Sufficiency of rank deficiency and support motion for finite-copy closure

Determine whether rank deficiency and first-order support motion are sufficient conditions for finite-copy attainment of the Holevo bound in multiparameter quantum estimation models.

Background

The paper constructs a rank-two, three-parameter quantum estimation model in which a strict single-copy gap above the Holevo bound closes exactly with a two-copy collective measurement. It also proves that full-rank models cannot exhibit such finite-copy closure when a strict single-copy gap exists. These results establish rank deficiency as necessary, while the discussion identifies first-order motion of the state support as part of the mechanism in the counterexample.

The authors explicitly leave unresolved whether these structural properties are sufficient in general. Resolving this question would characterize more precisely which rank-deficient quantum models can attain the Holevo bound with a finite number of copies.