I'm trying to work through a self-made exercise, which may be ill formed as a question. Any general advice in dealing with these types of problems is also much appreciated!
I'm looking at a quantum gate $U_f$ for a function $f$, that has the effect $$\sum_x \alpha_x \vert x\rangle\vert 0\rangle \mapsto \sum_x \alpha_x\vert x\rangle\vert f(x)\rangle. $$ This will in most cases be an entangled state: for instance, if $f(x) = x$, then I get what looks like a Bell state.
I want to consider a case where the first register is already maximally entangled with a third party, Eve.
One way to proceed is to write the first register as a mixed state which I obtain after tracing out Eve's part. The trouble now is, when we consider the action of the gate, that the gate entangles the two registers. I have no idea how to sort out the entanglement between Eve and the first register and the new entanglement between the first and second registers.
Alternatively, if I don't trace out Eve's register and instead implement the gate $\mathbb 1\otimes U_f$, then I'm still not sure what the outcome is. Before the gate, I have $$\sum_x \vert x \rangle_E\vert x\rangle\vert 0\rangle. $$ (I have marked Eve's register for clarity.) After the gate, I could naively write $$\sum_x \vert x\rangle_E\vert x\rangle\vert f(x)\rangle, $$ but this looks dubious to me. Particularly, this looks like Eve is now entangled with the second register but that seems wrong.
I'm not sure how entanglement monogamy fits in but I suspect my guess for the state isn't compatible with it. Can anyone clarify what's going on for me?