  Postive semidefinite (PSD) cone is the cone of positive semidefinite
matrices, and is the object of interest in semidefinite programming (SDP). A
computational efficient approximation of the PSD cone is the $k$-PSD closure,
$1 \leq k < n$, cone of $n\times n$ real symmetric matrices such that all of
their $k\times k$ principal submatrices are positive semidefinite. For $k=1$,
one obtains a polyhedral approximation, while $k=2$ yields a second order conic
(SOC) approximation of the PSD cone. These approximations of the PSD cone have
been used extensively in real-world applications such as AC Optimal Power Flow
(ACOPF) to address computational inefficiencies where SDP relaxations are
utilized for convexification the non-convexities. In a recent series of
articles Blekharman et al. provided bounds on the quality of these
approximations. In this work, we revisit some of their results and also propose
a new dominant bound on quality of the $k$-PSD closure approximation of the PSD
cone. In addition, we characterize the extreme rays of the $2$-PSD closure.
