# Comparing Hilbert spaces of coupled and uncoupled qubits

Imagine two situations. In one, there are two qubits that are next to each other, that is, they have non-zero coupling terms in their Hamiltonian, and thus suffer from cross-talk and energy can leak from one to the other. In the other situation, the same qubits are separated, unconnected.

Question: How would you describe the Hilbert spaces of these two situations? What mathematical properties would the space of the coupled qubits have that the separated qubits wouldn't have?

So the Hilbert spaces are the same in either case $$\mathbb{C^2}\otimes \mathbb{C^2}$$. What is different is not the Hilbert space, but rather the trajectories of states in these spaces, as they will depend on the unitary dynamics being driven by either a coupled or uncoupled Hamiltonian.
• it's going to depend greatly on the form of the hamiltonian. But if all the qubit 1 terms in your hamiltonian commute with all your qubit 2 terms, so that $H=H_{q1}+H_{q2}$ where $[H_{q1},H_{q2}]=0$, then your unitary $U(t)=\exp(iHt/\hbar)$ will factor into two commuting unitaries $U(t)=U_1(t)U_2(t)=U_2(t)U_1(t)$ with $U_1(t)=\exp(iH_{q1}t/\hbar)$ and $U_2(t)=\exp(iH_{q2}t/\hbar)$. Otherwise its tricky... Apr 8, 2022 at 18:26