Probabilistically Perfect Cloning of Two Pure States: Geometric Approach

We solve the long-standing problem of making $n$ perfect clones from $m$ copies of one of two known pure states with minimum failure probability in the general case where the known states have arbitrary a priori probabilities. The solution emerges from a geometric formulation of the problem. This formulation reveals that cloning converges to state discrimination followed by state preparation as the number of clones goes to infinity. The convergence exhibits a phenomenon analogous to a second-order symmetry-breaking phase transition.

Figure 1

Unitarity curves, $q_2$ versus $q_1$, from Eq. (3) and the associated sets $S_{\alpha }$ from Eq. (4) for $\alpha =1$ (solid light, medium, and dark gray), $\alpha =0.5$ (dotted medium and dark gray), and $\alpha =0$ (dashed dark gray). The figure also shows the optimal straight segment $Q={\eta }_1{q}_1+{\eta }_2{q}_2$ and its normal vector $({\eta }_1,{\eta }_2)$. For all plots, $s=0.5$, $m=1$, and $n=2$.

Figure 2

Unitarity curves, $q_2$ versus $q_1$, from Eq. (3) for different values of $s$ and for (a) $m=1$, $n=2$ and (b) $m=1$, $n=5$. The curves are symmetric under mirror reflection along the dotted line $q_1=q_2$. The dashed lines in (b) are the hyperbolas $q_1{q}_2=s^{2m}$.

Figure 3

Minimum cloning failure probability $Q_{\min }$ versus ${\eta }_1$ (solid lines) and UD failure probability $Q_{\mathrm{UD}}$ versus ${\eta }_1$ (dashed lines) for the same values of $m$, $n$, and $s$ used in Fig. 2.