Figure 1

The optimal Bayes risk ${\mathcal{R}}_B\left(p\right)$ in the discrimination of two Pauli channels versus the a priori probability $p$ will usually look like this. Notice that the rightmost and leftmost segments have slope 1 and $-1$, respectively. The minimal risk for the minimax discrimination corresponds to ${\mathcal{R}}_M={\max }_p{\mathcal{R}}_B\left(p\right)$, and is achieved at one of the break points $p^{\left(\alpha \right)}$.Reuse & Permissions

Figure 2

An example for the Bayes risks ${\mathcal{R}}_B^{\prime }(p,{\sigma }_i)$ with $i=x,y,z$ versus the a priori probability $p$, for discrimination without ancilla. Each of the three different dotted lines corresponds to the Bayes risk ${\mathcal{R}}_B^{\prime }(p,{\sigma }_i)$ when sending an eigenstate of the Pauli matrix ${\sigma }_i$ through the channel. The solid line is the optimal Bayes risk ${\mathcal{R}}_B^{\prime }\left(p\right)$ without ancillary assistance, and corresponds at any $p$ to the minimum of the three ${\mathcal{R}}_B^{\prime }(p,{\sigma }_i)$. The minimal risk for the minimax discrimination with no ancilla corresponds to ${\mathcal{R}}_M^{\prime }={\max }_p{\mathcal{R}}_B^{\prime }\left(p\right)$, and is achieved at one of the break points of ${\mathcal{R}}_B^{\prime }\left(p\right)$.Reuse & Permissions

Figure 3

Optimal Bayes risks versus the a priori probability $p$ for the discrimination of the Pauli channels with parameters given in Eq. (39). The solid line gives ${\mathcal{R}}_B\left(p\right)$ for an entanglement-assisted strategy; the dotted line gives ${\mathcal{R}}_B^{\prime }\left(p\right)$ for a strategy without ancilla. The minimal risk in the optimal minimax discrimination corresponds in both strategies to ${\mathcal{R}}_M^{\prime }={\max }_p{\mathcal{R}}_B^{\prime }\left(p\right)={\max }_p{\mathcal{R}}_B\left(p\right)={\mathcal{R}}_M$, namely, there is no need of an ancillary system.Reuse & Permissions