The distillable entanglement of multiple copies of Bell states

It is impossible to discriminate four Bell states through local operations and classical communication (LOCC), if only one copy is provided. To complete this task, two copies will suffice and be necessary. When n copies are provided, we show that the distillable entanglement is exactly n − 2. 03.67.Hk,03.65.Ud Typeset using REVTEX 1 The no-cloning theorem [1] asserts that it is impossible to discriminate nonorthogonal states with certainty. In general, orthogonal states may be distinguished perfectly only by means of global measurements since quantum information of orthogonality may be encoded in entanglement which may not be extracted by LOCC operations. If only LOCC operations are allowed, even product orthogonal states could not be discriminated exactly [2]. However, Walgate et al. [3] demonstrated that any two orthogonal multipartite states can be discriminated with certainty by only LOCC operations. To discriminate multiple orthogonal states, more copies are required. They also showed that n possible orthogonal states can be distinguished pefectly with n − 1 copies. It is an upper bound upon the number of copies required for local distinction of states. Further, they pointed out that there are sets of orthogonal states that can be distinguished using less than n−1 copies. In the case of four Bell states, two copies will suffice. Recently, using the existing inequality among the measures of entanglement, Ghosh et al. [4] proved that any three Bell states cannot be discriminated by LOCC operations. From a different point of view, Walgate and Hardy [5] arrived at the same conclusion and discussed the sufficent and necessary conditions to discriminate 2× n states. The question of local distinction of nonorthogonal states has also been investigated in recent papers [6,7]. In [4], Ghosh et al. calculated the distillable entanglement [8] of the mixed state comprising of two of the Bell basis with equal a priori probability. In this note, we prove that if n copies out of four Bell states are provided, the distillble entanglement is n− 2. Denote the four Bell states as |Φ1〉 = 1 √ 2 (|00〉+ |11〉), |Φ2〉 = 1 √ 2 (|00〉 − |11〉), |Φ3〉 = 1 √ 2 (|01〉+ |10〉), |Φ4〉 = 1 √ 2 (|01〉 − |10〉). (1)