7,895 research outputs found
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 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
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