Convex approximations of quantum channels

We address the problem of optimally approximating the action of a desired and unavailable quantum channel $\mathrm{\Phi }$ having at our disposal a single use of a given set of other channels $\left\{{\mathrm{\Psi }}_i\right\}$. The problem is recast to look for the least distinguishable channel from $\mathrm{\Phi }$ among the convex set ${\sum }_i{p}_i{\mathrm{\Psi }}_i$, and the corresponding optimal weights $\left\{p_i\right\}$ provide the optimal convex mixing of the available channels $\left\{{\mathrm{\Psi }}_i\right\}$. For single-qubit channels we study specifically cases where the available convex set corresponds to covariant channels or to Pauli channels, and the desired target map is an arbitrary unitary transformation or a generalized damping channel.

Figure 1

Optimal convex approximation of a unitary map $\mathcal{U}(\alpha ,\beta ,\delta )$ for qubits w.r.t. the set including the identity map and the map $\frac{1}{3}{\sum }_i{\sigma }_i(·){\sigma }_i$. The problem is solved by finding the closest covariant channel to the unitary map in the diamond norm, as in Eq. (15). The covariance distance $D_{\mathcal{C}}\left[\mathcal{U}(\alpha ,\beta ,\delta )\right]$ is a piecewise function of the diamond norm $x\equiv D_I\left[\mathcal{U}(\alpha ,\beta ,\delta )\right]={\parallel \mathcal{U}(\alpha ,\beta ,\delta )-\mathcal{I}\parallel }_{\diamond }$, as given in Eq. (17).

Figure 2

Optimal convex approximation of a unitary map $\mathcal{U}(\alpha ,\beta ,\delta )$ w.r.t. the set including the identity map and the three rotations by the three Pauli matrices. The problem is solved by finding the closest Pauli channel to the unitary map in the diamond norm. The solution provides the Pauli distance $D_{\mathcal{P}}\left[\mathcal{U}(\alpha ,\beta ,\delta )\right]$, here plotted vs $\alpha$ and $\beta$, for a fixed value of $\delta$, namely, $\delta \equiv \pi /8$.

Figure 3

Optimal convex approximation of a generalized damping channel $\mathrm{\Gamma }(q,\gamma )$ w.r.t. the set including the identity map and the three rotations by the three Pauli matrices. The problem is solved by finding the closest Pauli channel to $\mathrm{\Gamma }(q,\gamma )$ in the diamond norm. The solution provides the Pauli distance $D_{\mathcal{P}}\left[\mathrm{\Gamma }(q,\gamma )\right]$, here plotted vs $q$ and $\gamma$.

Figure 4

Optimal convex approximation of a generalized damping channel $\mathrm{\Gamma }(0.7,\gamma )$ vs the parameter $\gamma$, with respect to the set of Pauli channels. The Pauli distance $D_{\mathcal{P}}\left[\mathrm{\Gamma }(0.7,\gamma )\right]$ is plotted as the solid line; the upper and lower bounds given by Eq. (25), by the dashed lines.