  The weak value, the average result of a weak measurement, has proven useful
for probing quantum and classical systems. Examples include the amplification
of small signals, investigating quantum paradoxes, and elucidating fundamental
quantum phenomena such as geometric phase. A key characteristic of the weak
value is that it can be complex, in contrast to a standard expectation value.
However, typically only either the real or imaginary component of the weak
value is determined in a given experimental setup. Weak measurements can be
used to, in a sense, simultaneously measure non-commuting observables. This
principle was used in the direct measurement of the quantum wavefunction.
However, the wavefunction's real and imaginary components, given by a weak
value, are determined in different setups or on separate ensembles of systems,
putting the procedure's directness in question. To address these issues, we
introduce and experimentally demonstrate a general method to simultaneously
read out both components of the weak value in a single experimental apparatus.
In particular, we directly measure the polarization state of an ensemble of
photons using weak measurement. With our method, each photon contributes to
both the real and imaginary parts of the weak-value average. On a fundamental
level, this suggests that the full complex weak value is a characteristic of
each photon measured.
