On an extension problem for density matrices

We investigate the problem of the existence of a density matrix rho(123) on a Hilbert space H-1 circle times H-2 circle times H-3 with given partial traces rho(12) = Tr-3 rho(123) and rho(23) = Tr-1 rho(123). While we do not solve this problem completely, we offer partial results in the form of some necessary and some sufficient conditions on rho(12) and rho(23). The quantum case differs markedly from the classical (commutative) case, where the obvious necessary compatibility condition suffices, namely, Tr-1 rho(12) = Tr-3 rho(23). (C) 2013 AIP Publishing LLC.