Dimension dependence of the entanglement-resolution bounds

Determine whether the dimension-dependent gap between the constants c_d and C_d in the trace-distance entanglement-resolution bounds can be closed.

Background

Theorem 1 proves that, for sufficiently fine finite measurement families, the largest trace-distance entanglement left uncertified is Θ_d(ε), where ε is the covering infidelity and the constants supplied by the proof are c_d=1/[16d(d+1)] and C_d=2d.

The authors note that these constants differ by a factor of order d^3. The asymptotic scaling in ε is established, but the sharp dependence of the bounds on the local dimension is unresolved; in particular, it is unknown whether the existing constant gap can be eliminated.