Bounds on entanglement catalysts

Abstract

Given a finite dimensional pure state transformation restricted by entanglement assisted local operations and classical communication (ELOCC), we derive minimum and maximum bounds on the entanglement of an ancillary catalyst that allows that transformation. These bounds are non-trivial even when the Schmidt number of both the original and ancillary states becomes large. We identify a lower bound for the dimension of a catalyst allowing a particular ELOCC transformation. Along with these bounds, we present further constraints on ELOCC transformations by identifying restrictions on the Schmidt coefficients of the target state. In addition, an example showing the existence of qubit ELOCC transformations with multiple ranges of potential ancillary states is provided. This example reveals some additional difficulty in finding strict bounds on ELOCC transformations, even in the qubit case. Finally, a comparison of the bounds in this paper with previously discovered bounds is presented.

Necessary conditions on effective quantum entanglement catalysts

“…The following example illustrates that our result (8) behaviors better than the result ( 9) from [23] in determining whether a catalytic state can complete a certain catalytic process.…”

“…. , l} when q l r s q 1 r s+1 , q d r s−1 q l+1 r s and q d r s−1 q 1 r s+1 , which is true for ∀l ∈ L. Hence, Proposition 1 fills such loopholes in the Theorem 1 in [23], and also gives finer characterizations of r by increasing the number of constraints on r.…”

“…Recently, by using a very nice idea the authors in [23] derived elegant results on the minimum and maximum bounds about the auxiliary catalysts under ELOCC. It is showed that (Theorem 1 in [23]) if p ⊗ r ≺ q ⊗ r, r satisfies max v∈(1,2,...,k−1)…”

“…B. A remark on the relationship between Proposition 1 and the Theorem 1 in [23] In the proof of Theorem 1 in [23], it is mentioned that "q m r v q 1 r v +1 implies that the first (v − 1) d + m elements of (q ⊗ r) ↓ consist of q x r y , ∀x ∈ {1, 2, . .…”

“…We analyze the ELOCC conversion between 4 × 4 entangled quantum states catalyzed by 2 × 2 catalyst, and give both necessary and sufficient conditions of catalytic states. Detailed examples are given to show that our criteria work is better than the existing ones [23], .…”