By definition a separable state has the form \sum A_i \otimes B_i, where 0 \leq A_i, B_i for each i. In this paper we consider the class of states which admit such a decomposition with B_1, ..., B_p having independent images. We give a simple intrinsic characterization of this class of states, and starting with a density matrix in this class, describe a procedure to find such a decomposition with B_1, ..., B_p having independent images, and A_1, ..., A_p being distinct with unit trace. Such a decomposition is unique, and we relate this to the facial structure of the set of separable states.
A special subclass of such separable states are those for which the rank of the matrix matches one marginal rank. Such states have arisen in previous studies of separability (e.g., they are known to be a class for which the PPT condition is equivalent to separability).
The states investigated also include a class that corresponds (under the Choi-Jamiołkowski isomorphism) to the quantum channels called quantum-classical and classical-quantum by Holevo.
We construct one parameter families of three qubit separable states with length ten, which is strictly greater than the whole dimension eight. These states are located on the boundary of the convex set of all separable states, but they are in the interior of the convex set of all states with positive partial transposes. They are also decomposed into the convex sum of ten pure product states in a unique way.
We present a practical scheme for the decomposition of a bipartite mixed state into a sum of direct products of local density matrices, using the technique developed in Li and Qiao (Sci. Rep. 8: 1442, 2018). In the scheme, the correlation matrix which characterizes the bipartite entanglement is first decomposed into two matrices composed of the Bloch vectors of local states. Then we show that the symmetries of Bloch vectors are consistent with that of the correlation matrix, and the magnitudes of the local Bloch vectors are lower bounded by the correlation matrix. Concrete examples for the sep…
We prove that every symmetric separable state admits a convex decomposition into symmetric pure product states. While the result is not new in itself, here we focus on convex geometry. We discuss the decomposition in the context of numerical ranges and ground state problems of infinite bosonic systems.
In this paper, we consider a subclass of quantum states in the multipartite system, namely, the supersymmetric states. We investigate the problem whether they admit the symmetrically separable decomposition, i.e., each term in this decomposition is a supersymmetric pure product state $|x,x\rangle\langle x,x|$, which are called S-separable. We conjecture that any supersymmetric states are S-separable and we prove that this conjecture holds when the rank is less than or equal to 3 or $N$. Moreover, we propose another weaker conjecture that any separable supersymmetric states are S-separable. It …
Separable approximations of density matrices of composite quantum systems
本文研究了任意维MxN的二分量子系统A和B的态ρ的最佳可分近似（分解）。Lewenstein和A. Sanpera，Phys. Rev. Lett. 80，2261（1998）]。这样的近似允许以最佳的方式将任何密度算符表示为可分离态和某种形式的纠缠态的总和。对于两个量子比特系统（M=N=2），最佳可分近似具有可分态和投影到纯纠缠态的混合形式。给出了最佳可分近似下纯态不是最大纠缠态的必要条件。我们证明了在任意维的最佳可分近似下，纠缠态的权重提供了一个很好的纠缠度量。我们证明了在一般情况下，任意M和N的最佳可分近似对应于一个混合的可分和纠缠态都是唯一的。我们还开发了一个理论的最佳可分离近似的状态与正部分转置（PPT状态）。这种近似允许将具有正部分转置的任何密度算子分解为可分离状态和纠缠PPT状态的总和。我们讨论的过程中建设这样的分解。
The decompositions of separable Werner state, and also isotropic state, are well-known tough issues in quantum information theory, in this work we investigate them in the Bloch vector representation, exploring the symmetric informationally complete positive operator-valued measure (SIC-POVM) in the Hilbert space. We successfully get the decomposition for arbitrary $N\times N$ Werner state in terms of regular simplexes. Meanwhile, the decomposition of isotropic state is found to be related to the decomposition of Werner state via partial transposition. It is interesting to note that in the larg…
The separability detecting problem of mixed states is one of the fundamental problems in quantum information theory. In the last 20 years, almost all methods are based on the sufficient or necessary conditions for entanglement. However, in this paper, we only need one algorithm to solve the problem. We propose a tensor optimization method to check whether an $m$-partite quantum mixed state is separable or not and give a decomposition for it if it is. We first convert the separability discrimination problem of mixed states to the positive Hermitian decomposition problem of Hermitian tensors. Th…
Checking the optimality of entanglement witnesses: an application to structural physical approximations
在2008年，最佳纠缠见证者的结构物理近似是可分离态（通常是未归一化的）的猜想被提出。为了证明这一点，在[K。C.哈和S。H. Kye，Separable states with unique decompositions，arXiv：1210.1088v3]，Ha和Kye提出了一种可分解的纠缠见证，其SPA是纠缠的，并论证了它是最优的.本文是在对后一工作[R. Augusiak等人，评论“具有独特分解的可分离态”，arXiv：1304.2040v1]，我们表明，无论是分析和数值，这种纠缠见证不是最佳的，因此它不是一个反例的猜想。为此，我们利用一种方法来检查纠缠证人的最优性已经在[M。Lewenstein等人，A 62，052310（2000）]，然而，迄今为止在文献中几乎没有利用。
Smallest disentangling state spaces for general entangled bipartite quantum states
如果我们放宽局域算符是量子态的限制，纠缠量子态就可以得到一个可分离的分解。出于经典的模拟和本地隐藏变量模型的建设，我们构建“最小”的本地运营商集，实现这一点。换句话说，给定一个任意的二分量子态，我们构造了允许可分分解的局部算子的凸集，但是在继续这样做的同时不能使其变小。然后，我们考虑两个进一步的变种的问题，当地的状态空间需要包含当地的量子态，并获得解决方案的各种情况下，包括一个区域的纯态周围的最大纠缠态。这些方法涉及计算某些形式的交叉范数。两个变种的问题有很强的关系，定理系综分解的正算子，我们的结果，从而给这些定理一个额外的解释。这些结果概括了我们以前关于这个主题的工作[New J. Phys. 17，093047（2015）]中获得的结果。
We prove experimentally the predicted existence of a three-qubit quantum state with genuine multipartite entanglement which can be certified solely from its separable two-qubit reduced density matrices. The qubits are encoded into different degrees of freedom of a pair of correlated photons and the state is prepared by letting the photons to propagate through a linear optical circuit. The presence of genuine multipartite entanglement is verified by finding numerically a fully decomposable entanglement witness acting nontrivially only on the reductions of the global state. Our result confirms v…
