Bipartite quantum system (${\mathcal{H}}_{AB}={\mathcal{H}}_A\otimes {\mathcal{H}}_B$)

A quantum system with two specified subsystems has a tensor product Hilbert space. A joint density operator determines subsystem $A$'s measurement statistics through its partial trace ${\rho }_A={\mathrm{Tr}}_B{\rho }_{AB}$. A joint pure state need not give a pure reduced state: the Bell state $(|00⟩+|11⟩)/\sqrt{2}$ gives ${\rho }_A=I/2$.