Quantum key distribution for d-level systems with generalized Bell states

Using the generalized Bell states and controlled not gates, we introduce an enatanglement-based quantum key distribution (QKD) of d-level states (qudits). In case of eavesdropping, Eve's information gain is zero and a quantum error rate of (d-1)/d is introduced in Bob's received qudits, so that for large d, comparison of only a tiny fraction of received qudits with the sent ones can detect the presence of Eve.Comment: 8 pages, 3 figures, REVTEX, references added, extensive revision, to appear in Phys. Rev.

Random walk questions for linear quantum groups

We study the discrete quantum groups $\Gamma$ whose group algebra has an inner faithful representation of type $\pi:C^*(\Gamma)\to M_K(\mathbb C)$. Such a representation can be thought of as coming from an embedding $\Gamma\subset U_K$. Our main result, concerning a certain class of examples of such quantum groups, is an asymptotic convergence theorem for the random walk on $\Gamma$. The proof uses various algebraic and probabilistic techniques.Comment: 27 page

From SICs and MUBs to Eddington

This is a survey of some very old knowledge about Mutually Unbiased Bases (MUB) and Symmetric Informationally Complete POVMs (SIC). In prime dimensions the former are closely tied to an elliptic normal curve symmetric under the Heisenberg group, while the latter are believed to be orbits under the Heisenberg group in all dimensions. In dimensions 3 and 4 the SICs are understandable in terms of elliptic curves, but a general statement escapes us. The geometry of the SICs in 3 and 4 dimensions is discussed in some detail.Comment: 12 pages; from the Festschrift for Tony Sudber