  Quantum discord is a measure based on local projective measurements which
captures quantum correlations that may not be fully captured by entanglement. A
change in the measurement process, achieved by replacing rank-one projectors
with a weak positive operator-valued measure (POVM), allows one to define weak
variants of quantum discord. In this work, we experimentally simulate the
effect of a weak POVM on a nuclear magnetic resonance quantum information
processor. The two-qubit system under investigation is part of a three-qubit
system, where one of the qubits is used as an ancillary to implement the phase
damping channel. The strength of the weak POVM is controlled by varying the
strength of the phase damping channel. We experimentally observed two weak
variants of quantum discord namely, super quantum discord and weak quantum
discord, in two-qubit Werner and Bell-diagonal states. The resultant dynamics
of the states is investigated as a function of the measurement strength.
