Optimal waveform estimation for classical and quantum systems via time-symmetric smoothing

Classical and quantum theories of time-symmetric smoothing, which can be used to optimally estimate wave forms in classical and quantum systems, are derived using a discrete-time approach, and the similarities between the two theories are emphasized. Application of the quantum theory to homodyne phase-locked loop design for phase estimation with narrowband squeezed optical beams is studied. The relation between the proposed theory and weak value theory of Aharonov et al. is also explored.

Figure 1

(Color online) Four classes of estimation problems, depending on the observation time interval relative to $\tau$, the time at which the signal is to be estimated.

Figure 2

(Color online) The classical smoothing problem.

Figure 3

(Color online) Schematic of the hybrid classical-quantum smoothing problem.

Figure 4

(Color online) Homodyne phase-locked loop (PLL) for phase estimation with a narrowband squeezed optical beam produced from an optical parametric oscillator (OPO).

Figure 5

(Color online) The quantum smoothing problem.