  In this paper, we consider the local unitary classification of the class of
qudit bipartite mixed states for which no information can be obtained locally.
These states are represented by symmetrical density matrices in which both
tracial states are maximally mixed. Interestingly, this symmetry facilitates
the local unitary classification of two-qubit states. However, the same
formalism fails in the case of systems of higher dimensions. We consider a
broader set of states by introducing a family of qudit bipartite mixed states
with maximally mixed marginals. For this family of states, we determine several
constants which are in variant under local unitary transformations so can be
used for entanglement classification. Finally, we consider the two-qutrit case
and in particular, a two-parameter family of states for which the local unitary
classification is complete. We relate this classification to known entanglement
measures such as purity and negativity.
