Parity measurements, Heisenberg-limited phase estimation, and beyond

We review the use of parity measurements to obtain Heisenberg-limited phase shift measurements from interferometry performed with maximally path entangled two mode states containing N photons, contrasting this measure with the method of coincidence detection which requires a number of such detections that scales with N. For large N, coincidence measurements will become impractical. In contrast, the parity method we propose is independent of N. In addition, with the parity approach, one can use an entanglement of a coherent state and the vacuum to approach the Heisensberg limit of phase resolution without the requirement of first producing a Fock state of large photon number. We also show in a very simple way, using Maximally entangled states and parity measurements that it is possible to go beyond the Heisenberg limit if the phase shifts are generated by a nonlinear Kerr-like interaction.