Optimal random-unitary decomposition of symmetric Werner–Holevo channels
- Field
- Topic
Problem
For every odd integer \(d\geq5\), do there exist \(r=d(d+1)/2\) unitary operators \(U_1,\ldots,U_r\) on \(\mathbb C^d\) such that
The transpose in Eq. (1) is taken in a fixed orthonormal basis.
Source
Girard et al. explicitly conjecture that the symmetric Werner–Holevo channel has mixed-unitary rank equal to its Choi rank; see Section 6, final paragraph after Theorem 23, p. 29 of the preprint [GLL+22].
Progress
Let \(N(\Phi_d)\) denote the minimum number of unitary conjugations in a convex decomposition. Its Choi rank gives \(N(\Phi_d)\geq r\); for odd \(d\), Theorem 22 gives \(N(\Phi_d)\leq d(d+3)/2\) [GLL+22].
The optimum \(N(\Phi_d)=r\) holds for every even \(d\) and for \(d=3\) (Theorems 21 and 23) [GLL+22]. Levick and Rahaman prove it for every prime \(d\equiv7\pmod8\) (Theorem 3.8 and Corollary 3.12) [LR21].
Theorem 19 identifies \(N(\Phi_d)=r\) with the existence of a Hilbert–Schmidt orthogonal basis of the complex symmetric matrices consisting of unitaries. Every optimal \(r\)-term decomposition necessarily has weights \(1/r\), so the uniform formulation in Eq. (1) is equivalent to the rank conjecture [GLL+22].
Comment
The unresolved target is the universal odd-dimensional optimum. Numerical decompositions do not supply an exact proof. Literature audit: 9 September 2026. Searches under the channel, rank, and symmetric-unitary-basis formulations found no full resolution; indexed literature searches cannot exclude unindexed work. The cited partial results are peer-reviewed.