Information theory and optimal phase measurement
K. R. W. Jones · Physica Scripta · 1993
Using a single mode analysis, we assess the utility of Shannon information as a robust performance criterion for phase detection. To provide a sharp test, we apply it to the Shapiro, Shepard and Wong optimal phase state. This has well-known pathologies, and is thus a good hurdle. Numerical studies show that the SSW-state information gain converges rapidly to the finite value ⟨Δ I ⟩ ∼ 0.966 bits at infinite photon number. For large photon numbers N the truncated phase state scheme yields ⟨Δ I ⟩ ∼ log 2 (2 N + 1) − 1.220 bits whereas the coherent state scheme returns ⟨Δ I ⟩ ∼ log 2 N 1/2 − 1.604 bits. Numerical studies confirm these asymptotic formulae and reveal an interesting crossover regime where the coherent state scheme proves superior to the truncated phase state scheme at low photon numbers N ⩽ 10. The anomalous behaviour of the SSW-state is studied with the aid of a simple scaling argument.