The coloured quantum plane

We study the quantum plane associated to the coloured quantum group GL_{q}^{\lambda,\mu}(2) and solve the problem of constructing the corresponding differential geometric structure. This is achieved within the R-matrix framework generalising the Wess-Zumino formalism and leads to the concept of coloured quantum space. Both, the coloured Manin plane as well as the bicovariant differential calculus exhibit the colour exchange symmetry. The coloured h-plane corresponding to the coloured Jordanian quantum group GL_{h}^{\lambda,\mu}(2) is also obtained by contraction of the coloured q-plane.Comment: 10 pages, (AMS)LaTeX, to appear in J. Geom. Phy

Maximally discordant separable two qubit XX states

In a recent article S. Gharibian [\href{http://dx.doi.org/10.1103/PhysRevA.86.042106}{Phys. Rev. A {\bf 86}, 042106 (2012)}] has conjectured that no two qubit separable state of rank greater than two could be maximally non classical (defined to be those which have normalized geometric discord $1/4$) and asked for an analytic proof. In this work we prove analytically that among the subclass of $X$ states, there is a unique (up to local unitary equivalence) maximal separable state of rank two. Partial progress has been made towards the general problem and some necessary conditions have been derived.Comment: Preliminary draft, comment(s)/suggestion(s) welcome

Comment on "Sudden change in quantum discord accompanying the transition from bound to free entanglement"

In a recent article [http://dx.doi.org/10.1103/PhysRevA.87.022340, Phys. Rev. A 87, 022340 (2013)], Yan et al have studied geometric discord for a well known class of bound entangled states. Based on their calculation, they claim "It is found that there exists a nondynamic sudden change in quantum discord" for these states. In this Comment, we criticize their work by pointing out that what has been calculated is actually a bound, and not the exact value of discord. Since, generally, it is not possible to infer the exact value (or property) of a quantity just from its bound, we can not conclude about (exact) discord and its change. Thus, the above-mentioned conclusion can not be drawn from the calculations of the paper.Comment: 1 pag

Realisations of Quantum GL_p,q(2) and Jordanian GL_h,h'(2)

The quantum group GL_p,q(2) is known to be related to the Jordanian GL_h,h'(2) via a contraction procedure. It can also be realised using the generators of the Hopf algebra G_r,s. We contract the G_r,s quantum group to obtain its Jordanian analogue G_m,k, which provides a realisation of GL_h,h'(2) in a manner similar to the q-deformed case.Comment: 6 pages LaTex, Contribution to Proceedings of "8th International Colloquium on Quantum Groups and Integrable Systems", Prague, June 17 - 19, 199