Entanglement sudden death in qubit-qutrit systems

We demonstrate the existence of entanglement sudden death (ESD), the complete loss of entanglement in finite time, in qubit-qutrit systems. In particular, ESD is shown to occur in such systems initially prepared in a one-parameter class of entangled mixed states and then subjected to local dephasing noise. Together with previous results, this proves the existence of ESD for some states in all quantum systems for which rigorously defined mixed-state entanglement measures have been identified. We conjecture that ESD exists in all quantum systems prepared in appropriate bipartite states.Comment: 10 pages. To appear in Physics Letters

Sieving by large integers and covering systems of congruences

An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdos and Selfridge. We also prove a conjecture of Erdos and Graham, that, for each fixed number K>1, the complement in the integers of any union of residue classes r(n) mod n, for distinct n in (N,KN], has density at least d_K for N sufficiently large. Here d_K is a positive number depending only on K. Either of these new results implies another conjecture of Erdos and Graham, that if S is a finite set of moduli greater than N, with a choice for residue classes r(n) mod n for n in S which covers the integers, then the largest member of S cannot be O(N). We further obtain stronger forms of these results and establish other information, including an improvement of a related theorem of Haight.Comment: v3. 28 pages. Minor corrections and notational improvements. Added reference to recent discovery by Gibson of a covering system with least modulus 25. To appear in J. Amer. Math. So