A theorem of Glasner says that if X is an infinite subset of the torus T, then for any > 0, there exists an integer n such that the dilation nX = {nx : x ∈ T} is -dense (i.e, it intersects any interval of length 2 in T). Alon and Peres provided a general framework for this problem, and showed quantitatively that one can restrict the dilation to be of the form f(n)X where f ∈ Z[x] is not constan...

Grimmett, Kesten and Zhang (1993) showed that for d 3, simple random walk on the innnite cluster C 1 (Z d ; p) of supercritical percolation on Z d is a.s. transient. Their result is equivalent to the existence of a nonzero ow f on the innnite cluster such that the 2{energy P e f(e) 2 is nite. Here we sharpen this result, and show that if d 3 and p > p c (Z d), then C 1 (Z d ; p) supports a nonz...

Goldenberg and Vaidman Reply: Peres [1] claims that our protocol (GV) [2] does not present any novel feature, and it is very similar to the oldest protocol of Bennett and Brassard (BB84) [3]. We completely disagree with this claim and with other points raised in the Comment. The essential novelty of our protocol is that the carrier of information is in a quantum state belonging to a definite se...

We give an explicit tight lower bound for the entanglement of formation for arbitrary bipartite mixed states by using the convex hull construction of a certain function. This is achieved by revealing a novel connection among the entanglement of formation, the well-known Peres-Horodecki, and realignment criteria. The bound gives a quite simple and efficiently computable way to evaluate quantitat...