On an extension problem for density matrices
Eric A. Carlen, Joel L. Lebowitz, Élliott H. Lieb · Journal of Mathematical Physics · 2013
We investigate the problem of the existence of a density matrix ρ123 on a Hilbert space \documentclass[12pt]{minimal}\begin{document}${\mathcal {H} }_{1}\otimes {\mathcal {H} }_2\otimes {\mathcal {H} }_3$\end{document}H1⊗H2⊗H3 with given partial traces ρ12 = Tr3 ρ123 and ρ23 = Tr1 ρ123. While we do not solve this problem completely, we offer partial results in the form of some necessary and some sufficient conditions on ρ12 and ρ23. The quantum case differs markedly from the classical (commutative) case, where the obvious necessary compatibility condition suffices, namely, Tr1 ρ12 = Tr3ρ23.