Generalized measurement of the non-normal two-boson operator $Z_\gamma = a_1 + \gamma a_2^\dag$

We address the generalized measurement of the two-boson operator Z γ ≤ a1 + γa†2 which, for |γ|2 ≠ 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 analysed in detail. Results are exploited to revisit the Caves-Shapiro concept of generalized phase observable based on heterodyne detection.