Is it possible for two respectively closed systems to be entangled? Is it possible for two respectively closed systems to be entangled?
I was thinking something along the lines of,

*

*We have a 2-qubit system, prepared in e.g. a Bell state.

*Each qubit is carried off to a different laboratory, and their respective systems can be described by their own local Hamiltonian. Each qubit can evolve unitarily in its respective system.

Is that possible?
 A: Yes, this is possible if the systems start in an entangled state, and no measurements are performed on either subsystem.
The evolution of each closed system is described by its Hamiltonian $H_A, H_B$, with the global Hamiltonian given by
$$ H = H_A \otimes \mathbb{I}_B + \mathbb{I}_A \otimes H_B .$$
$$ U(t) = \mathrm{e}^{-\mathrm{i}H_A t} \otimes \mathbb{I}_B + \mathbb{I}_A \otimes \mathrm{e}^{-\mathrm{i}H_B t}  .$$
Of course this cannot create entanglement between $A$ and $B$. If the system starts in a product state $|\psi_0\rangle = |\psi_0^A\rangle|\psi_0^B\rangle$, it will remain in a product state at all times
$$ |\psi(t)\rangle = U(t)|\psi_0\rangle = \mathrm{e}^{-\mathrm{i}H_A t} |\psi_0^A\rangle\otimes \mathrm{e}^{-\mathrm{i}H_B t} |\psi_0^B\rangle .$$
On the other hand, if the initial state $|\psi_0\rangle$ is entangled, under unitary time evolution it will remain in general non-factorisable. Subsystem $A$ can be described as an isolated system with Hamiltonian $H_A$ and initial mixed state $\rho_A(0) = \mathrm{tr}_B(|\psi_0\rangle\langle\psi_0|)$. Its time evolution is given by
$$ \rho_A(t) = \mathrm{e}^{-\mathrm{i}H_A t} \rho_A(0) \mathrm{e}^{\mathrm{i}H_A t}.$$
Its purity is time-independent,
$$ \mathrm{tr}[ \rho_A(t)^2] =\mathrm{tr} \left[\mathrm{e}^{-\mathrm{i}H_A t} \rho_A(0)^2 \mathrm{e}^{\mathrm{i}H_A t}\right] = \mathrm{tr}\left[ \rho_A(0)^2\right], $$
therefore the state will remain mixed, and the two subsystems entangled.
Applications
This is an important topic in quantum computation and quantum information theory.
For example, the quantum teleportation protocol is based on the two parties sharing two qubits from an EPR pair, or Bell state, and each performing local operations on their qubit.

In particular, Alice performs operations that change her part of the state, without affecting Bob's state or the relative entanglement, until she measures her two qubits.
Quantum information
More generally, it is important to characterise which states can be reached by operating on each subsystem separately. This is formalised as evolution under local operations and classical communication (LOCC), which roughly speaking means the separate unitary evolution described above, plus measurements and classical communication between Alice and Bob.
Studying the evolution of entanglement has been an important part of this characterisation: the basic idea is that it can only decrease under LOCC. This is quite a broad topic, and has been the subject of research for over twenty years. Starting points to read more about this include this Wikipedia page, chapter 12 of Nielsen and Chuang, and the seminal paper by Nielsen.
A: The system as a whole evolves unitarily unless/until you make an observation.  Any observation on either qubit will disentangle the system.
