What does it mean for two quantum systems to be "independent"? If I have two independent particles in QM, with state spaces $V_{1}$ and $V_{2}$, then the state space of the combined system is the tensor product $V_{1} \otimes V_{2}$. In this case it is physically clear to me that the two systems are independent, but it isn't mathematically clear to me. 
That is, if someone gives me two Hamiltonians, with corresponding state spaces, how do I know whether the combined system (with, I think, $H=H_{1}+H_{2}$) is described by the state space $V_{1}\otimes V_{2}$? It's clear to me that we need some kind of "non-interacting condition", since we have the trivial situation $H=\frac12 H + \frac12 H$ while $V \ne V \otimes V$ in general. But I can't see what the condition should be.
 A: Mathematically, the operational meaning of independent systems (i.e. described by separable pure states) is that local measurements will lead to statistically uncorrelated results. For example if we measure the operator $A_1$ for system $1$ and $A_2$ for system $2,$ then (assuming the states are normalized and $| \psi_{12}\rangle=|\psi_1\rangle \otimes |\psi_2\rangle$)
\begin{align*}
\langle A_1\otimes A_2\rangle_{\psi_{12}}=\langle \psi_{12}|A_1 \otimes A_2 |\psi_{12}\rangle &= \langle \psi_{12}|A_1\otimes \mathbb{1 } |\psi_{12}\rangle \langle \psi_{12}|\mathbb{1}\otimes A_2|\psi_{12}\rangle \\
&= \langle \psi_1 | A_1 |\psi_1\rangle \langle \psi_2 | A_2 |\psi_2\rangle \\
&= \langle A_1\rangle_{\psi_1} \langle A_2\rangle_{\psi_2}
\end{align*}
meaning the average of the product equals the product of the averages. 
As for the general case with Hamiltonians, if there's no interaction term that couples the two systems, then they will evolve independently of one another. Maybe I've misunderstood your question, in any case, let me know. 
