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I have the following problem and have attempted to find a solution to it, but to no avail.

Alice and Bob have one qubit each, say $|\psi\rangle$ with Alice and $|\phi\rangle$ with Bob. They also share an entangled qubit state $$\beta_{00}=\frac{|00\rangle+|11\rangle}{\sqrt{2}}.$$ Bob then applies a unitary operator $U$ to both of his qubits. After that, Alice makes a measurement in the Bell basis on both her qubits.

How do I compute the reduced density matrix of Bob's qubits now? I'm not sure whether to take the tensor product of all three qubits as my starting point and apply $U$ to only the latter two terms. I don't think this is right. I'm having trouble figuring out how to even approach the problem. Any help would be appreciated.

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There are 4 qubits, and the total state can be written as $$ |\theta\rangle = |\phi\rangle \otimes |\beta_{00}\rangle \otimes |\psi\rangle = \frac{|\phi\rangle|00\rangle|\psi\rangle+|\phi\rangle|11\rangle|\psi\rangle}{\sqrt{2}}, $$ where the first two qubits belong to Bob. He then applies operation $U \otimes I$ on the whole system, so the state becomes $(U \otimes I)|\theta\rangle$.

Alice's measurement corresponds to the decomposition of identity $I$ on her qubits into a sum of projections $P_{1} = |B_{00}^+\rangle\langle B_{00}^+|$, $P_{2} = |B_{00}^-\rangle\langle B_{00}^-|$, $P_{3} = |B_{01}^+\rangle\langle B_{01}^+|$, $P_{4} = |B_{01}^-\rangle\langle B_{01}^-|$ on 4 Bell basis states: $$ P_{1} + P_{2} + P_{3} + P_{4} = I. $$

In essence, she applies a random projection $I \otimes P_{x}$ on the whole system. The resulting state will be $$ |\theta_{x}\rangle = \frac{(I \otimes P_{x}) (U\otimes I)|\theta\rangle} {||(I \otimes P_{x}) (U\otimes I)|\theta\rangle||} $$ for a random index $x$. If we don't know the result $x$, then we should assume that the resulting state is a mixture of $|\theta_{x}\rangle$, i.e. the mixed state with density matrix $$ \frac{1}{2}\sum_{x=1}^4|\theta_{x}\rangle\langle\theta_{x}|. $$

It's not clear what's the case in the formulation. Anyway, we then can compute partial trace (over Alice's qubits) of the density matrix to find out the reduced state of Bob's qubits.

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