  We consider the description of two independent quantum systems by a complete
atomistic ortho-lattice (cao-lattice) L. It is known that since the two systems
are independent, no Hilbert space description is possible, i.e. $L\ne P(H)$,
the lattice of closed subspaces of a Hilbert space (theorem 1). We impose five
conditions on L. Four of them are shown to be physically necessary. The last
one relates the orthogonality between states in each system to the
ortho-complementation of L. It can be justified if one assumes that the
orthogonality between states in the total system induces the
ortho-complementation of L. We prove that if L satisfies these five conditions,
then L is the separated product proposed by Aerts in 1982 to describe
independent quantum systems (theorem 2). Finally, we give strong arguments to
exclude the separated product and therefore our last condition. As a
consequence, we ask whether among the ca-lattices that satisfy our first four
basic necessary conditions, there exists an ortho-complemented one different
from the separated product.
