We define a polynomial measure of multiparticle entanglement which is scalable, i.e., applies to any number spin-1/2 particles. By evaluating it for three particle states, eigenstates the one dimensional Heisenberg antiferromagnet and on quantum error correcting code subspaces, we illustrate extent quantifies global entanglement. also apply track evolution during computation.

1. Introduction.The purpose of this note is to show that if T 1 a closed, connected one-dimensional subgroup SU(3) has no nonzero fixed points, then SUtyjT admits an SC/^-invariant Riemannian structure strictly positive curvature.This result implies the statement title since there are infinite number distinct homotopy types among spaces with as above (see Lemma 3.3).To prove we introduce what call condition II, which generalizes III [2] and [3].Although SUty/T only new satisfying II Theorem...

Abstract In this paper we announce a qualitative description of an important class closed n-dimensional submanifolds the m-dimensional sphere, namely, those which locally minimize n-area in same way that geodesics arc length and are themselves n-spheres constant radius r; r may appear called admissible. It is known for n = 2 each admissible determines unique element above class. The main result here ≥ 3 [unk]8 there exists continuum distinct such submanifolds.

This is the second in a series of papers on analytic continuation holomorphic discrete series. In this paper necessary and sufficient conditions for unitarizability are given case line bundles. The foundations vector valued begun.

We find an operational interpretation for the 4-tangle as a type of residual entanglement, somewhat similar to 3-tangle. Using this remarkable interpretation, we are able class maximally entangled four-qubits states which is characterized by four real parameters. The in sense that their average bipartite entanglement with respect all possible cuts maximal. show while maximize tangle, there only few Tsillas or Renyi α-entropy entanglement. Quite remarkably, up local unitaries, exists two...

Introduction.Let G be a connected unimodular Lie group.Let T discrete subgroup of so that T\G is compact.We fix Haar measure, dg, on G. Then dg induces G-invariant measure T\G.We can then form unitary representation (rrr, L 2 (F\G)) where (rrr(g)f)(x)= : f(xg) for feL (T\G), xeT\G, geG.If <f>eC7(G) (the space all C 00 compactly supported complex valued functions G) we formIt standard fact (see §2) 7r r (4>) trace class.In particular, 7Tr(4>) completely continuous 4>eC7(G).This implies (r\G)...

The main purpose of this article is to provide an alternate proof a result Perelman on gradient shrinking solitons.Moreover in dimension three our generalizes Perelman's by removing the κ-non-collapsing assumption and allowing general curvature growth.The method also allows us prove classification solitons with vanishing Weyl tensor high dimensions, which includes rotationally symmetric ones.

Abstract Let Δ n −1 denote the ( − 1)‐dimensional simplex. Y be a random k ‐dimensional subcomplex of obtained by starting with full skeleton and then adding each ‐simplex independently probability p . H ; R ) reduced homology group coefficients in finite abelian It is shown that for any fixed ≥ 1 function ω( tends to infinity © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

We consider global hypoellipticity of constant coefficient differential operators on the 2-torus, and prove that it is equivalent to an algebraic growth condition symbol.This applied give necessary sufficient conditions a vector field be globally hypoelliptic.Similar results are true compact homogeneous spaces.

We provide a systematic classification of multiparticle entanglement in terms equivalence classes states under stochastic local operations and classical communication (SLOCC). show that such SLOCC equivalency class is characterized by ratios homogenous polynomials are invariant action the special linear group. then construct complete set all SL-invariant (SLIPs). Our construction based on Schur-Weyl duality applies to any number qudits (finite) dimensions. In addition, we an elegant formula...

Local operations assisted by classical communication (LOCC) constitute the free in entanglement theory. Hence, determination of LOCC transformations is crucial for understanding entanglement. We characterize here almost all among pure multipartite multilevel states. Combined with analogous results qubit states shown Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a characterization local show that nontrivial generic, fully entangled, are never possible. Thus, isolated. They...

Article On quaternionic discrete series representations, and their continuations. was published on January 1, 1996 in the journal Journal für die reine und angewandte Mathematik (volume 1996, issue 481).