

Using a single-mode approximation, we carry out the entanglement measures, e.g., the negativity and von Neumann entropy when a tetrapartite generalized GHZ state is treated in a noninertial frame, but only uniform acceleration is considered for simplicity. In terms of explicit negativity calculated, we notice that the difference between the algebraic average π_(4) and geometric average Π_(4) is very small with the increasing accelerated observers and they are totally equal when all four qubits are accelerated simultaneously. The entanglement properties are discussed from one accelerated observer to all four accelerated observers. It is shown that the entanglement still exists even if the acceleration parameter r goes to infinity. It is interesting to discover that all 1-1 tangles are equal to zero, but 1-3 and 2-2 tangles always decrease when the acceleration parameter γ increases. We also study the von Neumann entropy and find that it increases with the number of the accelerated observers. In addition, we find that the von Neumann entropy S_(ABCDI), S_(ABCIDI), S_(ABICIDI) and S_(AIBICIDI) always decrease with the controllable angle θ, while the entropies S_(3-3 non), S_(3-2 non), S_(3-1 non) and S_(3-0 non) first increase with the angle θ and then decrease with it.

Algorithm for evaluating distance-based entanglement measures

Quantifying entanglement in quantum systems is an important yet challenging task due to its NP-hard nature.In this work,we propose an efficient algorithm for evaluating distance-based entanglement measures.Our approach builds on Gilbert's algorithm for convex optimization,providing a reliable upper bound on the entanglement of a given arbitrary state.We demonstrate the effectiveness of our algorithm by applying it to various examples,such as calculating the squared Bures metric of entanglement as well as the relative entropy of entanglement for GHZ states,W states,Horodecki states,and chessboard states.These results demonstrate that our algorithm is a versatile and accurate tool that can quickly provide reliable upper bounds for entanglement measures.

Extremal Entangled Four-Qubit Pure States with Respect to Multiple Entropy Measures

Four-qubit entanglement has been investigated using a recent proposed entanglement measure, multiple entropy measures （MEMS）. We have performed optimization for the nine different families of states of four-qubit system. Some extremal entangled states have been found.

Three-body entanglement induced by spontaneous emission in a three two-level atoms system

We study three-body entanglement induced by spontaneous emission in a three two-level atoms system by using the entanglement tensor approach. The results show that the amount of entanglement is strongly dependent on the initial state of the system and the species of atoms. The three-body entanglement is the result of the coherent superposition of the two-body entanglements. The larger the two-body entanglement is, the stronger the three-body entanglement is. On the other hand, if there exists a great difference in three two-body entanglement measures, the three-body entanglement is very weak. We also find that the maximum of the two-body entanglement obtained with nonidentical atoms is greater than that obtained with identical atoms via adjusting the difference in atomic frequency.

Optimized monogamy and polygamy inequalities for multipartite qubit entanglement

We investigate the monogamy and polygamy inequalities of arbitrary multipartite quantum states,and provide new classes of monogamy and polygamy inequalities of multiqubit entanglement in terms o f concurrence,entanglement of formation,negativity,and Tsallis-q entanglement,respectively.We show that these new monogamy and polygamy inequality relations are tighter than the existing ones with detailed examples.

Multiple Entropy Measures for Multi-particle Pure Quantum State

We propose to use a set of averaged entropies, the multiple entropy measures (MEMS), to partiallyquantify quantum entanglement of multipartite quantum state.The MEMS is vector-like with m = [N/2] components:[S_1, S_2,..., S_m], and the i-th component S_i is the geometric mean of i-qubits partial entropy of the system.The S_imeasures how strong an arbitrary i qubits from the system are correlated with the rest of the system.It satisfies theconditions for a good entanglement measure.We have analyzed the entanglement properties of the GHZ-state, theW-states, and cluster-states under MEMS.

Revised Geometric Measure of Entanglement for Multipartite and Continuous Variable Systems

Based on the revised geometric measure of entanglement （RGME） proposed by us [J. Phys. A： Math. Theor. 40 （2007） 3507], we obtain the RGME of multipartite state including three-qubit GHZ state, W state, and the generalized Smolin state （GSS） in the presence of noise and the two-mode squeezed thermal state. Moreover, we compare their RGME with geometric measure of entanglement （GME） and relative entropy of entanglement （RE）. The results indicate RGME is an appropriate measure of entanglement. Finally, we define the Gaussian GME which is an entangled monotone.

A Quantum Tanimoto Coefficient Fidelity for Entanglement Measurement

Fidelity plays an important role in quantum information processing,which provides a basic scale for comparing two quantum states.At present,one of the most commonly used fidelities is Uhlmann-Jozsa(U-J)fidelity.However,U-J fidelity needs to calculate the square root of the matrix,which is not trivial in the case of large or infinite density matrices.Moreover,U-J fidelity is a measure of overlap,which has limitations in some cases and cannot reflect the similarity between quantum states well.Therefore,a novel quantum fidelity measure called quantum Tanimoto coefficient(QTC)fidelity is proposed in this paper.Unlike other existing fidelities,QTC fidelity not only considers the overlap between quantum states,but also takes into account the separation between quantum states for the first time,which leads to a better performance of measure.Specifically,we discuss the properties of the proposed QTC fidelity.QTC fidelity is compared with some existing fidelities through specific examples,which reflects the effectiveness and advantages of QTC fidelity.In addition,based on the QTC fidelity,three discrimination coefficients d_(1)^(QTC),d_(2)^(QTC),and d_^(3)^(QTC)are defined to measure the difference between quantum states.It is proved that the discrimination coefficient d_(3)^(QTC)is a true metric.Finally,we apply the proposed QTC fidelity-based discrimination coefficients to measure the entanglement of quantum states to show their practicability.

Genuine tripartite entanglement and quantum phase transition

A new simplified formula is presented to characterize genuine tripartite entanglement of （2 2 n）-dimensional quantum pure states. The formula turns out equivalent to that given in （Quant. Inf. Comp. 7（7） 584 （2007））, hence it also shows that the genuine tripartite entanglement can be described only on the basis of the local （2 2）-dimensional reduced density matrix. In particular, the two exactly solvable models of spin system studied by Yang （Phys. Rev. A 71 030302（R） （2005）） are reconsidered by employing the formula. The results show that a discontinuity in the first derivative of the formula or in the formula itself of the ground state just corresponds to the existence of quantum phase transition, which is obviously different from the concurrence.

Quantifying entanglement in terms of an operational way

We establish entanglement monotones in terms of an operational approach,which is closely connected with the state conversion from pure states to the objective state by the local operations and classical communications.It is shown that any good entanglement quantifier defined on pure states can induce an entanglement monotone for all density matrices.Particularly,we show that our entanglement monotone is the maximal one among all those having the same form for pure states.In some special cases,our proposed entanglement monotones turn to be equivalent to the convex roof construction,which hence gain an operational meaning.Some examples are given to demonstrate different cases.

Concurrence and Wigner-Yanase Skew Information

A measure of entanglement on n qubits is defined in terms of Wigner-Yanase skew information. It is shown that the measure coincides essentially with the concurrence on two qubits. This uncovers the information-theoretic meaning of the concurrence of entangled states.

Application of quantum algorithms to direct measurement of concurrence of a two-qubit pure state

This paper proposes a method to measure directly the concurrence of an arbitrary two-qubit pure state based on a generalized Grover quantum iteration algorithm and a phase estimation algorithm. The concurrence can be calculated by applying quantum algorithms to two available copies of the bipartite system, and a final measurement on the auxiliary working qubits gives a better estimation of the concurrence. This method opens new prospects of entanglement measure by the application of quantum algorithms. The implementation of the protocol would be an important step toward quantum information processing and more complex entanglement measure of the finite-dimensional quantum system with an arbitrary number of qubits.

Direct measurement of two-qubit phononic entangled states via optomechanical interactions

We propose schemes of direct concurrence measurement for two-qubit phononic states from quantized mechanical vibration. By combining the Mach–Zehnder interferometer with the optomechanical cross-Kerr nonlinear effect, direct concurrence measurement schemes for two-qubit phononic entangled states are achieved with the help of photon detection with respect to the output of the interferometer. For different types of entangled states, diversified quantum devices and operations are designed accordingly. The final analysis shows reasonable performance under the current parameter conditions.Our schemes may be useful for potential phonon-based quantum computation and information in the future.

Unveiling the geometric meaning of quantum entanglement:Discrete and continuous variable systems

We show that the manifold of quantum states is endowed with a rich and nontrivial geometric structure.We derive the Fubini−Study metric of the projective Hilbert space of a multi-qubit quantum system,endowing it with a Riemannian metric structure,and investigate its deep link with the entanglement of the states of this space.As a measure,we adopt the entanglement distance E preliminary proposed in Phys.Rev.A 101,042129(2020).Our analysis shows that entanglement has a geometric interpretation:E(|ψ>)is the minimum value of the sum of the squared distances between|ψ>and its conjugate states,namely the statesυ^(μ).σ^(μ)|ψ>,whereυ^(μ)are unit vectors andμruns on the number of parties.Within the proposed geometric approach,we derive a general method to determine when two states are not the same state up to the action of local unitary operators.Furthermore,we prove that the entanglement distance,along with its convex roof expansion to mixed states,fulfils the three conditions required for an entanglement measure,that is:i)E(|ψ>)=0 iff|ψ>is fully separable;ii)E is invariant under local unitary transformations;iii)E does not increase under local operation and classical communications.Two different proofs are provided for this latter property.We also show that in the case of two qubits pure states,the entanglement distance for a state|ψ>coincides with two times the square of the concurrence of this state.We propose a generalization of the entanglement distance to continuous variable systems.Finally,we apply the proposed geometric approach to the study of the entanglement magnitude and the equivalence classes properties,of three families of states linked to the Greenberger−Horne−Zeilinger states,the Briegel Raussendorf states and the W states.As an example of application for the case of a system with continuous variables,we have considered a system of two coupled Glauber coherent states.

Some Tighter Monogamy Inequalities in N-Qubit Systems

We construct a piecewise function to investigate some monogamy inequalities in terms of theαth power of concurrence and negativity(α≥2),entanglement of formation(α≥√2),and Tsallis-q entanglement(α≥1).These inequalities are tighter than the existing results with detailed examples.Particularly,it is worth highlighting some classes of quantum states which can saturate these monogamy inequalities forα=2,4 and 6 in terms of concurrence and negativity and forα=1,2 and 3 in terms of Tsallis-q entanglement.

Computing Geometric Measure of Entanglement for Symmetric Pure States via the Jacobian SDP Relaxation Technique

The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue(US-eigenvalue)for symmetric complex tensors,which can be taken as a multilinear optimization problem in complex number field.In this paper,we convert the problem of computing the geometric measure of entanglement for symmetric pure states to a real polynomial optimization problem.Then we use Jacobian semidefinite relaxation method to solve it.Some numerical examples are presented.

Quantum entanglement swapping of two arbitrary biqubit pure states

In this paper, the issue of swapping quantum entanglements in two arbitrary biqubit pure states via a local bipartite entangledstate projective measure in the middle node is studied in depth, especially with regard to quantitative aspects. Attention is mainly focused on the relation between the measure and the final entanglement obtained via swapping. During the study, the entanglement of formation(EoF) is employed as a quantifier to characterize and quantify the entanglements present in all involved states. All concerned EoFs are expressed analytically; thus, the relation between the final entanglement and the measuring state is established.Through concrete analyses, the measure demands for getting a certain amount of a final entanglement are revealed. It is found that a maximally entangled final state can be obtained from any two given initial entangled states via swapping with a certain probability;however, a peculiar measure should be performed. Moreover, some distinct properties are revealed and analyzed. Such a study will be useful in quantum information processes.

Efficient preparation of a four-photon cluster state with homodyne measurement via cross-Kerr nonlinearity

A scheme is proposed for generating a multiphoton entangled cluster state among four modes. The scheme only uses Kerr medium, beam splitter and homodyne measurements on coherent light fields, which can be efficiently made in quantum optical laboratories. The photon in the signal mode is prepared in a superposition state of the vacuum state and one-photon state while the probe beam is initially set in a coherent state superposition. The strong probe mode interacts successively with multiple signal-mode photons, each causing a conditional phase rotation in the probe mode. Subsequent momentum quadrature homodyne measurement of the probe mode will project the photons in the signal mode into the desired entangled states. It is shown that under certain conditions, the four-photon cluster state can be generated with high fidelity and high success probability, and the scheme is feasible by current experimental technology.