Realignment criteria for recognizing multipartite entanglement of quantum states

By multiple realignments of density matrices, we present a new separability criterion for the multipartite quantum state, which includes the computable cross-norm or realignment criterion and the multipartite partial realignment criterion as special cases. An example is used to show that the new criterion can be more efficient than the corresponding multipartite realignment criteria given in Horodecki et al. (Open Syst Inf Dyn 13:103–111, 2006) and Shen et al. (Phys Rev A 92:042332, 2015).     

Characterization of Combinatorially Independent Permutation Separability Criteria

The so-called permutation separability criteria are simple operational conditions that are necessary for separability of mixed states of multipartite systems: (1) permute the indices of the density matrix and (2) check if the trace norm of at least one of the resulting operators is greater than one. If it is greater than one then the state is necessarily entangled. A shortcoming of the permutation separability criteria is that many permutations give rise to equivalent separability criteria. Therefore, we introduce a necessary condition for two permutations to yield independent criteria called combinatorial independence. This condition basically means that the map corresponding to one permutation cannot be obtained by concatenating the map corresponding to the second permutation with a norm-preserving map. We characterize completely combina-torially independent criteria, and determine simple permutations that represent all independent criteria. The representatives can be visualized by means of a simple graphical notation. They are composed of three basic operations: partial transpose, and two types of so-called reshufflings. In particular, for a four-partite system all criteria except one are composed of partial transpose and only one type of reshuffling; the exceptional one requires the second type of reshuffling. Furthermore, we show how to obtain efficiently a simple representative for every permutation. This method allows to check easily if two permutations are Combinatorially equivalent or not.     

Matched witness for multipartite entanglement

Entanglement criteria for multipartite entangled states are obtained by matching witnesses to multipartite entangled states. The necessary and sufficient criterion of separability for three qubit X states is given as an example to illustrate the procedure of finding a criterion. The result is utilized to obtain the noise tolerance of W state. The necessary and sufficient criteria of three partite separability and full separability for four qubit noisy cluster states, three partite separability for four qubit noisy GHZ states are obtained.     

基于量子态系数矩阵的纠缠判定准则

量子纠缠的判定问题（也称为可分性判定问题）是量子纠缠理论中的核心问题之一。越来越多的两体纠缠判定准则被提出,但其中大部分都难以理解和计算,或是难以应用到任意多体量子系统中。为此,对于一个任意的多体量子纯态,基于其系数矩阵提出了一个纠缠判定准则。通过考察一个量子态的系数矩阵的秩,就可以断定该状态是可分态还是纠缠态。通过具体的实例表明,所提出的方法可以找到一个多体量子态的具体可分形式,并且简单易懂、方便计算。     

Rotationally Invariant Multipartite States

We construct a class of multipartite states possessing rotational SO(3) symmetry – these are states of K spin-j _A particles and K spin-j _B particles. The construction of symmetric states follows our two recent papers devoted to unitary and orthogonal multipartite symmetry. We study basic properties of multipartite SO(3) symmetric states: separability criteria and multi-PPT conditions. Presented at the 38th Symposium on Mathematical Physics “Quantum Entanglement & Geometry”, Toruń, June 4–7, 2006.     

Realignment entanglement criterion of phase damped Gaussian states

We derive the realignment entanglement criterion for non-Gaussian states prepared by two mode symmetric Gaussian states undergoing phase damping. The entanglement detecting ability is compared with that of second moment criterion and Fock space criterion of positive partial transpose. New non-Gaussian entangled states are detected.     

Exploring degrees of entanglement

In spite of a long history, the quantification of entanglement still calls for exploration. What matters about entanglement depends on the situation, and so presumably do the numbers suitable for its quantification. Regardless of situational complications, a necessary first step is to make available for calculation some quantitative measure of entanglement. Here we define a geometric degree of entanglement, distinct from earlier definitions, but in the case of bipartite pure states related to that proposed by Shimony (Ann N Y Acad Sci 755:675–679, 1995). The definition offered here applies also to multipartite mixed states, and a variational method simplifies the calculation. We analyze especially states that are invariant under permutation of particles, states that we call bosonic. Of interest to quantum sensing, for bosonic states, we show that no partial trace can increase a degree of entanglement. For some sample cases we quantify the degree of entanglement surviving a partial trace. As a function of the degree of entanglement of a bosonic 3-qubit pure state, we show the range of degree of entanglement for the 2-qubit reduced density matrix obtained from it by a partial trace. Then we calculate an upper bound on the degree of entanglement of the mixed state obtained as a partial trace over one qubit of a 4-qubit bosonic state. As a reminder of the situational dependence of the advantage of entanglement, we review the way in which entanglement combines with scattering theory in the example of light-based radar.     

Further Results on the Cross Norm Criterion for Separability

In the present paper we develop and investigate a novel approach that aims to characterize quantum entanglement by using cross norms. In the first part of the paper we further develop the mathematical theory by determining the value of the greatest cross norm for Werner states, for isotropic states and for Bell diagonal states. In the second part we show that our techniques induce a novel powerful analytical and computable separability criterion for bipartite systems. This new criterion complements the well-known Peres positive partial transpose criterion in several aspects. It is a necessary but in general not a sufficient criterion for separability. We demonstrate the power of the new criterion by evaluating the criterion for a number of important examples. We also demonstrate that the new criterion is able to detect bound entangled states.     

An analytic approach to the problem of separability of quantum states based upon the theory of cones

Exploiting the cone structure of the set of unnormalized mixed quantum states, we offer an approach to detect separability independently of the dimensions of the subsystems. We show that any mixed quantum state can be decomposed as ρ = (1−λ)C _ρ  + λE _ρ , where C _ρ is a separable matrix whose rank equals that of ρ and the rank of E _ρ is strictly lower than that of ρ. With the simple choice C_r=M₁?M₂{C_{\rho}=M_{1}\otimes M_{2}} we have a necessary condition of separability in terms of λ, which is also sufficient if the rank of E _ρ equals 1. We give a first extension of this result to detect genuine entanglement in multipartite states and show a natural connection between the multipartite separability problem and the classification of pure states under stochastic local operations and classical communication. We argue that this approach is not exhausted with the first simple choices included herein.     

Detecting quantum entanglement

We review the criteria for separability and quantum entanglement, both in a bipartite as well as a multipartite setting. We discuss Bell inequalities, entanglement witnesses, entropic inequalities, bound entanglement and several features of multipartite entanglement. We indicate how these criteria bear on the experimental detection of quantum entanglement.     

基于萤火虫算法的高光谱遥感波段选择方法

高光谱图像在遥感领域中的应用越来越广泛,但由于自身的高数据维、波段间的高冗余度等特性给图像处理带来了一定困难,针对这个问题,提出一种基于类间可分性准则的改进萤火虫仿生算法,进行高光谱遥感波段选择。在分析萤火虫算法机理的基础上,阐述了利用该算法进行高光谱波段选择的思路,并构造波段相似性矩阵,选择欧氏距离、JM距离、光谱信息散度和离散度作为可分性准则来设置目标函数,根据目标函数值的优劣选择优势波段。最后,使用HYDICE Washington DC Mall和 HyMap Purdue Campus两个高光谱遥感影像数据进行实验验证,并利用支持向量机分类器对最佳波段组合进行精度评价,证明该算法的可行性和有效性。
     

A new concept for separability problems in blind source separation

The goal of blind source separation (BSS) lies in recovering the original independent sources of a mixed random vector without knowing the mixing structure. A key ingredient for performing BSS successfully is to know the indeterminacies of the problem-that is, to know how the separating model relates to the original mixing model (separability). For linear BSS, Comon (1994) showed using the Darmois-Skitovitch theorem that the linear mixing matrix can be found except for permutation and scaling. In this work, a much simpler, direct proof for linear separability is given. The idea is based on the fact that a random vector is independent if and only if the Hessian of its logarithmic density (resp. characteristic function) is diagonal everywhere. This property is then exploited to propose a new algorithm for performing BSS. Furthermore, first ideas of how to generalize separability results based on Hessian diagonalization to more complicated nonlinear models are studied in the setting of postnonlinear BSS.     

Schmidt number of bipartite and multipartite states under local projections

The Schmidt number is a fundamental parameter characterizing the properties of quantum states, and local projections are fundamental operations in quantum physics. We investigate the relation between the Schmidt numbers of bipartite states and their projected states. We show that there exist bipartite positive partial transpose entangled states of any given Schmidt number. We further construct the notion of joint Schmidt number for multipartite states and explore its relation with the Schmidt number of bipartite reduced density operators.     

Multipartite quantum and classical correlations in symmetric <Emphasis Type="Italic">n</Emphasis>-qubit mixed states

We discuss how to calculate genuine multipartite quantum and classical correlations in symmetric, spatially invariant, mixed n-qubit density matrices. We show that the existence of symmetries greatly reduces the amount of free parameters to be optimized in order to find the optimal measurement that minimizes the conditional entropy in the discord calculation. We apply this approach to the states exhibited dynamically during a thermodynamic protocol to extract maximum work. We also apply the symmetry criterion to a wide class of physically relevant cases of spatially homogeneous noise over multipartite entangled states. Exploiting symmetries we are able to calculate the non-local and genuine quantum features of these states and note some interesting properties.     

一类基于多目标规划的线性鉴别准则

两类Fisher鉴别准则、大间距线性投影准则以及最大散度差鉴别准则都是直接用于模式分类的两类线性鉴别准则,它们的共同点是将“投影后数据的可分性达到最大的方向”作为最优投影方向。区别在于它们对数据可分性的定义有所不同。过去的研究成果表明,大间距线性投影分类器与支持向量机之间、大间距线性投影准则与最大散度差鉴别准则之间以及最大散度差鉴别准则与两类Fisher鉴别准则之间,均存在着这样或那样的联系。论文试图在以往研究成果的基础上进一步理清这些两类线性鉴别准则之间的内在关系,并建立一个统一的理论框架从而将基于投影后数据可分性的这些两类线性鉴别准则都纳入其中。     

The Jamiołkowski Isomorphism and a Simplified Proof for the Correspondence Between Vectors Having Schmidt Number <Emphasis Type="Italic">k</Emphasis> and <Emphasis Type="Italic">k</Emphasis>-Positive Maps

Positive maps which are not completely positive are used in quantum information theory as witnesses for convex sets of states, in particular as entanglement witnesses and, more generally, as witnesses for states having Schmidt number not greater than k. Such maps and witnesses are related to k-positive maps, and their properties may be investigated by making use of the Jamiołkowski isomorphism. In this article we review the properties of this isomorphism, noting that there are actually two related mappings bearing that name. As a new result, we give a simplified proof for the correspondence between vectors having Schmidt number k and k-positive maps and thus for the Jamiołkowski criterion for complete positivity. Another consequence is a special case of a result by Choi, namely that k-positivity implies complete positivity, if k is the dimension of the smaller one of the Hilbert spaces on which the operators act.     

Generalized Choi states and 2-distillability of quantum states

We investigate the distillability of bipartite quantum states in terms of positive and completely positive maps. We construct the so-called generalized Choi states and show that it is distillable when it has negative partial transpose. We convert the distillability problem of 2-copy \(n\times n\) Werner states into the determination of the positivity of an Hermitian matrix. We obtain several sufficient conditions by which the positivity holds. Further, we investigate the case \(n=3\) by the classification of \(2\times 3\times 3\) pure states.     

A discrete differential evolution algorithm for the permutation flowshop scheduling problem

Very recently, Pan et al. [Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation, GECCO07, pp. 126–33] presented a new and novel discrete differential evolution algorithm for the permutation flowshop scheduling problem with the makespan criterion. On the other hand, the iterated greedy algorithm is proposed by [Ruiz, R., & Stützle, T. (2007). A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem. European Journal of Operational Research, 177(3), 2033–49] for the permutation flowshop scheduling problem with the makespan criterion. However, both algorithms are not applied to the permutation flowshop scheduling problem with the total flowtime criterion. Based on their excellent performance with the makespan criterion, we extend both algorithms in this paper to the total flowtime objective. Furthermore, we propose a new and novel referenced local search procedure hybridized with both algorithms to further improve the solution quality. The referenced local search exploits the space based on reference positions taken from a reference solution in the hope of finding better positions for jobs when performing insertion operation. Computational results show that both algorithms with the referenced local search are either better or highly competitive to all the existing approaches in the literature for both objectives of makespan and total flowtime. Especially for the total flowtime criterion, their performance is superior to the particle swarm optimization algorithms proposed by [Tasgetiren, M. F., Liang, Y. -C., Sevkli, M., Gencyilmaz, G. (2007). Particle swarm optimization algorithm for makespan and total flowtime minimization in permutation flowshop sequencing problem. European Journal of Operational Research, 177(3), 1930–47] and [Jarboui, B., Ibrahim, S., Siarry, P., Rebai, A. (2007). A combinatorial particle swarm optimisation for solving permutation flowshop problems. Computers & Industrial Engineering, doi:10.1016/j.cie.2007.09.006]. Ultimately, for Taillard’s benchmark suite, four best known solutions for the makespan criterion as well as 40 out of the 90 best known solutions for the total flowtime criterion are further improved by either one of the algorithms presented in this paper.     

New delay-range-dependent exponential estimates for singular systems with time-varying delay

This paper considers the problem of exponential stability for continuous-time singular systems with interval time-varying delay. By defining a novel Lyapunov-Krasovskii function and giving a tighter upper bound of its derivative, a new delay-range-dependent exponential admissibility criterion, which not only guarantees the regularity, absence of impulses and exponential stability of the system but also gives the estimates of decay rate and decay coefficient, is established in terms of linear matrix inequality (LMI). The resulting criterion has advantages over the result previously reported by Haidar et al. [17] in that it involves fewer matrix variables but has less conservatism, which is established theoretically. Examples are provided to demonstrate the advantage of the proposed criterion.     

Convolutional neural network feature maps selection based on LDA

Convolutional neural network (CNN), as widely applied to vision and speech, has developed lager and lager network size in last few years. In this paper, we propose a CNN feature maps selection method which can simplify CNN structure on the premise of stabilize the classifier performance. Our approach aims to cut the feature map number of the last subsampling layer and achieves shortest runtime on the basis of Linear Discriminant Analysis (LDA). We rebuild feature maps selection formula based on the between-class scatter matrix and within-class scatter matrix, because LDA can lead to information loss in the dimension-reduction process. Our experiments measure on two standard datasets and a dataset made by ourselves. According to the separability value of each feature map, we suggest the least number of feature maps which can keep the classifier performance. Furthermore, we prove that separability value is an effective indicator for reference to select feature maps.