  The multi-terminal rate-distortion problem has been studied extensively.
Notably, among these, Tung and Housewright have provided the best known inner
and outer bounds for the rate region under certain distortion constraints. In
this paper, we first propose an outer bound for the rate region, and show that
it is tighter than the outer bound of Tung and Housewright. Our outer bound
involves some $n$-letter Markov chain constraints, which cause computational
difficulties. We utilize a necessary condition for the Markov chain constraints
to obtain another outer bound, which is represented in terms of some
single-letter mutual information expressions evaluated over probability
distributions that satisfy some single-letter conditions.
