We characterize the algebraic structure of semi-direct product of cyclic groups, $Z_{N}rtimesZ_{p}$, where $p$ is an odd prime number which does not divide $q-1$ for any prime factor $q$ of $N$, and provide a polynomial-time quantum computational alg
orithm solving hidden symmetry subgroup problem of the groups.
It was shown that two distant particles can be entangled by sending a third particle never entangled with the other two [T. S. Cubitt et al., Phys. Rev. Lett. 91, 037902 (2003)]. In this paper, we investigate a class of three-qubit separable states t
o distribute entanglement by the same way, and calculate the maximal amount of entanglement which two particles of separable states in the class can have after applying the way.
The monogamy inequality in terms of the concurrence, called the Coffman-Kundu-Wootters inequality [Phys. Rev. A {bf 61}, 052306 (2000)], and its generalization [T.J. Osborne and F. Verstraete, Phys. Rev. Lett. {bf 96}, 220503 (2006)] hold on general
$n$-qubit states including mixed ones. In this paper, we consider the monogamy inequalities in terms of the fully entangled fraction and the teleportation fidelity. We show that the monogamy inequalities do not hold on general mixed states, while the inequalities hold on $n$-qubit pure states.
We study the explicit relation between violation of Bell inequalities and bipartite distillability of multi-qubit states. It has been shown that even though for $Nge 8$ there exist $N$-qubit bound entangled states which violates a Bell inequality [Ph
ys. Rev. Lett. {bf 87}, 230402 (2001)], for all the states violating the inequality there exists at least one splitting of the parties into two groups such that pure-state entanglement can be distilled [Phys. Rev. Lett. {bf 88}, 027901 (2002)]. We here prove that for all $N$-qubit states violating the inequality the number of distillable bipartite splits increases exponentially with $N$, and hence the probability that a randomly chosen bipartite split is distillable approaches one exponentially with $N$, as $N$ tends to infinity. We also show that there exists at least one $N$-qubit bound entangled state violating the inequality if and only if $Nge 6$.
