We show that higher order inter-group covariances involving even number of qubits are necessarily positive semidefinite for N qubit separable states, which are completely symmetric under permutations of the qubits. This identification leads to a family of sufficient conditions of inseparability based on the negativity of 2kth order inter-group covariance matrices of symmetric N-qubit systems. These conditions have a simple structure and detect entanglement in all even partitions of the symmetric multiqubit system. The observables involved are feasible experimental quantities and do not demand full state determination through quantum state tomography.
Algorithm for characterizing stochastic local operations and classical communication classes of multiparticle entanglement
众所周知，根据随机局域操作对纯多粒子纠缠态的分类导致了根据凸集对混合态的自然分类。我们提出了一个简单的算法过程来证明一个量子态位于一个给定的凸集。我们的算法推广了最近证明量子态可分性的算法[J.Barreiro et al.，6，943（2010）]。我们给出了几个例子，表明我们的方法的广泛适用性。我们还提出了一个程序，以确定附近的一个给定的量子态仍然属于所考虑的凸集。
A well-known strategy to characterize multiparticle entanglement utilizes the notion of stochastic local operations and classical communication (SLOCC), but characterizing the resulting entanglement classes is difficult. Given a multiparticle quantum state, we first show that Gilbert's algorithm can be adapted to prove separability or membership in a certain entanglement class. We then present two algorithms for convex optimization over SLOCC classes. The first algorithm uses a simple gradient approach, while the other one employs the accelerated projected-gradient method. For demonstration, t…
We introduce the concept of entanglement width as measure of the spatial distribution of entanglement in multiparticle systems. We develop criteria to detect the width of entanglement using global observables such as energy and magnetic susceptibility. Therefore, the introduced entanglement criteria can be applied to systems where addressing of single particles is not possible. We apply the criteria to different examples such as the J1-J2- Heisenberg model and point out the difference between entanglement depth and entanglement width.
Entanglement of high-dimensional quantum systems has become increasingly important for quantum communication and experimental tests of nonlocality. However, many effects of high-dimensional entanglement can be simulated by using multiple copies of low-dimensional systems. We present a general theory to characterize those high-dimensional quantum states for which the correlations cannot simply be simulated by low-dimensional systems. Our approach leads to general criteria for detecting multilevel entanglement in multiparticle quantum states, which can be used to verify these phenomena experimen…
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…
Characterizing Long-Range Entanglement in a Mixed State Through an Emergent Order on the Entangling Surface
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；Department of Physics, University of California at San Diego（加州大学圣地亚哥分校物理系）；Department of Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校物理系）
We consider the problems of maximizing the entanglement negativity of X-form qubit-qutrit density matrices with (i) a fixed spectrum and (ii) a fixed purity. In the first case, the problem is solved in full generality whereas, in the latter, partial solutions are obtained by imposing extra spectral constraints such as rank-deficiency and degeneracy, which enable a semidefinite programming treatment for the optimization problem at hand. Despite the technically-motivated assumptions, we provide strong numerical evidence that three-fold degenerate X states of purity $P$ reach the highest entangle…
While entanglement is believed to be an important ingredient in understanding quantum many-body physics, the complexity of its characterization scales very unfavorably with the size of the system. Finding super-sets of the set of separable states that admit a simpler description has proven to be a fruitful approach in the bipartite setting. In this paper we discuss a systematic way of characterizing multiparticle entanglement via various relaxations. We furthermore describe an operational witness construction arising from such relaxations that is capable of detecting every entangled state. Fin…
A basic diagnostic of entanglement in mixed quantum states is known as the positive partial transpose (PT) criterion. Such criterion is based on the observation that the spectrum of the partially transposed density matrix of an entangled state contains negative eigenvalues, in turn, used to define an entanglement measure called the logarithmic negativity. Despite the great success of logarithmic negativity in characterizing bosonic many-body systems, generalizing the operation of PT to fermionic systems remained a technical challenge until recently when a more natural definition of PT for ferm…
A basic diagnostic of entanglement in mixed quantum states is known as the partial transpose and the corresponding entanglement measure is called the logarithmic negativity. Despite the great success of logarithmic negativity in characterizing bosonic many-body systems, generalizing the partial transpose to fermionic systems remained a technical challenge until recently when a new definition that accounts for the Fermi statistics was put forward. In this paper, we propose a way to generalize the partial transpose to anyons with (non-Abelian) fractional statistics based on the apparent similari…
A scheme for characterizing entanglement using the statistical measure of correlation given by the Pearson correlation coefficient (PCC) was recently suggested that has remained unexplored beyond the qubit case. Towards the application of this scheme for the high dimensional states, a key step has been taken in a very recent work by experimentally determining PCC and analytically relating it to Negativity for quantifying entanglement of the empirically produced bipartite pure state of spatially correlated photonic qutrits. Motivated by this work, we present here a comprehensive study of the ef…
Twisted Rényi Negativity as a Reliable Proxy for Mixed-State Entanglement in Fermionic Systems
Tsung-Dao Lee Institute, Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University（上海交大李政道研究所、人工结构量子控制教育部重点实验室、物理与天文学院）；Hefei National Laboratory（合肥国家实验室）
Given a non-maximally entangled state, an operationally significant question is to quantitatively assess as to what extent the state is away from the maximally entangled state, which is of importance in evaluating the efficacy of the state for its various uses as a resource. It is this question which is examined in this paper for two-qubit pure entangled states in terms of different entanglement measures like Negativity (N), Logarithmic Negativity (LN), and Entanglement of Formation (EOF). Although these entanglement measures are defined differently, to what extent they differ in quantitativel…
