Phase sensitivity of a Mach-Zehnder interferometer

The best performance of a Mach-Zehnder interferometer is achieved with the input state parallel to N-T/2+1 >parallel to N-T/2-1 >+parallel to N-T/2-1 >parallel to N-T/2+1 >, N-T being the total number of atoms or photons. This gives (i) a phase-shift error confidence C-68%=2.67/N-T with (ii) a single interferometric measurement. Different input quantum states can achieve the Heisenberg scaling similar to 1/N-T but with higher prefactors and at the price of a statistical analysis of two or more independent measurements.