  Recently [Karimipour and Memarzadeh, Phys. Rev. A 73, 012329 (2006)] studied
the problem of finding a family of orthonormal bases in a bipartite space each
of dimension $D$ with the following properties: (i) The family continuously
interpolates between the product basis and the maximally entangled basis as
some parameter $t$ is varied, and (ii) for a fixed $t$, all basis states have
the same amount of entanglement. The authors derived a necessary condition and
provided explicit solutions for $D \leq 5$ but the existence of a solution for
arbitrary dimensions remained an open problem. We prove that such families
exist in arbitrary dimensions by providing two simple solutions, one employing
the properties of quadratic Gauss sums and the other using graph states. The
latter can be generalized to multipartite equientangled bases with more than
two parties.
