# Is a linear combination of unitaries unitary?

Suppose you have a pure state $$\vert\psi\rangle$$. Consider the following operation.

For unitaries $$U_1$$ and $$U_2$$, one can take complex numbers $$\alpha, \beta$$ where $$|\alpha|^2 + |\beta|^2 = 1$$ and construct the operator $$\alpha U_1 + \beta U_2$$. By linearity, applying this on $$\vert\psi\rangle$$ gives us $$\alpha U_1\vert\psi\rangle + \beta U_2\vert\psi\rangle$$. Since $$U_i\vert\psi\rangle$$ is a pure quantum state, we have by linearity that $$(\alpha U_1 + \beta U_2)\vert\psi\rangle$$ is a pure quantum state.

However, $$\alpha U_1 + \beta U_2$$ is not unitary itself. So what is the channel that corresponds to this operation?

$$(\alpha U_1 + \beta U_2)|\psi\rangle$$ is not in general a pure quantum state because pure quantum states need to be normalized. A matrix is unitary if and only if it maps pure quantum states to pure quantum states. But, as you say, $$\alpha U_1 + \beta U_2$$ is not unitary.
• I see, so for any $\alpha$ with $|\alpha|\leq 1$, there exist $\beta$ that satisfy the normalization requirement for all $\vert\psi\rangle$ - is this true or does the $\beta$ depend on the choice of $\vert\psi\rangle$? Dec 16, 2023 at 12:56
• Sorry, I was asking if for every choice of $\alpha, U_1, U_2$, there exists a fixed $\beta$ that makes $\alpha U_1 + \beta U_2$ a unitary. But having thought about it, I think this claim is false. Dec 17, 2023 at 13:06