I would like to compute the diamond norm of the difference of two single-qubit channels $\Phi_1$ and $\Phi_2$. This difference is equal to, for any $2\times2$ complex matrix $\rho$: $$\DeclareMathOperator{tr}{tr}\Phi_1(\rho)-\Phi_2(\rho)=\beta\rho+\gamma Z\rho Z+\delta\tr(\rho)|0\rangle\!\langle0|+\varepsilon\tr(\rho)|1\rangle\!\langle1|$$ with $\beta+\gamma+\delta+\varepsilon=0$.

Is it possible to obtain a closed-form expression of $\left\|\Phi_1-\Phi_2\right\|_{\diamond}$?

Since this difference is Hermitian preserving, I intended to use the following formula: $$\left\|\Phi_1-\Phi_2\right\|_{\diamond}=\max_{|\psi\rangle\in\mathbb{C}^4}\left\|\left({\rm id}\otimes(\Phi_1-\Phi_2\right))(|\psi\rangle\!\langle\psi|)\right\|_1$$ However, the computations get quite overwhelming, not to mention that I would have to optimize over the coefficients of $|\psi\rangle$ thereafter. Furthermore, since I'm interested in a closed form expression, I can't use the semidefinite problem the diamond norm is associated with.

Is there a way for me to compute this diamond norm? If there's not, is it possible if I set some coefficients to $0$?



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