quantum mechanics - What happens when measuring a subsystem of a product bipartite state?

I have an horrible doubt, as I found that in my books the question is never directly addressed. How is collapse defined to work for measures of composite system?

I always assumed that when we measure on a subsystem $I$ of a quantum system $I\otimes J$ we collapse the whole system unless the state is separable. For example if we have a state $\frac{1}{\sqrt{3}}\left|00\right\rangle +\frac{1}{\sqrt{3}}\left|01\right\rangle +\frac{1}{\sqrt{3}}\left|11\right\rangle$ and we measure the first q-bit finding $0$ we can say the state is collapsed in $\left|00\right\rangle$ or $\left|01\right\rangle$ with 50% probability each. The state can't be $\left|0\right\rangle(\frac{1}{\sqrt{2}}\left|0\right\rangle+\frac{1}{\sqrt{2}}\left|1\right\rangle)$ or something. It's either $\left|00\right\rangle$ or $\left|01\right\rangle$ am I right?

Answer

No -- the other option you propose is right: The result is the state obtained by applying $|0\rangle\langle0|\otimes1\!\!1$. In you example, this is $|0\rangle\otimes(|0\rangle+|1\rangle)/\sqrt{2}$.

Note that this is the only thing which makes sense if you don't want to have a preferred basis on the unmeasured system -- the scheme you propose is clearly basis-dependent.