We propose an inductive procedure to classify N-partite entanglement under stochastic local operations and classical communication (SLOCC) provided such a classification is known for N-1 qubits. The method is based upon the analysis of the coefficient matrix of the state in an arbitrary product basis. We illustrate this approach in detail with the well-known bi- and tripartite systems, obtaining as a by-product a systematic criterion to establish the entanglement class of a given pure state without resourcing to any entanglement measure. The general case is proved by induction, allowing us to find an upper bound for the number of N-partite entanglement classes in terms of the number of entanglement classes for N-1 qubits.
Witness operators are a central tool to detect entanglement or to distinguish among the different entanglement classes of multiparticle systems, which can be defined using stochastic local operations and classical communication (SLOCC). We show a one-to-one correspondence between general SLOCC witnesses and a class of entanglement witnesses in an extended Hilbert space. This relation can be used to derive SLOCC witnesses from criteria for full separability of quantum states; moreover, given SLOCC witness can be viewed as entanglement witnesses. As applications of this relation we discuss the c…
Multipartite pure states are equivalent under Stochastic Local Operations and Classical Communication (SLOCC) whenever they can be mapped into one another by Invertible Local Operations. It is shown that this is equivalent to the inter-convertibility through finite sequences of Elementary Local Operations. A multipartite version of the Gauss-Jordan elimination strategy is then obtained, enabling the analytical SLOCC-classification of previously unknown examples. It is argued that the problem of SLOCC-classification is equivalent to the problem of classifying the Multipartite Fully Reduced Form…
Classification of arbitrary-dimensional multipartite pure states under stochastic local operations and classical communication using the rank of coefficient matrix
L. Lamata et al. have presented an entanglement classification of four qubits under stochastic local operation and classical communication (SLOCC) based on inductive approach [Phys. Rev. A \textbf{75}, 022318 (2007)]. While it works well for three qubits, we show that some of the families for four qubits have overlaps and hence this approach is not valid for entanglement classification of four or more qubits under SLOCC.
Central in entanglement theory is the characterization of local transformations among pure multipartite states. As a first step towards such a characterization, one needs to identify those states which can be transformed into each other via local operations with a non-vanishing probability. The classes obtained in this way are called SLOCC classes. They can be categorized into three disjoint types: the null-cone, the polystable states and strictly semistable states. Whereas the former two are well characterized, not much is known about strictly semistable states. We derive a criterion for the …
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides threequbit states into separable, biseparable, W, and Greenberger-Horne-Zeilinger (GHZ) classes. The W and GHZ are two subclasses of genuine tripartite entanglement.We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have…
The study of multipartite entanglement is not only interesting but also important due to its wide application in quantum information processing. However, the complicated structure of the Hilbert space for many parties makes multipartite entanglement extremely complicated. It is then worth studying the structure of the Hilbert space itself. In this work, we provide a way to study the structure of SLOCC-equivalence and to determine the number free parameters for SLOCC-equivalent classes. Additionally, two different entanglement witnesses are introduced. The method matches well the existing resul…
In arXiv:1208.0365 entanglement polytopes where introduced as a coarsening of the SLOCC classification of multipartite entanglement. The advantages of classifying entanglement by entanglement polytopes are a finite hierarchy for all dimensions and a number of parameters linear in system size. In arXiv:1208.0365 a method to compute entanglement polytopes using geometric invariant theory is presented. In this thesis we consider alternative methods to compute them. Some geometrical and algebraical tools are presented that can be used to compute inequalities giving an outer approximation of the en…
Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically moti…
Optimising multipartite entanglement manipulation in quantum networks
Indian Institute of Technology Bombay（印度理工学院孟买分校）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Delft University of Technology（代尔夫特理工大学）；QuTech（奎泰克）
Entanglement manipulation of multipartite pure states with finite rounds of classical communication
我们研究了C语言中基于多轮经典通信（$LOCC_{\mathbb{N}}$）的局部操作的纯状态变换。Spee，J.I. de Vicente，D.绍尔温湾Kraus，arXiv：1606.04418（2016）.在这里，我们首先提出的一些证明的细节，并通过$LOCC_{\mathbb{N}}$，这需要一个概率的步骤，推广建设的状态转换的例子。然而，我们也提出了明确的SLOCC类的例子，其中任何可分离的转换可以实现的协议，其中每一步都是确定性的（所有-det- $LOCC_{\mathbb{N}}$）。这样的变换可以被认为是二分变换的自然推广。此外，我们提供了纯状态转换的例子，这是可能的，通过可分离的转换，但不是通过$LOCC_{\mathbb{N}}$。我们还分析了一个有趣的真正多方效应，我们称之为锁定或解锁其他各方的权力。这意味着一方可以阻止或允许其他方实施LOCC转换。此外，我们研究了限制在$LOCC_{\mathbb{N}}$的最大纠缠集，并展示了如何容易计算的限制$LOCC_{\mathbb{N}}$的一些纠缠措施的界限可以得到。
We present a fine-structure entanglement classification under stochastic local operation and classical communication (SLOCC) for multiqubit pure states. To this end, we employ specific algebraic-geometry tools that are SLOCC invariants, secant varieties, to show that for $n$-qubit systems there are $\lceil\frac{2^{n}}{n+1}\rceil$ entanglement families. By using another invariant, $\ell$-multilinear ranks, each family can be further split into a finite number of subfamilies. Not only does this method facilitate the classification of multipartite entanglement, but it also turns out to be operati…
When some of the parties of a multipartite entangled pure state are lost, the question arises whether the residual mixed state is also entangled, in which case the initial entangled pure state is said to be robust against particle loss. In this paper, we investigate this entanglement robustness for $N$-qubit pure states. We identify exhaustively all entangled states that are fragile, i.e., not robust, with respect to the loss of any single qubit of the system. We also study the entanglement robustness properties of symmetric states and put these properties in the perspective of the classificat…
Decoherence Mitigation with Local NOT Gates in Multipartite Systems
Qdit Labs Pvt. Ltd.（Qdit Labs有限公司）；Indian Institute of Technology Hyderabad（印度理工学院海德拉巴分校）；Poornaprajna Institute of Scientific Research（普尔纳普拉贾科学研究学院）；University of Calgary（卡尔加里大学）
Entanglement polytopes have been recently proposed as the way of witnessing the SLOCC multipartite entanglement classes using single particle information. We present first asymptotic results concerning feasibility of this approach for large number of qubits. In particular we show that entanglement polytopes of $L$-qubit system accumulate in the distance $\frac{1}{2\sqrt{L}}$ from the point corresponding to the maximally mixed reduced one-qubit density matrices. This implies existence of a possibly large region where many entanglement polytopes overlap, i.e where the witnessing power of entangl…
Walgate and Scott have determined the maximum number of generic pure quantum states that can be unambiguously discriminated by an LOCC measurement [Journal of Physics A: Mathematical and Theoretical, 41:375305, 08 2008]. In this work, we determine this number in a more general setting in which the local parties have access to pre-shared entanglement in the form of a resource state. We find that, for an arbitrary pure resource state, this number is equal to the Krull dimension of (the closure of) the set of pure states obtainable from the resource state by SLOCC. Surprisingly, a generic resourc…
