A Counterexample to the Optimality Conjecture in Convex Quantum Channel Optimization
Jianting Yang
TL;DR
The paper questions the sufficiency of spectral dual certificates for optimality in convex quantum-channel optimization. It constructs a concrete two-qubit counterexample with specific states $\rho$ and $\sigma$ and a corresponding $\Phi$ that violates the conjectured optimality conditions, even when a reduced problem is solved by $X$. The result demonstrates that the conjectured dual-certificate mechanism based on $Y=\operatorname{sign}(\sigma-(\Phi\otimes \operatorname{Id}_{\mathcal{Z}})(\rho))$ and $H=(\operatorname{Id}_{L(\mathcal{Y})}\otimes \Psi_{\rho}^*)(Y)$ does not universally certify optimality. This highlights limitations of Choi-spectral certificates and motivates developing alternative optimality certificates for nuclear-norm minimization in quantum-channel settings.
Abstract
This paper presents a counterexample to the optimality conjecture in convex quantum channel optimization proposed by Coutts et al. The conjecture posits that for nuclear norm minimization problems in quantum channel optimization, the dual certificate of an optimal solution can be uniquely determined via the spectral calculus of the Choi matrix. By constructing a counterexample in 2-dimensional Hilbert spaces, we disprove this conjecture.