Exact value of the complex Grothendieck constant

Determine the exact value of the complex Grothendieck constant KG that arises in Grothendieck’s inequality for Schur multipliers.

Background

Grothendieck’s inequality can be formulated for Schur multipliers as an inclusion S_d ⊆ K·Conv(D_d), where S_d is the unit ball of Schur multipliers (in any of several equivalent norms) and D_d is the set of rank-one Schur multipliers. The smallest universal constant K satisfying this for all dimensions d is the Grothendieck constant KG.

While bounds 1 < KG ≤ 1.4049 are known, the exact value of KG remains an open problem, and resolving it has implications across functional analysis and quantum information contexts.

References

The exact value of KG 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