We report here on the results of numerical searches for PPT states with specified ranks for density matrices and their partial transpose. The study includes several bipartite quantum systems of low dimensions. For a series of ranks extremal PPT states are found. The results are listed in tables and charted in diagrams. Comparison of the results for systems of different dimensions reveal several regularities. We discuss lower and upper bounds on the ranks of extremal PPT states.
Characterizing 3-qubit UPB states: violations of LHV models, preparation via nonlocal unitaries and PPT entangled nonlocal orbits
对于3量子位UPB状态，即，从Bennett等人的不可扩展的乘积基（Phys. Rev. Lett. 82：5385，1999），我们基于在该状态中编码的特定类型的反射对称性提供了一组局部隐藏变量（LHV）模型的违反。给出了从纯态的反射可分混合态中制备态所需的显式非局域幺正操作，以及一个具有正部分转置（PPT）的非局域单参数态轨道，该轨道在一个周期内将一个态与其反射之间的纠缠交换两次。
On the structure of low-rank PPT entangled states violating Peres conjecture
Indian Institute of Science Education and Research Mohali（印度科学教育与研究学院莫哈利分校）；Harish-Chandra Research Institute（哈里什-钱德拉研究所）；S. N. Bose National Centre for Basic Sciences（S.N. 玻色基础科学国家中心）
Bipartite entangled quantum states with a positive partial transpose (PPT), i.e., PPT entangled states, are usually considered very weakly entangled. Since no pure entanglement can be distilled from them, they are also called bound entangled. In this paper we present two classes of ($2d\times 2d$)-dimensional PPT entangled states for any $d\ge 2$ which outperform all separable states in metrology significantly. We present strong evidence that our states provide the maximal metrological gain achievable by PPT states for a given system size. When the dimension $d$ goes to infinity, the metrologi…
We compare the classification as entangled or separable of Bell diagonal bipartite qudits with positive partial transposition (PPT) and their properties for different dimensions. For dimension $d \geq 3$, a form of entanglement exists that is hard to detect and called bound entanglement due to the fact that such entangled states cannot be used for entanglement distillation. Up to this date, no efficient solution is known to differentiate bound entangled from separable states. We address and compare this problem named separability problem for a family of bipartite Bell diagonal qudits with spec…
Entangled symmetric states of N qubits with all positive partial transpositions
从理论和实验的观点来看，对称态构成了一类重要的多体态。然而，这些态的纠缠特性，特别是正部分转置态（PPT）的纠缠特性还缺乏系统的研究。为了填补这一空白，我们最近肯定地回答了存在具有正部分转置的四量子比特纠缠对称态的公开问题，并彻底表征了这种态的纠缠特性[J. Tura et al.，A 85，060302（R）（2012）]利用本贡献，我们继续表征PPT纠缠对称态。一方面，我们详细介绍了我们以前工作的所有结果。另一方面，我们将它们推广到由任意数量的量子比特组成的系统。特别是，我们提供了这些国家制定的职级方面的可分性的标准。有趣的是，在大多数情况下，对称态要么是可分的，要么是典型可分的。然后，在这些系统中的边缘状态进行了研究，特别是显示，一般的PPT纠缠态的特征与四个和五个量子比特，这是足够的，只研究那些假设很少（分别为两个和三个）的具体配置的行列。最后，我们数值搜索极端的PPT纠缠态在这样的系统组成的多达23个量子比特。人们可以清楚地注意到这种极端状态的秩背后的规律性，特别是对于由奇数个量子比特组成的系统，我们发现存在极端状态的秩的单一配置。
In quantum information theory, it is a fundamental problem to construct multipartite unextendible product bases (UPBs). We show that there exist two families UPBs in Hilbert space $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^4$ by merging two different systems of an existing $7$-qubit UPB of size $11$. Moreover, a new family of $7$-qubit positive-partial-transpose (PPT) entangled states of rank $2^7-11$ is constructed. We analytically derive a geometric measure of entanglement of a special PPT entangled states. Also an upper bound a…
With a probability of success of $95 \%$ we solve the separability problem for Bell diagonal qutrit states with positive partial transposition (PPT). The separability problem, i.e. distinguishing separable and entangled states, generally lacks an efficient solution due to the existence of bound entangled states. In contrast to free entangled states that can be used for entanglement distillation via local operations and classical communication, these states cannot be detected by the Peres-Horodecki criterion or PPT criterion. We analyze a large family of bipartite qutrit states that can be sepa…
It is known that beyond $2 \otimes 2$ and $2 \otimes 3$ dimensional quantum systems, Peres-Hordecki criterion is no longer sufficient as an entanglement detection criterion as there are entangled states with both positive and negative partial transpose (PPT and NPT). Further, it is also true that all PPT entangled states are bound entangled states. However, in the class of NPT states, there can exist bound entangled states as well as free entangled states. All free/useful/distillable entanglement is a part of the class of NPT entangled states. In this article, we ask the question that given an…
We construct a parameterized family of $n\otimes n$ PPT (positive partial transpose) states of corank one for each $n\ge 3$. With a suitable choice of parameters, we show that they are $n\otimes n$ PPT entangled edge states of corank one for $3\le n\le 1000$. They violate the range criterion for separability in the most extreme way. Note that corank one is the smallest possible corank for such states. The corank of the partial transpose is given by $2n-3$, which is also the smallest possible corank for the partial transposes of PPT entangled edge states of corank one. They provide the first ex…
Statistical Characterization of Entanglement Degradation Under Markovian Noise in Composite Quantum Systems
Scuola Normale Superiore（高等师范学院）；Center for Life Nano- & Neuro-Science, Italian Institute of Technology (IIT)（意大利理工学院生命纳米与神经科学中心）；Department of Physics, Harvard University（哈佛大学物理系）；Department of Mathematics, University of Bologna（博洛尼亚大学数学系）；NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR（NEST，高等师范学院和CNR纳米科学研究所）
The Minimal Dimension of Entangled-Noise Advantage
National Laboratory of Solid State Microstructures and School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University（南京大学固体微结构物理国家重点实验室和物理学院，先进微结构协同创新中心）
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。
Equivalence classes and canonical forms for two-qutrit entangled states of rank four having positive partial transpose
设E'表示具有正部分转置（PPT）的秩为4的非归一化二元纠缠态集合。我们证明了E '中具有商拓扑的SLOCC等价态类集合通过交错群A_5的作用与欧氏空间R^4中开矩形盒R的商R/A_5同胚。我们构造了一个显式映射Ω：Ω-> E '，其中Ω是R^4中的开正正交，它的像满足E'中包含的所有SLOCC等价类E.虽然Ω和E的像的交集不一定是单例集，但它总是基数最多为60的有限集。滥用语言，我们说，任何国家在这个交集是一个典型形式的状态ρ在E。特别是，我们表明，所有棋盘PPT纠缠态可以参数化到SLOCC等价的只有两个真正的参数。我们还总结了已知的两个quutrit极端PPT状态和边缘状态的结果，并检查其他有趣的性质，他们可能有。因此，我们找到了第一个极端PPT状态的例子，其秩不同于其部分转置的秩。
