Even though a method to perfectly sign quantum messages has not been known, the arbitrated quantum signature scheme has been considered as one of good candidates. However, its forgery problem has been an obstacle to the scheme being a successful meth
od. In this paper, we consider one situation, which is slightly different from the forgery problem, that we check whether at least one quantum message with signature can be forged in a given scheme, although all the messages cannot be forged. If there exist only a finite number of forgeable quantum messages in the scheme then the scheme can be secure against the forgery attack by not sending the forgeable quantum messages, and so our situation does not directly imply that we check whether the scheme is secure against the attack. But, if users run a given scheme without any consideration of forgeable quantum messages then a sender might transmit such forgeable messages to a receiver, and an attacker can forge the messages if the attacker knows them in such a case. Thus it is important and necessary to look into forgeable quantum messages. We here show that there always exists such a forgeable quantum message-signature pair for every known scheme with quantum encryption and rotation, and numerically show that any forgeable quantum message-signature pairs do not exist in an arbitrated quantum signature scheme.
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 $Nge 4$.
