Generalized measurement of the non-normal two-boson operator $Z= a_1 +gamma a_2 ^{dag}$

We address the generalized measurement of a non-normal two-boson operator, which 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.