Weak Measurement

A weak measurement extracts a little information while barely disturbing the state, trading precision for gentleness.

Tuning measurement strength

Measurement is not all-or-nothing. By coupling a system weakly to a probe, one performs a weak measurement that yields only partial information and causes correspondingly small back-action. The strong projective limit and the trivial no-measurement limit are two ends of a continuum; weak measurement lives in between, parametrized by a coupling strength that sets how much the state is disturbed per unit information gained.

Formalism

A weak measurement is a generalized measurement whose operators are close to the identity, for instance M_k = sqrt(p_k)(I + epsilon A_k) for small epsilon. Each individual result is noisy and nearly uninformative, but averaging many runs recovers expectation values while leaving each system only slightly perturbed. Formally the measurement operators have broad, overlapping distributions rather than sharp projectors.

Weak values

Combining a weak measurement with pre- and post-selection yields a weak value, _w = / . Weak values can lie outside the ordinary eigenvalue range and can even be complex, which has made them a tool for amplifying small signals in precision metrology. They must be interpreted carefully: a weak value is a conditioned average over a rare post-selected subensemble, not a property any single system possessed.