2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/130186We address the generalized measurement of the two-boson operator $Z_γ= a_1 + γa_2^†$ which, for $|γ|^2 \neq 1$, is not normal and cannot be detected by a joint measurement of quadratures on the two bosons. We explicitly construct the minimal Naimark extension, which involves a single additional bosonic system, and present its decomposition in terms of two-boson linear SU(2) interactions. The statistics of the measurement and the added noise are analyzed in details. Results are exploited to revisit the Caves-Shapiro concept of generalized phase observable based on heterodyne detection.8 pages, 2 figures, slightly revised versionQuantum Physicstext