Shankar on Hamiltonian Separability, Entanglement

Question

In Shankar's Principles of Quantum Mechanics, I recently came across the attached section.
I am not sure I agree with equation 10.3.38 in particular. What is clear is that, if the Hamiltonian (operator) of the universe is separable as in 10.3.37 (but promoted to operators), then a basis for the Hilbert space of the universe can be formed from direct products of eigenstates of $H_{sys}$ and $H_{rest}$, because these would be eigenstates of the Hermitian operator $H$. But I do not see why that a priori implies that the state of the universe ought to be in the form 10.3.38 (10.3.38 is of course projected into position space, but that is of no consequence to what I'm asking about). Why couldn't the universe be in an entangled state in general in this example (ie. a state that does not factorize as in 10.3.38). It seems to me that Shankar is making an extra assumption -- that we are assuming the state is not entangled -- in this case. Is my understanding correct?

Dvij D.C. · Accepted Answer

You are correct in recognizing that entanglement between two systems is different from the existence of an interaction between two systems. However, I think is a two-step argument that makes what Shankar is saying perfectly Kosher:

If two systems start out as unentangled then if there is no interaction between them (i.e., if the Hamiltonian of the full system is separable in the way Shankar has described) then the two systems would remain unentangled. In other words, a product state evolves to a product state under a separable Hamiltonian.
One can still say that our system may start out with entanglement with the rest of the universe and then it is not appropriate to do what Shankar did. However, even if you have a system that is entangled with the other parts of the universe, you can simply measure the system, and then the state of the universe will become a product state. Now, since there is no interaction between the system and the rest of the universe, you can happily study the system because you can be sure that it won't get entangled with the rest of the universe (assuming our no-interaction/separable Hamiltonian assumption is correct).

Shankar on Hamiltonian Separability, Entanglement

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