An Equivalence of Entanglement-Assisted Transformation and Multiple-Copy Entanglement Transformation

TL;DR

This paper establishes an equivalence between entanglement-assisted transformation and multiple-copy entanglement transformation, providing conditions for their usefulness and implications for quantum state manipulation.

Contribution

It proves the equivalence of entanglement-assisted and multiple-copy transformations, offering explicit conditions and generalizations to probabilistic scenarios.

Findings

A sufficient condition for catalyst usefulness in state transformation.

Explicit construction of states using catalysts for non-maximally entangled states.

Equivalence between entanglement-assisted and multiple-copy transformations based on Schmidt coefficients.

Abstract

We examine the powers of entanglement-assisted transformation and multiple-copy entanglement transformation. First, we find a sufficient condition of when a given catalyst is useful in producing another specific target state. As an application of this condition, for any non-maximally entangled bipartite pure state and any integer $n$ not less than 4, we are able to explicitly construct a set of $n\times n$ quantum states which can be produced by using the given state as a catalyst. Second, we prove that for any positive integer $k$, entanglement-assisted transformation with $k\times k$-dimensional catalysts is useful in producing a target state if and only if multiple-copy entanglement transformation with $k$ copies of state is useful in producing the same target. Moreover, a necessary and sufficient condition for both of them is obtained in terms of the Schmidt coefficients of the target. This equivalence of entanglement-assisted transformation and multiple-copy entanglement transformation implies many interesting properties of entanglement transformation. Furthermore, these results are generalized to the case of probabilistic entanglement transformations.