Economical adjunction of square roots to groups

How large must an overgroup of a given group be in order to contain a square root of any element of the initial group? We give an almost exact answer to this question (the obtained estimate is at most twice worse than the best possible) and state several related open questions.Comment: 5 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V2: minor correction

General entanglement

The paper contains a brief review of an approach to quantum entanglement based on analysis of dynamic symmetry of systems and quantum uncertainties, accompanying the measurement of mean value of certain basic observables. The latter are defined in terms of the orthogonal basis of Lie algebra, corresponding to the dynamic symmetry group. We discuss the relativity of entanglement with respect to the choice of basic observables and a way of stabilization of robust entanglement in physical systems.Comment: 7 pages, 1 figure,1 tabe, will be published in special issue of Journal of Physics (Conference Series) with Proceedings of CEWQO-200

The invariant-comb approach and its relation to the balancedness of multipartite entangled states

The invariant-comb approach is a method to construct entanglement measures for multipartite systems of qubits. The essential step is the construction of an antilinear operator that we call {\em comb} in reference to the {\em hairy-ball theorem}. An appealing feature of this approach is that for qubits (or spins 1/2) the combs are automatically invariant under SL(2,\CC), which implies that the obtained invariants are entanglement monotones by construction. By asking which property of a state determines whether or not it is detected by a polynomial SL(2,\CC) invariant we find that it is the presence of a {\em balanced part} that persists under local unitary transformations. We present a detailed analysis for the maximally entangled states detected by such polynomial invariants, which leads to the concept of {\em irreducibly balanced} states. The latter indicates a tight connection with SLOCC classifications of qubit entanglement. \\ Combs may also help to define measures for multipartite entanglement of higher-dimensional subsystems. However, for higher spins there are many independent combs such that it is non-trivial to find an invariant one. By restricting the allowed local operations to rotations of the coordinate system (i.e. again to the SL(2,\CC)) we manage to define a unique extension of the concurrence to general half-integer spin with an analytic convex-roof expression for mixed states.Comment: 17 pages, revtex4. Substantially extended manuscript (e.g. proofs have been added); title and abstract modified

Entanglement, local measurements, and symmetry

A definition of entanglement in terms of local measurements is discussed. Viz, the maximum entanglement corresponds to the states that cause the highest level of quantum fluctuations in all local measurements determined by the dynamic symmetry group of the system. A number of examples illustrating this definition is considered.Comment: 10 pages. to be published in Journal of Optics

Generalized Involution Models for Wreath Products

We prove that if a finite group $H$ has a generalized involution model, as defined by Bump and Ginzburg, then the wreath product $H \wr S_n$ also has a generalized involution model. This extends the work of Baddeley concerning involution models for wreath products. As an application, we construct a Gelfand model for wreath products of the form $A \wr S_n$ with $A$ abelian, and give an alternate proof of a recent result due to Adin, Postnikov, and Roichman describing a particularly elegant Gelfand model for the wreath product \ZZ_r \wr S_n. We conclude by discussing some notable properties of this representation and its decomposition into irreducible constituents, proving a conjecture of Adin, Roichman, and Postnikov's.Comment: 29 page

Pentagrams and paradoxes

Klyachko and coworkers consider an orthogonality graph in the form of a pentagram, and in this way derive a Kochen-Specker inequality for spin 1 systems. In some low-dimensional situations Hilbert spaces are naturally organised, by a magical choice of basis, into SO(N) orbits. Combining these ideas some very elegant results emerge. We give a careful discussion of the pentagram operator, and then show how the pentagram underlies a number of other quantum "paradoxes", such as that of Hardy.Comment: 14 pages, 4 figure

On entanglement in neutrino mixing and oscillations

We report on recent results about entanglement in the context of particle mixing and oscillations. We study in detail single-particle entanglement arising in two-flavor neutrino mixing. The analysis is performed first in the context of Quantum Mechanics, and then for the case of Quantum Field Theory.Comment: 14 pages, 2 figures. Presented at "Symmetries in Science Symposium - Bregenz 2009"

Stein's Method and Characters of Compact Lie Groups

Stein's method is used to study the trace of a random element from a compact Lie group or symmetric space. Central limit theorems are proved using very little information: character values on a single element and the decomposition of the square of the trace into irreducible components. This is illustrated for Lie groups of classical type and Dyson's circular ensembles. The approach in this paper will be useful for the study of higher dimensional characters, where normal approximations need not hold.Comment: 22 pages; same results, but more efficient exposition in Section 3.

Dynamics of a Bose-Einstein condensate in a symmetric triple-well trap

We present a complete analysis of the dynamics of a Bose-Einstein condensate trapped in a symmetric triple-well potential. Our classical analogue treatment, based on a time-dependent variational method using SU(3) coherent states, includes the parameter dependence analysis of the equilibrium points and their local stability, which is closely related to the condensate collective behaviour. We also consider the effects of off-site interactions, and how these "cross-collisions" may become relevant for a large number of trapped bosons. Besides, we have shown analytically, by means of a simple basis transformation in the single-particle space, that an integrable sub-regime, known as twin-condensate dynamics, corresponds in the classical phase space to invariant surfaces isomorphic to the unit sphere. However, the quantum dynamics preserves the twin-condensate defining characteristics only partially, thus breaking the invariance of the associated quantum subspace. Moreover, the periodic geometry of the trapping potential allowed us to investigate the dynamics of finite angular momentum collective excitations, which can be suppressed by the emergence of chaos. Finally, using the generalized purity associated to the su(3) algebra, we were able to quantify the dynamical classicality of a quantum evolved system, as compared to the corresponding classical trajectory.Comment: 22 pages, 10 figure