Explicit error bounds for entanglement transformations between sparse multipartite states

TL;DR

This paper derives explicit error bounds for entanglement transformations between sparse multipartite states, providing new formulas and bounds that improve understanding of success probabilities and convergence rates in quantum information processing.

Contribution

It introduces a regularised formula for entanglement measures, establishes bounds on convergence rates, and identifies cases where single-letter formulas suffice for evaluation.

Findings

Derived a new regularised formula for entanglement measures.

Established bounds on the convergence rate of entanglement transformations.

Identified states satisfying sparsity constraints where evaluation simplifies to a single-letter formula.

Abstract

The trade-off relation between the rate and the strong converse exponent for probabilistic asymptotic entanglement transformations between pure multipartite states can in principle be characterised in terms of a class of entanglement measures determined implicitly by a set of strong axioms. A nontrivial family of such functionals has recently been constructed, but their previously known characterisations have so far only made it possible to evaluate them in very simple cases. In this paper we derive a new regularised formula for these functionals in terms of a subadditive upper bound, complementing the previously known superadditive lower bound. The upper and lower bounds evaluated on tensor powers differ by a logarithmically bounded term, which provides a bound on the convergence rate. In addition, we find that on states satisfying a certain sparsity constraint, the upper bound is equal to the value of the corresponding additive entanglement measure, therefore the regularisation is not needed for such states, and the evaluation is possible via a single-letter formula. Our results provide explicit bounds on the success probability of transformations by local operations and classical communication and, due to the additivity of the entanglement measures, also on the strong converse exponent for asymptotic transformations.