Quantum cloning machines for equatorial qubits are studied. For 1 to 2 phase-covariant quantum cloning machine, using Hilbert-Schmidt norm and Bures fidelity, we show that our transformation can achieve the bound of the fidelity. Networks consisting of quantum gates are presented to realize the quantum cloning transformations. The copied equatorial qubits are shown to be separable by using Pere...

We present a survey of results in the field of graph algorithms. We begin with an All Pairs Shortest Path algorithm with some constraints that runs in O(n) by Peres. We continue with a Replacement Paths algorithm by Williams and conclude with a smoothed analysis of a Single Source Shortest Paths algorithm by Banderier. Each of these problems has connections to randomization in graph algorithms.

We review entanglement in the context of two qubits. Equivalence under appropriate local operations leads to a representation of the notions of entangled and separable states and entanglement witnesses geometrically in 3 dimensions. The corresponding four dimensional description captures, in addition, the entanglement measure and en-tanglement distillation geometrically. We give a neat formula ...

Let Z and R be the sets of all integers and real numbers, respectively. For a,b ∈ Z, define Z(a)= {a,a+1, . . .} and Z(a,b)= {a,a+1, . . . ,b} when a≤ b. Let A be an n×m matrix. Aτ denotes the transpose ofA.When n=m, σ(A) and det(A) denote the set of eigenvalues and the determinant of A, respectively. In this paper, we study the existence of multiple p-periodic solutions to the following discre...

Benjamini, Häggström, Peres and Steif introduced the model of dynamical random walk on Z [2]. This is a continuum of random walks indexed by a parameter t. They proved that for d = 3,4 there almost surely exist t such that the random walk at time t visits the origin infinitely often, but for d ≥ 5 there almost surely do not exist such t. Hoffman showed that for d = 2 there almost surely exists ...

Benjamini, Häggström, Peres and Steif introduced the model of dynamical random walk on Z [2]. This is a continuum of random walks indexed by a parameter t. They proved that for d = 3, 4 there almost surely exist t such that the random walk at time t visits the origin infinitely often, but for d ≥ 5 there almost surely do not exist such t. Hoffman showed that for d = 2 there almost surely exists...

What follows is a paper based entirely on the paper "Thick Points of the Gaussian Free Field" by Xiaoyu Hu, Jason Miller, and Yuval Peres [1]. The purpose of this paper is both to convey that I read and understood the paper [1], and also to hopefully serve as a more intuitive/less rigorous explanation of the statements and proofs of Theorems 1.1 and 1.2 in [1]. After reading this paper, the rea...

Consider independent bond percolation with retention probability p on a spherically symmetric tree Γ. Write θΓ(p) for the probability that the root is in an infinite open cluster, and define the critical value pc = inf{p : θΓ(p) > 0}. If θΓ(pc) = 0, then the root may still percolate in the corresponding dynamical percolation process at the critical value pc, as demonstrated recently by Häggströ...

Recently in Reference [ quant-ph/0202121] a computational criterion of separability induced by greatest cross norm is proposed by Rudolph. There, Rudolph conjectured that the new criterion is not weaker than positive partial transpose criterion for separability. We show that there exist counterexample to this claim, that is, proposed criterion is weaker than the positive partial transpose crite...