  A quasi-entropy is constructed for tensors averaged by a density function on
$SO(3)$ using the log-determinant of a covariance matrix. It serves as a
substitution of the entropy for tensors derived from a constrained minimization
that involves integrals. The quasi-entropy is an elementary function that
possesses the essential properties of the original entropy. It constrains the
covariance matrix to be positive definite, is strictly convex, and is invariant
under rotations. Moreover, when reduced by symmetries, it keeps the vanishing
tensors of the symmetry zero. Explicit expressions are provided for axial
symmetries up to four-fold, as well as tetrahedral and octahedral symmetries.
The quasi-entropy is utilized to discuss phase transitions in several systems.
The results are consistent with using the original entropy. Besides, some novel
results are presented.
