  This paper develops a systematic approach to realising linear detectors with
an optimised sensitivity, allowing for the detection of extremely weak signals.
First, general constraints are derived on a specific class of input-output
transfer functions of a linear detector. Then a physical realization of
transfer functions in that class is found using the quantum network synthesis
technique, which allows for the inference of the physical setup directly from
the input-output transfer function. By exploring a minimal realization which
has the minimum number of internal modes, it is shown that the optimal such
detectors are internal squeezing schemes. Then, investigating non-minimal
realizations, which is motivated by the parity-time symmetric systems, a
quantum non-demolition measurement is systematically recovered.
