We propose a general scheme to measure the concurrence of an arbitrary two-qubit pure state in atomic systems. The protocol is based on one- and two-qubit operations acting on two available copies of the bipartite system, and followed by a global qubit readout. We show that it is possible to encode the concurrence in the probability of finding all atomic qubits in the ground state. Two possible scenarios are considered: atoms crossing 3D microwave cavities and trapped ion systems.
To explore the properties of a two-qubit mixed state, we consider quantum teleportation. The fidelity of a teleported state depends on the resource state purity and entanglement, as characterized by concurrence. Concurrence and purity are functions of state parameters. However, it turns out that a state with larger purity and concurrence, may have comparatively smaller fidelity. By computing teleportation fidelity, concurrence and purity for two-qubit X-states, we show it explicitly. We further show that fidelity changes monotonically with respect to functions of parameters - other than concur…
We present a new kind of monogamous relations based on concurrence and concurrence of assistance. For $N$-qubit systems $ABC_1...C_{N-2}$, the monogamy relations satisfied by the concurrence of $N$-qubit pure states under the partition $AB$ and $C_1...C_{N-2}$, as well as under the partition $ABC_1$ and $C_2...C_{N-2}$ are established, which give rise to a kind of restrictions on the entanglement distribution and trade off among the subsystems.
Concurrence, as one of entanglement measures, is a useful tool to characterize quantum entanglement in various quantum systems. However, the computation of the concurrence involves difficult optimizations and only for the case of two qubits an exact formula was found. We investigate the concurrence of four-qubit quantum states and derive analytical lower bound of concurrence using the multiqubit monogamy inequality. It is shown that this lower bound is able to improve the existing bounds. This approach can be generalized to arbitrary qubit systems. We present an exact formula of concurrence fo…
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.
The $k$-ME concurrence as a measure of multipartite entanglement (ME) unambiguously detects all $k$-nonseparable states in arbitrary dimensions, and satisfies many important properties of an entanglement measure. Negativity is a simple computable bipartite entanglement measure. Invariant and tangle are useful tools to study the properties of the quantum states. In this paper we mainly investigate the internal relations among the $k$-ME concurrence, negativity, polynomial invariants, and tangle. Strong links between $k$-ME concurrence and negativity as well as between $k$-ME concurrence and pol…
Entanglement and coherence are two essential quantum resources for quantum information processing. A natural question arises of whether there are direct link between them. And by thinking about this question, we propose a new measure for quantum state that contains concurrence and is called intrinsic concurrence. Interestingly, we discover that the intrinsic concurrence is always complementary to coherence. Note that the intrinsic concurrence is related to the concurrence of a special pure state ensemble. In order to explain the trade-off relation more intuitively, we apply it in some composit…
Quantum resources lie at the core of quantum computation as they are responsible for the computational advantage in many tasks. The complementary relations are a pathway to understanding how the physical quantities are related. Here it was shown that there is a complementary relationship that evolves quantumness of a two qubits state and the degree of entanglement, respectively measured by Roughness and Concurrence. It was shown that there is a $\mathbb{R}^2$ surface that characterizes these quantities and can be understood as a projection of the Bloch Sphere. The distribution of states on the…
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…
We study the dynamics of four-qubit W state under various noisy environments by solving analytically the master equation in the Lindblad form in which the Lindblad operators correspond to the Pauli matrices and describe the decoherence of states. Also, we investigate the dynamics of the entanglement using the lower bound to the concurrence. It is found that while the entanglement decreases monotonically for Pauli-Z noise, it decays suddenly for other three noises. Moreover, by studying the time evolution of entanglement of various maximally entangled four-qubit states, we indicate that the fou…
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.
One-qubit pure states, living on the surface of Bloch sphere, can be mapped onto the usual complex plane by using stereographic projection. In this paper, after reviewing the entanglement of two-qubit pure state, it is shown that the \emph{quaternionic} stereographic projection is related to concurrence measure. This is due to the fact that every two-qubit state, in ordinary complex field, corresponds to the one-qubit state in quaternionic skew field, called quaterbit. Like the one-qubit states in complex field, the stereographic projection maps every quaterbit onto a quaternion number whose c…
We investigate the monogamy relations related to the concurrence and the entanglement of formation. General monogamy inequalities given by the αth power of concurrence and entanglement of formation are presented for N-qubit states. The monogamy relation for entanglement of assistance is also established. Based on these general monogamy relations, the residual entanglement of concurrence and entanglement of formation are studied. Some relations among the residual entanglement, entanglement of assistance, and three tangle are also presented.
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 demonstrate that for every two-qubit state there is a X-counterpart, i.e., a corresponding two-qubit X-state of same spectrum and entanglement, as measured by concurrence, negativity or relative entropy of entanglement. By parametrizing the set of two-qubit X-states and a family of unitary transformations that preserve the sparse structure of a two-qubit X-state density matrix, we obtain the parametric form of a unitary transformation that converts arbitrary two-qubit states into their X-counterparts. Moreover, we provide a semi-analytic prescription on how to set the parameters of this uni…
In this work we study the steady state entanglement between two qubits interacting asymetrically with a common non-Markovian environment. Depending on the initial two-qubit state, the asymmetry in the couplings between each qubit and the non-Markovian environment may lead to enhanced entanglement in the steady state of the system, measured in terms of the two-qubit concurrence. Our results indicate that, if a qubit-qubit interaction is also present, the two-qubit steady state concurrence is always favored by the symmetric or anti-symmetric coupling configuration. Although finite, the steady co…
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…
Time Reversal and n-qubit Canonical Decompositions
对于n是偶数个量子比特和v是酉演化，酉群的矩阵分解v=k1 a k2是明确可计算的，并且允许研究并发纠缠单调的动力学。该并发正则分解（CCD）的边因子k1和k2是并发对称的，因此动力学问题归结为对因子a的考虑。在这项工作中，我们提供了一个显式的数值算法计算v=k1 k2为n奇数。此外，在奇数情况下，我们将单调性提升为双参数函数，从而允许奇数量子比特中的并发动力学理论。一般化也可以使用CCD来研究，再次导致大多数酉的最大并发能力。关键的技术是考虑自旋翻转作为一个时间反演对称运营商在维格纳的公理化，原来的CCD推导可以重述完全在这个时间反演。在途中，我们观察到Kramers的非简并性：任何时间反演对称n-量子比特哈密顿量的非简并本征态的存在要求（i）n偶数和（ii）所述本征态的最大并发。我们提供了如何应用这项工作的例子来研究自旋链哈密顿纠缠的运动学和动力学。
Adding the maximally mixed state with some weight to the entanglement system leads to disentanglement of the latter. For each predefined entangled state there exists a minimal value of this weight for which the system loses its entanglement properties. These values were proposed to be used as a quantitative measure of entanglement called robustness [G. Vidal and R. Tarrach, Phys. Rev. A 59, 141 (1999)]. Using the concurrence, we propose the derivation of this measure for the system of two-qubit. Namely, for a two-qubit pure state, an exact expression of robustness is obtained. Finally, in the …
Anisotropy and Magnetic Field Effects on the Genuine Multipartite Entanglement of Multi-Qubit Heisenberg {\it XY} Chains
研究表明，对于两量子比特海森堡XY模型，通过调节外部磁场超过某个有限的临界强度，各向异性和磁场可以一起用于产生任何有限温度下的纠缠。这个有趣的结果来自于使用Wootters并发度的分析，Wootters并发度是两个量子比特状态的纠缠的可计算度量。最近，Mintert { \em et al.}提出了Wootters并发多体状态的推广。这些MKB并发具有数学性质，使人们能够理解这种特征行为的起源。本文首先利用MKB共存研究了各向异性和磁场对四量子比特海森堡XY链多体热纠缠的影响。我们表明，该模型表现出的特征行为类似于双量子比特模型。此外，我们表明，这可以再次理解使用相同的数学属性。接下来，我们证明了六量子比特海森堡XY链也具有使其具有特征行为所必需的性质。最重要的是，可以直接测量纯态的多体MKB共存。这可能为我们的猜想提供了实验验证，即对于任何偶数量子比特的海森堡XY链，通过施加足够大的磁场，总是可以在任何有限温度下获得非零的真正的多体纠缠。
We investigate the dynamics of entanglement given by the concurrence of a two-qubit system in the non-Markovian setting. A quantum master equation is derived which is solved in the eigen basis of the system Hamiltonian for X-type initial states. A closed formula for time evolution of concurrence is presented for a pure state. It is shown that under the influence of dissipation that non-zero entanglement is created in unentangled two qubit states which decay in the same way as pure entangled states. We also show that under real circumstances, the decay rate of concurrence is strongly modified b…
Two-Qubit Separabilities as Piecewise Continuous Functions of Maximal Concurrence
一般的实（b=1）和复（b=2）两量子比特态本质上分别是9维和15维的。它们相对于希尔伯特-施密特和布雷斯度量所占据的空间的总体积可以作为Zyczkowski和Sommers公式的特殊情况获得。我们声称，如果人们可以确定某些度量无关的三维“特征值参数化可分性函数”（EPSF），那么这些公式可以很容易地修改，以产生希尔伯特-施密特和布雷斯体积只占用可分离的两个量子比特状态（因此相关的可分性概率）。类似的早期分析的“对角入口参数化可分性函数”的动机，我们进一步探讨的可能性，这种3维EPSF可能，反过来，可表示为一些特殊的相关变量的单变量函数-我们假设是最大的并发（0 < C <1）的频谱轨道。大量的数值结果，我们得到了相当密切的支持这一假设。实数和复数估计的EPSF在C=1/2处都表现出明显的幅度跳跃，大约为50%，以及一些额外的匹配不连续性。
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 …
我们讨论的纠缠措施称为I-并发和纠缠的性质。对于两体纯态，I-纠缠和纠缠只与边缘密度算符的纯度有关。二分混合态的I-共点（缠结）是联合密度算符的纯态系综分解的最小平均I-共点（缠结）。Terhal和Vollbrecht [Phys. Rev. Lett. 85，2625（2000）]给出了任意维各向同性态形成的纠缠的显式公式。我们使用他们的形式主义，以获得比较表达式的I-并发和纠缠的各向同性状态。
