The hypotheses considered here are chosen so that they share the same center of brightness. This is compatible with a standard exoplanet detection scenario, in which the only prior knowledge on the system is the position of its center [40]. In some other works, a different binary decision problem is formulated, in which the position of the star is known and common in the two hypotheses, while the centers of brightness are slightly offset [36]. This different choice of hypotheses leads to a different scaling in $d_a$ of the direct imaging relative entropy since the two hypotheses can be distinguished more easily via their offset centres. In Sec. S1 of Supplemental Material we discuss more in depth this discrepancy.

Notice that the two sources, if present, might not be aligned with the orientation of the demultiplexer, yielding nonvanishing and different probabilities of observing photons in modes ${\mathrm{HG}}_{01}$ and ${\mathrm{HG}}_{10}$, proportional to the projections of the relative position of the exoplanet onto the demultiplexer reference frame. Measuring these probabilities separately is not required to reach quantum optimality. However, if the position of the exoplanet relative to the star is unknown, observing only one between the ${\mathrm{HG}}_{01}$ and ${\mathrm{HG}}_{10}$ would discard useful information and, in the worst case scenario, render the measurement completely blind to the presence of the exoplanet.

The effective crosstalk matrix generally depends on the physical crosstalk between the modes of the demultiplexer due to manufacturing imperfections and on the properties of the imaged system. We present in Sec. S3 of Supplemental Material a model to retrieve the effective crosstalk from the instrument crosstalk.

We recently came across a preprint article performing an on-sky experimental realization of parameter-dependent asymmetric hypothesis testing using less precise double clad fiber as a binary mode sorter [50]. Beside demonstrating the timeliness of our results, this work showcases the feasibility of on-sky SPADE measurements, and it is a further evidence that SPADE can achieve 1 or 2 order-of-magnitude increases in performance compared with direct imaging, as we demonstrate in our experiment, if state-of-the-art detection setups and optimal hypothesis testing are employed with real observational data.