In the paper under the same title, we have constructed entangled states with positive partial transposes using indecomposable positive linear maps between matrix algebras. In this paper, we show that every entanglement with positive partial transpose arises in this way.

Let $p=\frac{1+\varepsilon}{n}$. It is known that if $N=\varepsilon^3n\to\infty$, then with high probability (w.h.p.) $G_{n,p}$ has a unique giant largest component. We show in addition, $\varepsilon=\varepsilon(n)\to 0$, w.h.p. the cover time of asymptotic to $n\log^2N$; previously Barlow, Ding, Nachmias, and Peres had shown this up constant multiplicative factors.

We sharpen the Evolving set methodology of Morris and Peres and extend it to study convergence in total variation, relative entropy, L2 and other distances. Bounds in terms of a modified form of conductance are given which apply even for walks with no holding probability. These bounds are found to be strictly better than earlier Evolving set bounds, may be substantially better than conductance ...

We give a polynomial time approximation scheme (PTAS) for computing the supremum of a Gaussian process. That is, given a finite set of vectors V ⊆ R, we compute a (1 + ε)-factor approximation to EX←Nd [supv∈V |〈v,X〉|] deterministically in time poly(d)·|V |Oε(1). Previously, only a constant factor deterministic polynomial time approximation algorithm was known due to the work of Ding, Lee and Pe...

A bipartite spin-1/2 system having the probabilities 1+3x 4 of being in the Einstein-Podolsky-Rosen entangled state |Ψ−>≡ 1 √ 2 (|↑>A |↓>B−|↓>A |↑>B) and 3(1−x) 4 of being orthogonal, is known to admit a local realistic description if and only if x < 1/3 (Peres criterion). We consider here a more general case where the probabilities of being in the entangled states |Φ±>≡ 1 √ 2 (|↑>A |↑>B ±|↓>A ...

The dynamical discrete web (DDW), introduced in recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical time parameter s. The evolution is by independent updating of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed s. In this paper, we study ...

The dynamical discrete web (DyDW), introduced in recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical time parameter τ . The evolution is by independent updating of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed τ . In this paper, we stu...

In loving memory of Asher Peres, we discuss a most important and influential paper written in 1935 by his thesis supervisor and mentor Nathan Rosen, together with Albert Einstein and Boris Podolsky. In that paper, the trio known as EPR questioned the completeness of quantum mechanics. The authors argued that the then-new theory should not be considered final because they believed it incapable o...

We present a construction of new bound entangled states from given bound entangled states for arbitrary dimensional bipartite systems. One way to construct bound entangled states is to show that these states are PPT (positive partial transpose) and violate the range criterion at the same time. By applying certain operators to given bound entangled states or to one of the subsystems of the given...