  We propose "super one-way deficit" by weak measurement as the generalization
of one-way deficit defined for standard projective measurement. The super
one-way deficit for Werner state is obtained analytically. We find that super
one-way deficit is smaller than the standard one-way deficit, which contrasts
with a straightforward expectation based on the known fact that super-quantum
discord by weak measurement is always larger than the quantum discord defined
by projective measurement. On the other hand, by tuning the weak measurement
continuously to the projective measurement, both super one-way deficit and
super-quantum discord converge to the same value, which is either the one-way
deficit or the quantum discord both quantifying quantum correlation. In this
sense, weak measurement does not necessarily capture more quantumness of
correlations. We also give the geometry of super- quantum discord of the
Bell-diagonal states with explicit geometrical figures. As an application, the
dynamic behavior of super- quantum correlation including super-quantum discord
and super one-way deficit under decoherence is investigated. We find that the
order relation of the super-quantum correlation and the quantum correlation
keep unchanged under the phase flipping channel for the Bell-diagonal states
and the Werner states.
