Addition modulo 2 notation (Quantum Computing) I am studying Quantum Computing using the book Quantum Computation and Quantum Information of Michael Nielsen and Isaac Chuang. I am confused by the meaning of the $\oplus$ (addition modulo 2) symbol in the notation.

Given a starting state $|\psi\rangle|\phi\rangle$ and a unitary gate $U$ such that $U|\psi\rangle|\phi\rangle = |\psi\rangle|\phi \oplus f(\psi)\rangle$. What does in this case $\oplus$ mean, in the general case for $|f(\psi)\rangle = \alpha_0 |0\rangle + \alpha_1|1\rangle$ and $|\phi\rangle = \beta_1 |0\rangle + \beta_2 |1\rangle$?
The definition shows how $U$ is defined on the standard basis (that is, when $\psi$ and $\phi$ are pure states). You can extend it to general states in a linear way.

Two more comments:


$\psi$ could be one qubit, but it could also be longer.
$f$ is probably a deterministic function.