Sharp finite-block norm equality for photon-sector correlation thresholds

Prove or disprove the conjectured equality between the operator norm of the partial transpose of every photon-sector correlation-threshold projector and its separable acceptance probability, namely establish whether ${T_{s,m,k,h}^Gamma}_\infty=p_{s,m,k,h}$ holds for all admissible integers $s,m,k,h$.

Background

The energy-constrained coding construction uses correlation-threshold projectors indexed by photon-number sectors. The paper proves a lower bound and a polynomial-factor upper bound relating the partial-transpose operator norm of each threshold projector to the corresponding acceptance probability under an explicitly constructed separable comparison state.

The sharper equality would improve finite-block performance estimates, although the paper states that it is not needed for the already established asymptotic capacity theorem. The authors reduce the conjecture to a finite family of inequalities involving the eigenvalues of partially transposed partial-pair operators on irreducible representation sectors.