I know that if we have two qubits in states differing by a global phase it does not matter that they are in the same state.

But if we have a qubit in state_1 and another in state_2 does the global phase need to be the same for these qubits to interact with each other?


1 Answer 1


It is not always clear what one means by some set of words, so let's look at some equations. If one has two states $|\psi\rangle$ and $|\phi\rangle$, they should respectively be equivalent to $e^{i\varphi_1}|\psi\rangle$ and $e^{i\varphi_2}|\phi\rangle$. If we "put them together" by considering any tensor product, the global phases do not matter: the states $$|\psi\rangle\otimes|\phi\rangle\qquad\mathrm{and}\qquad e^{i\varphi_1}|\psi\rangle\otimes e^{i\varphi_2}|\phi\rangle=e^{i(\varphi_1+\varphi_2)}(|\psi\rangle\otimes |\phi\rangle)$$ are equivalent. Then any operation acting on the joint state will be independent from these phases, so, indeed, the "global phase" of each individual state is immaterial.

If you instead meant that you are putting the two states in superposition, such as $(|\phi\rangle+|\psi\rangle)/\sqrt{2}$, then each state's phase matters, because $$(e^{i\varphi_1}|\phi\rangle+e^{i\varphi_2}|\psi\rangle)/\sqrt{2}\neq (|\phi\rangle+|\psi\rangle)/\sqrt{2}$$ unless $e^{i\varphi_1}=e^{i\varphi_2}$.


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