Quantum Hypothesis Testing for Exoplanet Detection

Detecting the faint emission of a secondary source in the proximity of the much brighter one has been the most severe obstacle for using direct imaging in searching for exoplanets. Using quantum state discrimination and quantum imaging techniques, we show that one can significantly reduce the probability of error for detecting the presence of a weak secondary source, especially when the two sources have small angular separations. If the weak source has intensity $\epsilon \ll 1$ relative to the bright source, we find that the error exponent can be improved by a factor of $1/\epsilon$. We also find linear-optical measurements that are optimal in this regime. Our result serves as a complementary method in the toolbox of optical imaging, with applications ranging from astronomy to microscopy.

Figure 1

An optical imaging system (modeled as a thin lens) is used to discern between two hypotheses. Hypothesis $H_0$ is that only one source is present of intensity $N$. Hypothesis $H_1$ is that two sources are present, with total intensity $N$ and relative intensity $\epsilon /(1-\epsilon )\ll 1$. The field focused on the image screen can be measured by DI, or by applying an interferometric measurement, for example, SPADE [9] or SLIVER [10, 11].

Figure 2

Comparison between the quantum relative entropy and the relative entropy for DI. Top: the two quantities plotted vs $\epsilon$, for $s/\sigma =0.05$. The log-log scale emphasizes the different scaling for small $\epsilon$. Bottom: the quantum and classical entropy plotted vs $s/\sigma$ for different values of $\epsilon$.

Figure 3

A multimode wave guide can be used as Hermite-Gauss mode sorter [9]. The optical center of mass of the two sources, $\overline{x}$, is aligned with the center of the wave guide.