First dimension where the Grothendieck constant Kd exceeds 1

Ascertain the smallest integer d for which Kd, the least constant satisfying the dimension-dependent inclusion S_d ⊆ Kd·Conv(D_d) in the complex formulation of Grothendieck’s inequality, is strictly greater than 1.

Background

Let Kd denote the least constant K such that S_d ⊆ K·Conv(D_d) holds for a fixed dimension d. It is known that KG = sup_d Kd and that K2 = 1. Identifying the first dimension at which Kd exceeds 1 would refine understanding of the geometry of Schur multipliers and the onset of genuinely nontrivial Grothendieck behavior in finite dimensions.

The authors highlight that this threshold dimension is not currently known, contrasting it with related thresholds in the positive/unital subclass where strictness is known to begin at d ≥ 4.

References

It seems, however, that the first integer d for which Kd > 1 is still unknown.

— Bounds on the distance between a unital quantum channel and the convex hull of unitary channels, with applications to the asymptotic quantum Birkhoff conjecture  (1201.1172 - Yu et al., 2012) in Section VII. A Connection to Grothendieck's Inequality