Coutts et al.'s spectral-calculus optimality conjecture for quantum channel optimization

Let $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ be finite-dimensional complex Hilbert spaces, let $\rho \in \mathsf{D}(\mathcal{X}\otimes \mathcal{Z})$ and $\sigma \in \mathsf{D}(\mathcal{Y}\otimes \mathcal{Z})$ be density operators, and let $\Phi \in \mathsf{C}(\mathcal{X},\mathcal{Y})$ be a completely positive map. Write $J(\Phi)$ for its Choi representation, let $\operatorname{Id}_{L(\mathcal{Z})}$ denote the identity map on $L(\mathcal{Z})$, and let $\Psi_{\rho}\in\mathsf{C}(\mathcal{X},\mathcal{Z})$ satisfy

$$(\Phi\otimes \operatorname{Id}_{L(\mathcal{Z})})(\rho)=(\operatorname{Id}_{L(\mathcal{Y})}\otimes\Psi_{\rho})(J(\Phi)).$$

Define

$$Y=\operatorname{sign}\left(\sigma-(\Phi\otimes \operatorname{Id}_{\mathcal{Z}})(\rho)\right),$$

and $H=(\operatorname{Id}_{L(\mathcal{Y})}\otimes\Psi_{\rho}^*)(Y)$. Coutts et al.'s optimality conjecture. The map $\Phi$ is an optimal solution to

$$\min_{\Phi}\ \left\|\sigma-(\Phi\otimes \operatorname{Id}_{\mathcal{Z}})(\rho)\right\|_*$$

subject to $\Phi\in\mathsf{C}(\mathcal{X},\mathcal{Y})$, if and only if

$$\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi))\in\operatorname{Herm}(\mathcal{X})$$

and

$$H\succeq \mathbf{1}_{\mathcal{Y}}\otimes\operatorname{Tr}_{\mathcal{Y}}(HJ(\Phi)).$$

The conjecture concerns certification of optimality in the trace-distance, or nuclear-norm, optimization of quantum channels: it claims that the dual certificate can be obtained from the spectral calculus of the Choi matrix of the candidate optimal channel. The paper constructs a counterexample in two-dimensional Hilbert spaces, so the conjecture is disproved.