A solution of the generalised quantum Stein’s lemma

I will discuss the solution of the generalised quantum Stein’s lemma presented in [Lami, arXiv:2408.06410] (see also [Hayashi/Yamasaki, arXiv:2408.02722] for related work), which establishes that the Stein exponent associated with entanglement testing, namely, the quantum hypothesis testing task of distinguishing between n copies of an entangled state and a generic separable state, equals the regularised relative entropy of entanglement. To solve the problem I will briefly introduce two techniques. The first is a procedure called “blurring”, which, informally, transforms a permutationally symmetric state by making it more evenly spread across nearby type classes. I will discuss this technique extensively in the classical case, where it already suffices to prove the generalised Stein’s lemma. Depending on time, I will then present a second technical innovation, which is needed to prove the quantum version of the statement. This consists in a second quantisation step, which lifts the problem from a finite-dimensional system to an infinite-dimensional bosonic quantum system, where it can then be solved with techniques from continuous-variable quantum information. Rather remarkably, the second-quantised action of the blurring map corresponds to a pure loss channel.

This initiative is part of the "PhD Lectures" activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.