Entanglement Cost of Three-Level Antisymmetric States

The concept of entanglement is the key for quantum information processing. To quantify the resource of entanglement, its measures should be additive, such as bits for classical information. One candidate for such additive measures is entanglement of formation. In [1], it is shown that the entanglement cost Ec to create some state can be asymptotically calculated from the entanglement of formation. In this sense, the entanglement cost has an important physical meaning. Since the known results are, nevertheless, not so much [5, 6], we pay attention to antisymmetric states that are easy to deal with. As is already shown[2], the entanglement of formation for two states in S (H−) is additive. Furthermore, the lower bound for entanglement cost of density matrices in d-level antisymmetric space, obtained in [3], is log2 d d−1 ebit. In this paper, we show that the entanglement cost of three-level antisymmetric states (d = 3) in S (H−) is exactly one ebit. We first define the three-level antisymmetric states. Let us consider a bipartite qutrit system, HA = HB = C. The antisymmetric subspace H− on HA ⊗HB is defined as follows: H− := spanC {|01〉 − |10〉, |12〉 − |21〉, |20〉 − |02〉} ⊂ HA ⊗HB. Then, the antisymmetric state on H − shared with Alice and Bob is, in general,