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arXiv:1008.3486 · Physical Review A

Maximal overlap with a fully separable state and translational invariance for multipartite entangled states

H. T. Cui, Di Yuan, Junlong Tian

Published Dec 27, 2010• 6 citations
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Abstract

The maximal overlap with the fully separable state for the multipartite entangled pure state with translational invariance is studied explicitly by some exact and numerical evaluations, focusing on the one-dimensional qubit system and some representative types of translational invariance. The results show that the translational invariance of the multipartite state could have an intrinsic effect on the determination of the maximal overlap and the nearest fully separable state for multipartite entangled states. Furthermore, a hierarchy of the basic entangled states with translational invariance is found, from which one could readily find the maximal overlap and a related fully separable state for the multipartite state composed of different translational invariance structures.

DOI
10.1103/physreva.82.062116

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The maximal overlap with the fully separable state for the multipartite entangled pure state with translational invariance is studied explicitly by some exact and numerical evaluations, focusing on the one-dimensional qubit system and some representative types of translational invariance. The results show that the translational invariance of the multipartite state could have an intrinsic effect on the determination of the maximal overlap and the nearest fully separable state for multipartite entangled states. Furthermore, a hierarchy of the basic entangled states with translational invariance is found, from which one could readily find the maximal overlap and a related fully separable state for the multipartite state composed of different translational invariance structures.
In quantum physics, multiparticle systems are described by quantum states acting on tensor products of Hilbert spaces. This product structure leads to the distinction between product states and entangled states; moreover, one can quantify entanglement by considering the distance of a quantum state to the set of product states. The underlying optimization problem occurs frequently in physics and beyond, for instance in the computation of the injective tensor norm in multilinear algebra. Here, we introduce a method to determine the maximal overlap of a pure multiparticle quantum state with product states based on considering several copies of the pure state. This leads to three types of hierarchical approximations to the problem, all of which we prove to converge to the actual value. Besides allowing for the computation of the geometric measure of entanglement, our results can be used to tackle optimizations over stochastic local transformations, to find entanglement witnesses for weakly entangled bipartite states, and to design strong separability tests for mixed multiparticle states. Finally, our approach sheds light on the complexity of separability tests.
Abstract We analyze tight informationally complete measurements for arbitrarily large multipartite systems and study their configurations of entanglement. We demonstrate that tight measurements cannot be exclusively composed neither of fully separable nor maximally entangled states. We establish an upper bound on the maximal number of fully separable states allowed by tight measurements and investigate the distinguished case in which every measurement operator carries the same amount of entanglement. Furthermore, we introduce the notion of nested tight measurements, i.e. multipartite tight informationally complete measurements such that every reduction to a certain number of parties induces a lower dimensional tight measurement, proving that they exist for any number of parties and internal levels.
We present a comprehensible introduction to tight informationally complete measurements for arbitrarily large multipartite systems and study their allowed entanglement configurations, focusing on those requiring simplest possible quantum resources. We demonstrate that tight measurements cannot be exclusively composed neither by fully separable nor maximally entangled states, according to a selected measure of entanglement. We establish an upper bound on the maximal number of fully separable states allowed by tight measurements and we study the distinguished case in which every measurement operator has the same amount of entanglement. Furthermore, we introduce the notion of nested tight measurements, i.e. multipartite tight informationally complete measurements such that every reduction to a certain number of parties induces a lower dimensional tight measurement, proving that they exist for any number of parties and internal levels.
The structural study of entanglement in multipartite systems is hindered by the lack of necessary and sufficient operational criteria able to discriminate among the various entanglement properties of a given mixed state. Here, we pursue a different route to the study of multipartite entanglement based on the closeness of a multipartite state to the set of separable ones. In particular, we analyze multipartite diagonal symmetric N qubit states and provide the analytical expression for their Best Separable Approximation (BSA [Phys. Rev. Lett. 80, 2261 (1998)]), that is, their unique convex decomposition into a separable part and an entangled one with maximal weight of the separable one.
We consider translationally invariant states of an infinite one-dimensional chain of qubits or spin-12\frac{1}{2}21​ particles. We maximize the entanglement shared by nearest neighbors via a variational approach based on finitely correlated states. We find an upper bound of nearest-neighbor concurrence equal to C=0.434095\mathcal{C}=0.434\phantom{\rule{0.2em}{0ex}}095C=0.434095 which is 0.090.09%0.09 away from the bound CW=0.434467{\mathcal{C}}_{\mathrm{W}}=0.434\phantom{\rule{0.2em}{0ex}}467CW​=0.434467 obtained by a completely different procedure. The obtained state-maximizing nearest-neighbor entanglement seems to approximate the maximally entangled mixed states. Further we investigate in detail several other properties of the so-obtained optimal state.
We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the Rényi–Ingarden–Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case q = 0, this quantity becomes a function of the rank of the tensor representing the state, while in the limit q → ∞, the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system, the entropy S1 coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three- and four-qubit systems. In the former case, the distribution of the three-tangle is studied and some of its moments are evaluated, while in the latter case, we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit.