A solution of the generalised quantum Stein's lemma

TL;DR

This paper proves the generalized quantum Stein's lemma, establishing the optimal rate for entanglement testing and resource conversion in quantum information, using innovative 'blurring' and second quantisation techniques.

Contribution

It introduces a novel approach combining blurring and second quantisation to solve the generalized quantum Stein's lemma, linking entanglement testing to the regularised relative entropy of entanglement.

Findings

Stein exponent equals the regularised relative entropy of entanglement.

Achieves the Stein exponent with approximately i.i.d. null hypotheses.

Establishes reversibility of quantum resource theories under asymptotic operations.

Abstract

We solve the generalised quantum Stein's lemma, proving that the Stein exponent associated with entanglement testing, namely, the quantum hypothesis testing task of distinguishing between $n$ copies of an entangled state $\rho_{AB}$ and a generic separable state $\sigma_{A^n:B^n}$, equals the regularised relative entropy of entanglement. Not only does this determine the ultimate performance of entanglement testing, but it also establishes the reversibility of all quantum resource theories under asymptotically resource non-generating operations, with the regularised relative entropy of resource governing the asymptotic transformation rate between any two quantum states. As a by-product, we prove that the same Stein exponent can also be achieved when the null hypothesis is only approximately i.i.d., in the sense that it can be modelled by an 'almost power state'. To solve the problem we introduce two techniques. The first is a procedure that we call 'blurring', which, informally, transforms a permutationally symmetric state by making it more evenly spread across nearby type classes. Blurring alone suffices to prove the generalised Stein's lemma in the fully classical case, but not in the quantum case. Our second technical innovation, therefore, is to perform a second quantisation step to lift the problem to an infinite-dimensional bosonic quantum system; we then solve it there by using techniques from continuous-variable quantum information. Rather remarkably, the second-quantised action of the blurring map corresponds to a pure loss channel. A careful examination of this second quantisation step is the core of our quantum solution.