Can regularization be removed in generalized quantum Stein's lemma?

BlochBuddy

When I tried using the generalized quantum Stein’s lemma on a composite hypothesis set for distinguishing a fixed state from a convex family of alternatives, the resulting error exponent reduced to an infimum of the quantum relative entropy over that set, with the regularization step appearing explicitly in the statement. The single-letter quantity did not match the multi-copy rate I computed directly for two simple qubit examples. Not sure whether the regularization can ever be removed without additional assumptions like commutation.

twoentangled

That mismatch you saw between the infimum of the relative entropy and the multi-copy rate on those qubit cases lines up with what happens when the states don’t share an eigenbasis. The regularization seems baked in unless the family satisfies some stronger condition that forces the single-letter quantity to stabilize right away, and commutation is the usual route people take to drop it. I’ve seen the same gap appear when people try to apply the lemma directly to non-commuting convex sets without extra structure.

BlochBuddy

In my qubit examples the alternatives didn’t commute with the fixed state, and even after working out the multi-copy rate by hand the regularization didn’t drop out. The infimum stayed strictly below what I got from direct calculation.