Relation Between Stereographic Projection and Concurrence Measure in Bipartite Pure States

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 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 complex and quaternionic parts are related to Schmidt and concurrence terms respectively. Rather, the same relation is established for three-qubit state under octonionic stereographic projection which means that if the state is bi-separable then, quaternionic and octonionic terms vanish. Finally, we generalize recent consequences to 2?N and 4?N dimensional Hilbert spaces (N ≥ 2) and show that, after stereographic projection, the quaternionic and octonionic terms are entanglement sensitive. These trends are easily confirmed by direct computation for general multi-particle W- and GHZ-states.     

二非正交纯态相混合的concurrence

导出了两量子比特中二非正交纯态相混合的concurrence 的明显表达式, 并作了验证. 推算表明，正交情形下的concurrence公式可以直接推广到非正交的情形. 讨论了两纯态相混合的最大纠缠混合态和一些重要特殊情形. 关键词： 量子纠缠 纯态 混合态 concurrence     

Simple Scheme for Directly Measuring Concurrence of Two-Qubit Pure States in One Step

In the present work, a simple scheme for the direct measurement of the concurrence of two-qubit pure states is proposed. The scheme is based on trapped ions and only needs one step when the two identical pure states are given. The vibrational mode in our proposal is only virtually excited, which is important in view of decoherence.Furthermore, the scheme is feasible based on current technologies.     

Entanglement dynamics of two interacting qubits under the influence of local dissipation

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 eigenbasis 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 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 by the non-Markovianity of the evolution.     

Concurrence, tangle and fully entangled fraction

We show that although we cannot distil a singlet from many pairs of bound entangled states, the concurrence and the tangle of two entangled quantum states are always strictly larger than those of one of them, even both entangled quantum states are bound entangled. We present a relation between the concurrence and the fidelity of optimal teleportation. We also give new upper and lower bounds for concurrence and tangle.     

Monogamy and entanglement in tripartite quantum states

We present an interesting monogamy equation for (2⊗2⊗n)-dimensional pure states, by which a quantity is found to characterize the tripartite entanglement with the GHZ type and W type entanglements as a whole. In particular, we, for the first time, reveals that for any quantum state of a pair of qubits, the difference between the two remarkable entanglement measures, concurrence and negativity, characterizes the W type entanglement of tripartite pure states with the two-qubit state as reduced density.     

Concurrence of arbitrary dimensional bipartite quantum states

We derive an analytical lower bound for the concurrence of a bipartite quantum state in arbitrary dimension. A functional relation is established relating concurrence, the Peres-Horodecki criterion, and the realignment criterion. We demonstrate that our bound is exact for some mixed quantum states. The significance of our method is illustrated by giving a quantitative evaluation of entanglement for many bound entangled states, some of which fail to be identified by the usual concurrence estimation method.     

两量子比特非线性相互作用模型中的量子关联

运用量子态变换的方法论证了两个qubit纯态中量子关联与纠缠的等价性.并利用三种带有横场的非线性相互作用模型研究了两个qubit体系中的量子关联.发现合适的横场对于最大量子关联态的获得、平均量子关联的提高都有着积极的作用.两个qubit体系获得最大量子关联时,不同模型,不同的横场,对应的量子态却各不相同.     

Direct measurement of the concurrence for two-qubit electron spin entangled pure state based on charge detection

We propose a protocol for directly measuring the concurrence of a two-qubit electronic pure entangled state. To complete this task, we first design a parity-check measurement(PCM) which is constructed by two polarization beam splitters(PBSs) and a charge detector. By using the PCM for three rounds, we can achieve the concurrence by calculating the total probability of picking up the odd parity states from the initial states. Since the conduction electron may be a good candidate for the realization of quantum computation, this protocol may be useful in future solid quantum computation.     

Monogamous nature of symmetric N-qubit states of the W class: Concurrence and negativity tangle

Using Majorana representation of symmetric N-qubit pure states, we have examined the monogamous nature of the family of states with two-distinct spinors, the W class of states. We have evaluated the N-concurrence tangle and showed that all the states in this family have vanishing concurrence tangle. The negativity tangle for the W class of states is shown to be non-zero, illustrating the fact that the concurrence tangle is always lesser than or equal to the negativity tangle in a pure N-qubit state.     

Complementary relation between quantum entanglement and entropic uncertainty

Quantum entanglement is regarded as one of the core concepts,which is used to describe the nonclassical correlation between subsystems,and entropic uncertainty relation plays a vital role in quantum precision measurement.It is well known that entanglement of formation can be expressed by von Neumann entropy of subsystems for arbitrary pure states.An interesting question is naturally raised:is there any intrinsic correlation between the entropic uncertainty relation and quantum entanglement?Or if the relation can be applied to estimate the entanglement.In this work,we focus on exploring the complementary relation between quantum entanglement and the entropic uncertainty relation.The results show that there exists an inequality relation between both of them for an arbitrary two-qubit system,and specifically the larger uncertainty will induce the weaker entanglement of the probed system,and vice versa.Besides,we use randomly generated states as illustrations to verify our results.Therefore,we claim that our observations might offer and support the validity of using the entropy uncertainty relation to estimate quantum entanglement.     

Transformation relation between coherence and entanglement for two-qubit states

Entanglement and coherence are two important resources in quantum information theory. A question naturally arises: Is there some connection between them? We prove that the entanglement of formation and the first-order coherence of two-qubit states satisfy an inequality relation. Two-qubit pure state reaches the upper bound of this inequality. A large number of randomly generated states are used to intuitively verify the complementarity between the entanglement of formation and the first-order coherence. We give the maximum accessible coherence of two-qubit states. Our research results will provide a reliable theoretical basis for conversion of the two quantum resources.     

Standard Teleportation of One-Qubit State and Partial Teleportation of Two-Qubit State via X-States

In this article,we derive the explicit entanglement and fidelity expressions for a larger class of two-qubit states,namely,a seven-parameter family of so called X-states.The analytical expressions of the entanglement,the output entanglement and the average fidelity are obtained for this general model by using the concept of concurrence and average fidelity.We study the relations between quantum entanglement,the output entanglement,and the average fidelity for standard teleportation of one-qubit and partial teleportation of two-qubit state.We discover that the average fidelity and entanglement is not a simple dependence relation.The higher entangled system is helpful for teleportation.     

The influence of non-Kolmogorov turbulence on the entanglement of spatial two-qubit states in a slant channel

The effects of atmospheric turbulence on the entanglement of spatial two-qubit states that are prepared using the signal and idler photons produced by parametric down-conversion are studied. Utilizing the non-Kolmogorov model for atmospheric turbulence and Rytov approximation method, we quantify the effects of atmospheric turbulence on the entanglement of the two-qubit state in terms of Wootters's concurrence. Our results show that the effects of the zenith angle of communication channel and the outer scale of turbulence on the concurrence of a spatial two-qubit state can be ignored and the smaller inner scale of turbulence, the smaller refractive-index power α, the shorter wavelength of beams and the longer propagation distance will lead to the larger fluctuations of the concurrence of a spatial two-qubit state.     

Robustness of two-qubit and three-qubit states in correlated quantum channels

We investigate how the correlated actions of quantum channels affect the robustness of entangled states. We consider the Bell-like state and random two-qubit pure states in the correlated depolarizing, bit flip, bit-phase flip, and phase flip channels. It is found that the robustness of two-qubit pure states can be noticeably enhanced due to the correlations between consecutive actions of these noisy channels, and the Bell-like state is always the most robust one. We also consider the robustness of three-qubit pure states in correlated noisy channels. For the correlated bit flip and phase flip channels, the result shows that although the most robust and most fragile states are locally unitary equivalent, they exhibit different robustness in different correlated channels, and the effect of channel correlations on them is also significantly different. However, for the correlated depolarizing and bit-phase flip channels, the robustness of two special three-qubit pure states is exactly the same. Moreover, compared with the random three-qubit pure states, they are neither the most robust states nor the most fragile states.     

Concurrence vectors for entanglement of high-dimensional systems

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 systems of arbitrary dimension. 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 there exists an entanglement edge. This is illustrated in terms of the concurrence surfaces of several concrete examples.     

Entanglement transformation between two-qubit mixed states by LOCC

Based on a new set of entanglement monotones of two-qubit pure states, we give sufficient and necessary conditions that one two-qubit mixed state is transformed into another one by local operations and classical communication (LOCC). This result can be viewed as a generalization of Nielsen's theorem Nielsen (1999) [1]. However, we find that it is more difficult to manipulate the entanglement transformation between single copy of two-qubit mixed states than to do between single copy of two-qubit pure ones.     

Maximal and total skew information for a two-qubit system using nonlinear interaction models

Maximal and total skew information is studied. For symmetric pure states of two-qubit, they are closely related to the linear entropy, the concurrence, and the spin squeezing parameter. For a two-qubit system implemented in three nonlinear interaction models with an external field, we give the exact state vectors and the expectation value 〈S_z〉 at any time t. Based on 〈S_z〉², we give the maximal and the total skew information and a condition in which the maximal and the total skew information can reach 1 and 2, respectively.     

Concurrence vectors for entanglement of high dimensional systems

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 systems of arbitrary dimension. 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 there exists an entanglement edge. This is illustrated in terms of the concurrence surfaces of several concrete examples.