  Quantum correlation is a key to our understanding of quantum physics. In
particular, it is essential for the powerful applications to quantum
information and quantum computation. There exist quantum correlations beyond
entanglement, such as quantum discord (QD) and measurement-induced nonlocality
(MiN) [Phys. Rev. Lett. \textbf{106}, 120401(2011)]. In [Phys. Rev. A
\textbf{77}, 022113(2008)], a subclass of PPT states so-called strong positive
partial transposition (SPPT) states was introduced and it was conjectured there
that SPPT states are separable. However, it was illustrated with examples in
[Phys. Rev. A \textbf{81}, 064101(2010)] that this conjecture is not true.
Viewing the original SPPT as SPPT up to part B, in the present paper, we define
SPPT state up to part A and B respectively and present a separable class of
SPPT states, that is the super SPPT (SSPPT) states, in terms of local
commutativity. In addition, classical-quantum (CQ) states and nullity of MiN
are characterized via local commutativity. Based on CQ states, the geometric
measure of quantum discord (GMQD) for infinite-dimensional case is proposed.
Consequently, we highlight the relation among MiN, QD(GMQD), SSPPT and
separability through a unified approach for both finite- and
infinite-dimensional systems: zero MiN implies zero QD(GMQD), zero QD(GMQD)
signals SSPPT and SSPPT guarantees separability, but the converses are not
valid.
