Multicopy programmable discrimination of general qubit states

Quantum state discrimination is a fundamental primitive in quantum statistics where one has to correctly identify the state of a system that is in one of two possible known states. A programmable discrimination machine performs this task when the pair of possible states is not a priori known but instead the two possible states are provided through two respective program ports. We study optimal programmable discrimination machines for general qubit states when several copies of states are available in the data or program ports. Two scenarios are considered: One in which the purity of the possible states is a priori known, and the fully universal one where the machine operates over generic mixed states of unknown purity. We find analytical results for both the unambiguous and minimum error discrimination strategies. This allows us to calculate the asymptotic performance of programmable discrimination machines when a large number of copies are provided and to recover the standard state discrimination and state comparison values as different limiting cases.

Figure 1

Error probability $P^{\mathrm{ME}}$ for $n=m=3$ (blue dashed line)$,11$ (green circles) and $29$ (yellow squares) versus purity. The fit $P^{\mathrm{ME}}\simeq 0.882/(n{r}^2)$ in the regime of high purities for $n=11$ and $n=29$ and the Gaussian approximation $P^{\mathrm{ME}}\simeq 1/2\exp [-n{r}^2/(2\sqrt{3})]$ in the regime of low purities for all cases is represented (solid lines).

Figure 2

Error probability $P^{\mathrm{ME}}$ for $r=0.2$ (thin blue solid line), $r=0.5$ (brown dashed line), $r=0.7$ (green dotted line) and $r=1$ (red dot-dashed line) versus $n$ ($n=m$ is assumed). Numerical points have been joined for an easier visualization. The asymptotic approximation $0.088/({\mathit{nr}}^2)$ is represented (thick solid line).

Figure 3

Error probability $P^{\mathrm{ME}}$ for $n=20$ (yellow circles) and $n=79$ (green squares) versus purity. The asymptotic behavior given by Eq. (70) is represented for both cases.

Figure 4

Error probability $P^{\mathrm{ME}}$ for the hard sphere (green solid line), Bures (blue dotted line), and Chernoff (red dashed line) priors versus $n$ ($n=m$ is assumed). The points correspond to the error probability for a fixed $r=0.93$; its proximity to the Chernoff curve exposes the fact that this prior gives larger weights to states of high purity.