Kai Chen, Sergio Albeverio, Shao-Ming Fei
We give explicit expressions for entanglement measures of Werner states in arbitrary dimensions in terms of concurrence and tangle. We show that an optimal ensemble decomposition for a joint density matrix of a Werner state can achieve the minimum average concurrence and tangle simultaneously. Furthermore, the same decomposition also attains entanglement of formation for Werner states.
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The genuine concurrence is a standard quantifier of multipartite entanglement, detection and quantification of which still remains a difficult problem from both theoretical and experimental point of view. Although many efforts have been devoted toward the detection of multipartite entanglement (e.g., using entanglement witnesses), measuring the degree of multipartite entanglement, in general, requires some knowledge about an exact shape of a density matrix of the quantum state. An experimental reconstruction of such density matrix can be done by full state tomography which amounts to having th…
In this report, a scheme different from the PT and Wootters concurrence is developed to acquire a criterion to investigate the bipartite separability of the Werner state.
In this article we investigate the unitary dynamics of squashed entanglement and concurrence measures in Werner state and maximally entangled mixed states (MEMS) under two different Hamiltonians. The aim of the present study is two fold. The first part of the study deals with the dynamics under Heisenberg Hamiltonian and the second part deals under bi-linear bi-quadratic Hamiltonian which is the extension of first Hamiltonian. In both the parts we investigate the dynamical trade off and balancing points for squashed entanglement and concurrence. During the study, we also found the results of e…
Bell-inequality violation and entanglement, measured by Wootters' concurrence and negativity, of two qubits initially in Werner or Werner-like states coupled to thermal reservoirs are analyzed within the master equation approach. It is shown how this simple decoherence process leads to generation of states manifesting the relativity of two-qubit entanglement measures.
We consider a special kind of mixed states -- a {\it Werner derivative}, which is the state transformed by nonlocal unitary -- local or nonlocal -- operations from a Werner state. We show the followings. (i) The amount of entanglement of Werner derivatives cannot exceed that of the original Werner state. (ii) Although it is generally possible to increase the entanglement of a single copy of a Werner derivative by LQCC, the maximal possible entanglement cannot exceed the entanglement of the original Werner state. The extractable entanglement of Werner derivatives is limited by the entanglement …
We propose a novel parameterized entanglement measure $α$-concurrence for bipartite systems. By employing positive partial transposition and realignment criteria, we derive analytical lower bounds for the $α$-concurrence. Moreover, we calculate explicitly the analytic expressions of the $α$-concurrence for isotropic states and Werner states.
A new entanglement measure, which is called D-concurrence, is proposed. Then the upper and lower bounds for D-concurrence are obtained and the relationship between D-concurrence and the usual concurrence of Wootters was established. In addition, comparing with the usual concurrence, D-concurrence has some special merits.
The Generalized Werner-Like states (GWLs) are a class of non--X states in which the exchange operator is replaced for a generic one-rank projector in the Werner states. We obtained an exact analytical expression of Quantum Discord for these states. The optimization problem involved is solved by giving an analytical expression, in exact form for the conditional entropy. We compared the Quantum Discord (QD) with the Entanglement of Formation (EoF) for the same states. The pure states of GWLs with equal concurrence have the same QD and EoF.
We derive an experimentally observable lower bound on concurrence of mixed quantum states in terms of an entanglement witness, relating measurements on single states with those on two copies.
We use Robust Semidefinite Programs and Entanglement Witnesses to study the distillability of Werner states. We perform exact numerical calculations which show 2-undistillability in a region of the state space which was previously conjectured to be undistillable. We also introduce bases which yield interesting expressions for the {\em distillability witnesses} and for a tensor product of Werner states with arbitrary number of copies.
We show that each entanglement witness detecting given bipartite entangled state provides an estimation of its concurrence. We illustrate our result with several well known examples of entanglement witnesses and compare the corresponding estimation of concurrence with other estimations provided by the trace norm of partial transposition and realignment.
Two measures of entanglement, negativity and concurrence are studied for two arbitrary qudits. We obtain negativity as an expectation value of an operator. The differences of the squares of negativity and concurrence are invariants of multilevel entanglement. Explicit results for qutrits and quadrits are obtained.
The verification of quantum entanglement under the influence of realistic noise and decoherence is crucial for the development of quantum technologies. Unfortunately, a full entanglement characterization is generally not possible with most entanglement criteria such as entanglement witnesses or the partial transposition criterion. In particular, so called bound entanglement cannot be certified via the partial transposition criterion. Here we present the full entanglement verification of dephased qubit and qutrit Werner states via entanglement quasiprobabilities. Remarkably, we are able to reve…
A note on the lower bounds of genuine multipartite entanglement concurrence
本文研究真正多体纠缠并发度的下界，通过推导其与二分及多体纠缠并发度的关系，提出更优下界，并探讨其与其他纠缠度量的联系。
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.
We obtained analytical expressions for Entangled of Formation (EoF) and Quantum Discord (QD) of Werner states and Generalized Werner-Like states. The optimization problem involved under the exact analytical form is obtained for both quantities. In order to illustrate the importance of our results we studied the EoF and the QD of these states. Using discrete formulation of continue states with the $f-$deformed coherent states obtained as deformed annihilation operator coherent states and as deformed displacement operator coherent states. The EoF and QD of bipartite Werner-Like states $f-$deform…
An entanglement monotone, which is invariant under the determinant 1 SLOCC operations and measures the true quadripartite entanglement, is explicitly constructed.
We discuss a model of two nonlinear Kerr-like oscillators, mutually coupled and excited by parametric process. We show that the system's evolution, starting from Werner-like states, remains closed within a small set of two-mode n-photon states the system, and pure two-qubit entangled state can be generated. For some initial Werner-like states delayed entanglement generation can be observed. We investigate the influence of two damping mechanisms on the system's evolution. We show that for the both cases, the entanglement can survive despite the presence of damping, and the effects of sudden ent…
Probabilities of measurement outcomes of two-particle entangled states give a physically transparent interpretation of the concurrence and of the I-concurrence as entanglement measures. The (I)-concurrence can thus be measured experimentally. The tight connection between these measures and Bell inequalities is highlighted.
We discuss a kind of generalized concurrence for a class of high dimensional quantum pure states such that the entanglement of formation is a monotonically increasing convex function of the generalized concurrence. An analytical expression of the entanglement of formation for a class of high dimensional quantum mixed states is obtained.
Genuine-multipartite-entanglement (GME) concurrence is a measure of genuine multipartite entanglement that generalizes the well-known notion of concurrence. We define an observable for GME concurrence. The observable permits us to avoid full state tomography and leads to different analytic lower bounds. By means of explicit examples we show that entanglement criteria based on the bounds have a better performance with respect to the known methods.
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.
We consider the maximum bipartite entanglement that can be distilled from a single copy of a multipartite mixed entangled state, where we focus mostly on $d\times d\times n$-dimensional tripartite mixed states. We show that this {\em assisted entanglement}, when measured in terms of the generalized concurrence (named G-concurrence) is (tightly) bounded by an entanglement monotone, which we call the G-concurrence of assistance. The G-concurrence is one of the possible generalizations of the concurrence to higher dimensions, and for pure bipartite states it measures the {\em geometric mean} of t…
We discuss the monotonicity under local operations and classical communication (LOCC) of systematically constructed quantities aiming at quantification of entanglement properties of multipartite quantum systems. The so-called generalized multipartite concurrences can qualify as legitimate entanglement measures if they are monotonous under LOCC. In the paper we give a necessary and sufficient criterion for their monotonicity.
The problems of genuine multipartite entanglement detection and classification are challenging. We show that a multipartite quantum state is genuine multipartite entangled if the multipartite concurrence is larger than certain quantities given by the number and the dimension of the subsystems. This result also provides a classification of various genuine multipartite entanglement. Then, we present a lower bound of the multipartite concurrence in terms of bipartite concurrences. While various operational approaches are available for providing lower bounds of bipartite concurrences, our results …
We define the concurrence hierarchy as d-1 independent invariants under local unitary transformations in d-level quantum system. The first one is the original concurrence defined by Wootters et al in 2-level quantum system and generalized to d-level pure quantum states case. We propose to use this concurrence hierarchy as measurement of entanglement. This measurement does not increase under local quantum operations and classical communication.
We obtain a simplified and obvious expression of "concurrence" in Wootters' measure of entanglement of a pair of qubits having no more than two non-zero eigenvalues in terms of concurrences of eigenstates and their simple combinations. It not only simplifies the calculation of Wootters' measure of entanglement, but also reveals some its general and important features. Our conclusions are helpful to understand and use quantum entanglement further.
Assuming the validity of a conjecture in quant-ph/9910026 and quant-ph/9910022 we show that the distillable entanglement for two bipartite states, each of which individually has zero distillable entanglement, can be nonzero. We show that this also implies that the distillable entanglement is not a convex function. Our example consists of the tensor product of a bound entangled state based on an unextendible product basis with a Werner state which lies in the class of conjectured undistillable states.
Transfer of entanglement and information is studied for quantum teleportation of an unknown entangled state through noisy quantum channels. We find that the quantum entanglement of the unknown state can be lost during the teleportation even when the channel is quantum correlated. We introduce a fundamental parameter of correlation information which dissipates linearly during the teleportation through the noisy channel. Analyzing the transfer of correlation information, we show that the purity of the initial state is important in determining the entanglement of the replica state.
Monogamy of entanglement is generally discussed using a bipartite entanglement measure as an upper bound. Here we discuss a new kind of monogamous relation where the upper bound is given by a multipartite measure of entanglement, the generalized concurrence. We show a new monogamous equality involving the multipartite concurrence, all the bipartite concurrences and the genuine tripartite entanglement for pure three qubits system. The result extends to mixed states in an inequality involving the generalized concurrence and all the bipartite concurrences. We provide a counter-example showing tha…
We study generalized concurrences as a tool to detect the entanglement of bipartite quantum systems. By considering the case of 2 X 4 states of rank 2, we prove that generalized concurrences do not, in general, give a necessary and sufficient condition of separability. We identify a set of entangled states which are undetected by this method.
In this paper we investigate two different entanglement measures in the case of mixed states of two qubits. We prove that the negativity of a state can never exceed its concurrence and is always larger then $\sqrt{(1-C)^2+C^2}-(1-C)$ where $C$ is the concurrence of the state. Furthermore we derive an explicit expression for the states for which the upper or lower bound is satisfied. Finally we show that similar results hold if the relative entropy of entanglement and the entanglement of formation are compared.
The symmetric Werner states for $n$ qubits, important in the study of quantum nonlocality and useful for applications in quantum information, have a surprisingly simple and elegant structure in terms of tensor products of Pauli matrices. Further, each of these states forms a unique local unitary equivalence class, that is, no two of these states are interconvertible by local unitary operations.
The concurrence vectors are proposed by employing the fundamental representation of $A_n$ Lie algebra, which provides a clear criterion to evaluate the entanglement of bipartite system of arbitrary dimension for both pure and mixed states. Accordingly, a state is separable if the norm of its concurrence vector vanishes. The state vectors related to SU(3) states and SO(3) states are discussed in detail. The sign situation of nonzero components of concurrence vectors of entangled bases presents a simple criterion to judge whether the whole Hilbert subspace spanned by those bases is entangled, or…
Unique entanglement time evolution of two-qubit product separable and extended Werner-like states in a discrete qubit environment
本文研究两量子比特乘积可分离态与扩展Werner-like态在均匀、白噪声及混合环境中的纠缠时间演化，揭示两者纠缠子空间不相交、环境与子系统耦合的时间反比关系，以及白噪声耗散由相互作用强度分布宽度决定等规律。
Concurrence and further entanglement quantifyers can be computed explicitly for channels of rank two if representable by just two Kraus operators. Almost all details are available for the subclass of rank two 1-qubit-channels. There is a simple geometric picture beyond, explaining nicely the role of anti-linearity.
We use matched quantum entanglement witnesses to study the separable criteria of continuous variable states. The witness can be written as an identity operator minus a Gaussian operator. The optimization of the witness then is transformed to an eigenvalue problem of a Gaussian kernel integral equation. It follows a separable criterion not only for symmetric Gaussian quantum states, but also for non-Gaussian states prepared by photon adding to or/and subtracting from symmetric Gaussian states. Based on Fock space numeric calculation, we obtain an entanglement witness for more general two-mode s…
We study the role of average concurrence in entanglement swapping in quantum networks. We begin with qubit pure states, and there is a very simple rule governing the propagation of average concurrence in multiple swaps. We look at examples of mixed qubit states, and find the relation for pure states gives an upper bound on what is possible with mixed states. We then move on to qudits, where we make use of the I-concurrence. Here the situation is not as simple as for qubits, but in some cases relatively straightforward results can be obtained.
Quantum entanglement approaching with concurrence in the presence of chaos
本研究利用并发度表征二分系统动力学，通过量子 kicked top 模型分析纠缠的混沌特性，并基于约化密度矩阵特征值区分规则与混沌行为，依赖于激发强度。
We extend the definition of concurrence into a family of entanglement monotones, which we call concurrence monotones. We discuss their properties and advantages as computational manageable measures of entanglement, and show that for pure bipartite states all measures of entanglement can be written as functions of the concurrence monotones. We then show that the concurrence monotones provide bounds on quantum information tasks. As an example, we discuss their applications to remote entanglement distributions (RED) such as entanglement swapping and remote preparation of bipartite entangled state…
In this paper we explore the non-monotonic nature of entanglement of formation with respect to concurrence for pure bipartite states. For pure bipartite system, one of the basic physical reason of this non-monotonicity character is due to the existence of incomparable states, i.e., the pure bipartite states which are not convertible to each other by LOCC with certainty.
A new entanglement measure based on the total concurrence
本文提出基于总纠缠度的新纠缠度量$\mathcal{C}^{t}_q$和$\mathcal{C}^{t}_α$，并推导了其解析表达式和下界，研究了其在双ubit系统中的结合关系。
Tetrahedron genuine entanglement measure of four-qubit systems
针对四量子比特系统，本文利用九类SLOCC封闭态的concurrence构造concurrence四面体，并以其体积定义真纠缠测度，可分类双可分纠缠且不同于现有真四体concurrence。
We propose two classes of the generalized concurrence vectors of the multipartite systems consisting of qubits. Making use of them, we are able to, respectively, describe and quantify GHZ-class and W-class entanglement both in total and between arbitrary two partite in the multipartite system consisting of qubits. In the case of pure state of three qubits that one partite is separable, it is shown to exactly back to the usual Wootters' concurrence after introduce a whole concurrence vector. In principle, our method is applicable to any $N$-partite systems consisting of $N$ qubits.
Quantum coherence is a fundamental manifestation of the quantum superposition principle. Recently, Baumgratz \emph{et al}. [Phys. Rev. Lett. \textbf{113}, 140401 (2014)] presented a rigorous framework to quantify coherence from the view of theory of physical resource. Here we propose a new valid quantum coherence measure which is a convex roof measure, for a quantum system of arbitrary dimension, essentially using the generalized Gell-Mann matrices. Rigorous proof shows that the proposed coherence measure fulfills all the requirements dictated by the resource theory of quantum coherence measur…
Entanglement estimation of Werner states with a quantum extreme learning machine
提出一种基于量子极限学习机（QELM）的协议，通过结合随机Werner态与储备池态并利用Ising哈密顿量演化，实现对Werner态纠缠量的估计，并验证了其鲁棒性及磁场参数对精度的影响。
G_q-concurrence and entanglement constraints in multiqubit systems
本文提出$G_q$-并发度作为并发度的推广，证明其满足纠缠度量公理，并利用其平方的单夫不等式构造纠缠指示器，以检测多量子比特系统中的真正纠缠态。
Single-qubit measurement of two-qubit entanglement in generalized Werner states
本文推广了一种无需同时探测两个量子比特的干涉式纠缠测量方法，使其适用于广义Werner态双量子比特混合态，并通过定量分析实验不完美性验证了方法的可实现性与抗损耗能力。
Entanglement and steering are used to describe quantum inseparabilities. Steerable states form a strict subset of entangled states. A natural question arises concerning how much territory steerability occupies entanglement for a general two-qubit entangled state. In this work, we investigate the constraint relation between steerability and concurrence by using two kinds of evolutionary states and randomly generated two-qubit states. By combining the theoretical and numerical proofs, we obtain the upper and lower boundaries of steerability. And the lower boundary can be used as a sufficient cri…
In this work we propose the geometric mean of bipartite concurrences as a genuine multipartite entanglement measure. This measure achieves the maximum value for absolutely maximally entangled states and has desirable properties for quantifying potential quantum resources. The simplicity and symmetry in the definition facilitates its computation for various multipartite entangled states including the GHZ states and the $W$ states. With explicit examples we show that our measure results in distinct entanglement orderings from other measures, and can detect differences in certain types of genuine…
Explicit expressions for the concurrence of all positive and trace-preserving ("stochastic") 1-qubit maps are presented. We construct the relevant convex roof patterns by a new method. We conclude that two component optimal decompositions always exist. Our results can be transferred to 2xN-quantumsystems providing the concurrence for all rank two density operators as well as lower and upper bounds for their entanglement of formation. We apply these results to a study of the entanglement entropy of 1-qubit stochastic maps which preserve axial symmetry. Using analytic and numeric results we anal…
Entanglement, a non-trivial phenomena manifested in composite quantum system, can be served as a new type of physical resource in the emerging technology of quantum information and quantum computation. However, a quantum entanglement is fragile to the environmental-induced decoherence. Here, we present a novel way to prepare a high-concurrence steady-state entanglement of two hole spins in a quantum dot molecular via optical pumping of trion levels. In this scheme, the spontaneous dispassion is used to induce and stabilize the entanglement with rapid rate. It is firstly shown that under certai…
We study the entanglement of a multipartite quantum state. An inequality between the bipartite concurrence and the multipartite concurrence is obtained. More effective lower and upper bounds of the multipartite concurrence are obtained. By using the lower bound, the entanglement of more multipartite states are detected.
Concurrence, introduced by Hill and Wootters [Phys. Rev. Lett. 78, 5022 (1997)], provides an important measure of entanglement for a general pair of qubits that is faithful: strictly positive for entangled states and vanishing for all separable states. Such a measure captures the entire content of entanglement, providing necessary and sufficient conditions for separability. We present an extension of concurrence to multiparticle pure states in arbitrary dimensions by a new framework using the Lagrange's identity and wedge product representation of separability conditions, which coincides with …
Dynamics of Entanglement in Schwarzschild Black Holes
研究Hawking辐射对Schwarzschild黑洞中纠缠的影响，通过纠缠度分析其随Hawking温度的变化，发现可访问区域的纠缠度随Hawking加速度增加而减小，不可访问区域则单调增加，并探讨了三种噪声通道下纠缠度的动力学行为。
We propose a new classification for the entanglement in graph states based on generalized con- currence. The numerical results indicate that the eight different three-qubit graph states in three categories, 64 four-qubit graph states in five categories and 1024 five-qubit graph states are in ten classes. We also compare this classification with equivalence classes of these graph states under local complementation (LC) operator, and the obtained result suggests that classification by generalized concurrence is not in contradiction with the LC-rule.
We classify the completely-positive maps acting on two $d$-dimensional systems which commute with all $U\otimes U$ unitaries, where $U\in SU(d)$. This set of operations map Werner states to Werner states. We find a simple condition for a map being implementable by stochastic local operations and classical communication (SLOCC). We show that all PPT-preserving maps can be implemented by SLOCC. This can be used to prove that the entanglement of Werner states cannot be stochastically increased, even if we allow PPT entanglement for free.
We provide detailed proofs of triangle inequalities in coherence measures and entanglement concurrence. If a rank-$2$ state $\varrho$ can be expressed as a convex combination of two pure states, i.e., $\varrho=p_{1}|ψ_{1}\rangle\langleψ_{1}|+p_{2}|ψ_{2}\rangle\langleψ_{2}|$, a triangle inequality can be established as $\big{|}E(|Ψ_{1}\rangle)-E(|Ψ_{2}\rangle)\big{|}\leq E(\varrho)\leq E(|Ψ_{1}\rangle)+E(|Ψ_{2}\rangle)$, where $|Ψ_{1}\rangle=\sqrt{p_{1}}|ψ_{1}\rangle$ and $|Ψ_{2}\rangle=\sqrt{p_{2}}|ψ_{2}\rangle$, $E$ can be considered either coherence measures or entanglement concurrence. This…
Interaction with environment may lead to the transition of quantum system from pure state to the mixed one. In this case, the problem of definition of entanglement may arise. In particular, quantitative measure of entanglement concurrence is introduced differently for pure and mixed states. To solve this problem, alternative definition of concurrence of mixed states is introduced in present work. In particular, concurrence of mixed states is determined as an average of pure states concurrences. Averaging is carried out over the quantum ensemble, corresponding to all possible realization of ran…
It is an interesting problem to construct genuine tripartite entangled states based on the collective use of two bipartite entangled states. We consider the case that the states are two-qubit Werner states, we construct the interval of parameter of Werner states such that the tripartite state is genuine entangled. Further, we present the way of detecting the tripartite genuine entanglement using current techniques in experiments. We also investigate the lower bound of genuine multipartite entanglement concurrence.
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