Validity of Brandão–Plenio Lemma III.7 in full sublinear-defect generality

Ascertain whether Brandão–Plenio Lemma III.7 holds in its original generality for (n,r)-almost power states with sublinear defect r=o(n). Concretely, prove that for a fixed state \rho and a family of free-state sets (F_n)_n obeying the Brandão–Plenio axioms, the asymptotic equality lim_{n→∞} (1/n) inf_{\omega_n∈A(\rho,r_n),\, \sigma_n∈F_n} D(\omega_n\|\sigma_n)=D^\infty(\rho\|F) remains valid when r_n grows sublinearly with n, thus confirming the full strength of Lemma III.7.

Background

The original proof of the generalised quantum Stein’s lemma by Brandão and Plenio contained a gap (Lemma III.9), which undermined subsequent steps including Lemma III.7 concerning almost power states. In this paper, the author recovers a weaker variant where the number of defective sites r is fixed (independent of n), and shows the generalised Stein’s lemma under that restriction.

The full statement of Lemma III.7 (allowing r=o(n)) remains unsettled; confirming it would align the almost-i.i.d. extension with physical intuition that a sublinear number of defects should not affect extensive asymptotic quantities.