Naimark extension for the single-photon canonical phase measurement

Understand

We address the implementation of the positive operator-valued measure (POVM) describing the optimal M-outcomes discrimination of the polarization state of a single photon.

• Initially, the POVM elements are extended to projective operators by Naimark theorem, then the resulting projective measure is implemented by a Knill-Laflamme-Milburn scheme involving an optical network and photon counters.
• We find the analytical expression of the Naimark extension and the detection scheme that realise it for an arbitrary number of outcomes M = 2^N.