We study a relation between the concurrence of assistance and the Mermin inequality on three-qubit pure states. We find that if a given three-qubit pure state has the minimal concurrence of assistance greater than 1/2 then the state violates some Mermin inequality.
D{u}r [Phys. Rev. Lett. {bf 87}, 230402 (2001)] constructed $N$-qubit bound entangled states which violate a Bell inequality for $Nge 8$, and his result was recently improved by showing that there exists an $N$-qubit bound entangled state violating t
he Bell inequality if and only if $Nge 6$ [Phys. Rev. A {bf 79}, 032309 (2009)]. On the other hand, it has been also shown that the states which D{u}r considered violate Bell inequalities different from the inequality for $Nge 6$. In this paper, by employing different forms of Bell inequalities, in particular, a specific form of Bell inequalities with $M$ settings of the measuring apparatus for sufficiently large $M$, we prove that there exists an $N$-qubit bound entangled state violating the $M$-setting Bell inequality if and only if $Nge 4$.
In this work, we investigate what kinds of quantum states are feasible to perform perfectly secure secret sharing, and present its necessary and sufficient conditions. We also show that the states are bipartite distillable for all bipartite splits, a
nd hence the states could be distillable into the Greenberger-Horne-Zeilinger state. We finally exhibit a class of secret-sharing states, which have an arbitrarily small amount of bipartite distillable entanglement for a certain split.
There is an interesting property about multipartite entanglement, called the monogamy of entanglement. The property can be shown by the monogamy inequality, called the Coffman-Kundu-Wootters inequality [Phys. Rev. A {bf 61}, 052306 (2000); Phys. Rev.
Lett. {bf 96}, 220503 (2006)], and more explicitly by the monogamy equality in terms of the concurrence and the concurrence of assistance, $mathcal{C}_{A(BC)}^2=mathcal{C}_{AB}^2+(mathcal{C}_{AC}^a)^2$, in the three-qubit system. In this paper, we consider the monogamy equality in $2otimes 2 otimes d$ quantum systems. We show that $mathcal{C}_{A(BC)}=mathcal{C}_{AB}$ if and only if $mathcal{C}_{AC}^a=0$, and also show that if $mathcal{C}_{A(BC)}=mathcal{C}_{AC}^a$ then $mathcal{C}_{AB}=0$, while there exists a state in a $2otimes 2 otimes d$ system such that $mathcal{C}_{AB}=0$ but $mathcal{C}_{A(BC)}>mathcal{C}_{AC}^a$.
We develop a three-party quantum secret sharing protocol based on arbitrary dimensional quantum states. In contrast to the previous quantum secret sharing protocols, the sender can always control the state, just using local operations, for adjusting
the correlation of measurement directions of three parties and thus there is no waste of resource due to the discord between the directions. Moreover, our protocol contains the hidden value which enables the sender to leak no information of secret key to the dishonest receiver until the last steps of the procedure.
