Extend the analysis to multiple signal modes

Extend the worst-case error-exponent analysis for quantum illumination with an unknown return phase from a single bosonic signal mode to multiple modes, including the case in which the return phase differs from mode to mode.

Background

The paper restricts the signal to a single bosonic mode and treats the return phase as one unknown constant shared by all trials. In multimode settings, such as frequency-mode illumination, the phase may vary across modes.

The authors note that mode-dependent phases create a multidimensional unknown parameter and alter the structure of the worst-case evaluation. Establishing the corresponding performance limits for independent and identical measurements is therefore an explicitly unresolved extension.