The problem of estimating a generic phase-shift experienced by a quant
um state is addressed for a generally degenerate phase-shift operator.
The optimal positive operator-valued measure is derived along with th
e optimal input state. Two relevant examples are analyzed: (i) a multi
-mode phase-shift operator for multi-path interferometry; (ii) the two
-mode heterodyne phase detection. (C) 1998 Elsevier Science B.V.