Entanglement and symmetry in permutation symmetric states

We investigate the relationship between multipartite entanglement and symmetry, focusing on permutation symmetric states. We use the Majorana representation, where these states correspond to points on a sphere. Symmetry of the representation under rotation is equivalent to symmetry of the states under products of local unitaries. The geometric measure of entanglement is thus phrased entirely as a geometric optimisation, and a condition for the equivalence of entanglement measures written in terms of point symmetries. Finally we see that different symmetries of the states correspond to different types of entanglement with respect to SLOCC interconvertibility.Comment: 4 pages, 2 figures. Preliminary versions of some of these results were presented in the QIT 16 workshop in Japan, D. Markham, Proceedings of QIT 16, Japan (2007). Updated to reflect changes for publication: expanded proofs and some new examples give

The maximally entangled symmetric state in terms of the geometric measure

The geometric measure of entanglement is investigated for permutation symmetric pure states of multipartite qubit systems, in particular the question of maximum entanglement. This is done with the help of the Majorana representation, which maps an n qubit symmetric state to n points on the unit sphere. It is shown how symmetries of the point distribution can be exploited to simplify the calculation of entanglement and also help find the maximally entangled symmetric state. Using a combination of analytical and numerical results, the most entangled symmetric states for up to 12 qubits are explored and discussed. The optimization problem on the sphere presented here is then compared with two classical optimization problems on the S^2 sphere, namely Toth's problem and Thomson's problem, and it is observed that, in general, they are different problems.Comment: 18 pages, 15 figures, small corrections and additions to contents and reference

Another look at nagyagite from the type locality, Sacarimb, Romania: Replacement, chemical variation and petrogenetic implications

Extensive compositional heterogeneity is displayed by Pb-Sb-Au tellurides from the type locality at S $\check{\rm{a}}$ c $\check{\rm{a}}$ rîmb. These phases are collectively considered as varieties of nagyágite in the absence of crystal chemical data confirming the presence of distinct, but topologically closely related compounds. Chemical heterogeneity is seen relative to ‘normal’ nagyágite, with close to the ideal composition Pb3[Pb1.8(Sb1.1,As0.1)1.2]Σ3S6 (AuTe2), which is the primary and common type in the deposit. A modified formula, (Pb3S3)[(Pb2−x )(Sb,As,Te b )1+x (S3−y Te y )]Σ6(Au1−z−w Te2+z S w ), accounts for the chemical variation observed. Values of x (0.2 to 1.15) express substitution of Pb by Sb+As for Me2 in sulfosalt modules in the case of Au-depleted and low-Au nagyágite, and by Sb+As+Te b in high-As and low-Pb varieties (b = x+1−(Sb+As) = 0.24 to 0.29). Excess Te compensates for Au deficiency in the telluride layer, with substitution by S also observed; empirical values of z and w are 0 to 0.45 and 0 to 0.32, respectively. Minor substitution of Te for S (y < 0.17) is noted in all varieties except low-Au. These varieties are formed during replacement of the ‘normal’ type as seen in overprinting relationships in those veins reactivated during rotation of the duplex fault-system responsible for vein formation. Replacement is by coupled dissolution-reprecipitation reactions, as indicated by pseudomorphism of one nagyágite type by another in all cases. Variable rates of both molar-excess and -deficit reaction are invoked to explain the observed chemical and textural modifications. Low-Pb nagyágite is also present in zoned platelets where it grows over resorbed cores of ideal composition. Such platelets are instead interpreted as products of self-patterning in a residual precipitate. A marked depletion in the Au content of some nagyágite lamellae is considered to be a diffusion driven Te for Au substitution in the presence of Te-bearing fluid. Replacement of ‘normal’ nagyágite by other varieties can be linked to high fluid acidity, whereas replacement by galena-altaite symplectites relates to changes in the fTe2/fS2 within a narrow domain defined by coexistence of these two minerals. Nagyágite is a mineral with modular crystal chemistry and is able to adjust to variable rates of fluid infiltration by subtle chemical substitutions. The behavior of nagyágite will map and assist coupling between dissolution and precipitation during such reactions.C. L. Ciobanu, N. J. Cook, A. Pring, G. Damian and N. Căprar