Computable entanglement cost under positive partial transpose operations

TL;DR

This paper introduces an efficient algorithm to compute the asymptotic entanglement cost of quantum states under PPT operations, overcoming regularisation challenges and providing the first such measure that is computationally feasible.

Contribution

It constructs a hierarchy of semi-definite programs that converges exponentially fast to the true entanglement cost, enabling practical approximation without regularisation.

Findings

Hierarchy of semi-definite programs converges exponentially fast.

Algorithm approximates entanglement cost within additive error in polynomial time.

First demonstration of an efficiently computable asymptotic entanglement measure.

Abstract

Quantum information theory is plagued by the problem of regularisations, which require the evaluation of formidable asymptotic quantities. This makes it computationally intractable to gain a precise quantitative understanding of the ultimate efficiency of key operational tasks such as entanglement manipulation. Here we consider the problem of computing the asymptotic entanglement cost of preparing noisy quantum states under quantum operations with positive partial transpose (PPT). By means of an analytical example, a previously claimed solution to this problem is shown to be incorrect. Building on a previous characterisation of the PPT entanglement cost in terms of a regularised formula, we construct instead a hierarchy of semi-definite programs that bypasses the issue of regularisation altogether, and converges to the true asymptotic value of the entanglement cost. Our main result establishes that this convergence happens exponentially fast, thus yielding an efficient algorithm that approximates the cost up to an additive error $\varepsilon$ in time $\mathrm{poly}(D,\,\log(1/\varepsilon))$, where $D$ is the underlying Hilbert space dimension. To our knowledge, this is the first time that an asymptotic entanglement measure is shown to be efficiently computable despite no closed-form formula being available.