Separability criterion for three-qubit states with a four dimensional norm

Abstract

PAPERSeparability criterion for three-qubit states with a four dimensional norm AbstractWe give a separability criterion for three qubit states in terms of diagonal and anti-diagonal entries. This gives us a complete characterization of separability when all the entries are zero except for diagonal and anti-diagonals. The criterion is expressed in terms of a norm arising from anti-diagonal entries. We compute this norm in several cases, so that we get criteria with which we can decide the separability by routine computations.

Separability of multi-qubit states in terms of diagonal and anti-diagonal entries

“…The number∆ 3 (a) for the three qubit case appears in Gühne's separability criterion [15]. We show that the equality ∆ 3 (a) =∆ 3 (a) holds for three qubit case, which recovers the main result in [20]. The notion of multisets is also useful to deal with the anti-diagonal part, and will be used to characterize separability of half rank states X(a, c) in Theorem 5.3.…”

“…This number appears in the Gühne's separability criterion [15]. We show that the equality ∆ 3 (a) =∆ 3 (a) holds for the three qubit case, from which we recover the main result in [20]. It would be nice to know if the identity ∆ n (a) =∆ n (a) holds for n ≥ 4.…”

“…Motivated by the characterization of block-positivity of three qubit X-shaped matrices in [23], the half of the last number was taken as the definition of B(u) in [18], and has been calculated [20,18] in terms of entries u i in several cases.…”

“…by Theorem 5.3. Therefore, θ k j 1 − θ ℓ j 1 may be replaced by θ ℓ j 2 − θ k j 2 in (20), and so we have the identity |c j 1 | = |c j 2 |.…”

“…Xn ≥ c ∞ in (9). Now, we proceed to get another lower bound of the dual norm c ′ Xn for c ∈ V sa n in terms of the norms · X n of some elements 20 in V sa n . To do this, we take φ ∈ V ph n to get…”