One-shot quantum blurring lemma

Develop a one-shot (non-asymptotic) quantum blurring lemma for the blurring map \widebar{B}_{n,\delta}^\rho that appends \delta n copies of a state \rho, symmetrises over n+\delta n systems, and traces out \delta n systems. Specifically, establish a finite-n bound analogous to the asymptotic Lemma \ref{quantum_blurring_lemma}, providing explicit one-shot control that would enable corresponding one-shot bounds on hypothesis testing relative entropy in quantum resource testing.

Background

The paper’s central technical tool is a quantum blurring map that mixes in additional copies of a reference state, symmetrises, and discards systems. The authors prove an asymptotic variant (the quantum blurring lemma) sufficient to resolve the generalised quantum Stein’s lemma, but unlike the classical case, they do not obtain a one-shot version.

A one-shot quantum blurring lemma would mirror the classical one-shot result and yield finite-copy guarantees, including tighter one-shot bounds on hypothesis testing relative entropy for resource testing.