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.