Computing the evolution of expectation values of the Pauli observable

Question

I have a quantum state described by $N$ qubits, and I don't know anything about this quantum state except the expectation value of the single-qubit Pauli observables of the $i$-th qubit ($\langle X_i \rangle$, $\langle Y_i \rangle$ and $\langle Z_i \rangle$). Then I apply a single-qubit quantum gate $U$ to the $i$-th qubit. I know the $2\times 2$ matrix describing this quantum gate.

Is it possible to determine $\langle X_i \rangle^f$, $\langle Y_i \rangle^f$ and $\langle Z_i \rangle^f$ after the application of $U$ by knowing $\langle X_i \rangle$, $\langle Y_i \rangle$, $\langle Z_i \rangle$ and the elements of $U$?

Could it be useful to find a way to do that?

Answer 1

$\langle X\rangle=\text{Tr}(X\rho)$. So, if we update by a single qubit unitary, $\rho\mapsto U\rho U^\dagger$, we have that $$ \langle X\rangle\mapsto \text{Tr}(XU\rho U^\dagger)=\text{Tr}(U^\dagger XU\rho). $$ Now, $$ U^\dagger XU=n_XX+n_YY+n_ZZ. $$ (Basically, the Pauli matrices with identity form a basis, but since $U^\dagger XU$ is traceless, it has no identity component.) You should be able to calculate the values $n_X,n_Y,n_Z$. This means that $$ \langle X\rangle\mapsto n_X\langle X\rangle+n_Y\langle Y\rangle+n_Z\langle Z\rangle $$