A solution of the generalised quantum Stein's lemma

Ludovico Lami

Published 2024-08-12Version 1

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 the theory of entanglement manipulation under asymptotically non-entangling operations, with the regularised relative entropy of entanglement governing the asymptotic transformation rate between any two quantum states. 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. In the fully classical case, blurring alone suffices to prove the generalised Stein's lemma. To solve the quantum problem, however, it does not seem to suffice. 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.