  Weak values and measurements have been proposed as means to achieve dramatic
enhancements in metrology based on the greatly increased range of possible
measurement outcomes. Unfortunately, the very large values of measurement
outcomes occur with highly suppressed probabilities. This raises three vital
questions in weak-measurement-based metrology, namely, (Q1) Does post-selection
enhance the measurement precision? (Q2) Does weak measurement offer better
precision than strong measurement? (Q3) Is it possible to beat the standard
quantum limit or to achieve the Heisenberg limit with weak measurement using
only classical resources? We analyse these questions for two prototypical, and
generic, measurement protocols and show that while the answers to the first two
questions are negative for both protocols, the answer to the last is
affirmative for measurements with phase-space interactions, and negative for
configuration space interactions. Our results, particularly the ability of weak
measurements to perform at par with strong measurements in some cases, are
instructive for the design of weak-measurement-based protocols for quantum
metrology.

Quantum Physics