# How do I find the reduced density matrix of a system where two people share one qubit and have one qubit of their own?

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.

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 $$\frac{1}{2}\sum_{x=1}^4|\theta_{x}\rangle\langle\theta_{x}|.$$