# Probabilities of entangled state. Quantum measurement [duplicate]

I confused about how to calculate the PROBABILITIES and getting a certain result of measuring Bell's states with Pauli matrices as the operator. When you measure something, the state involved would be projected onto an eigenstate of the observable.

Given $$|ψ⟩ = \frac{1}{\sqrt2} (|01⟩ + |10⟩)$$ as the state and $$σy = \left(\begin{matrix}0&-i\\i&0\\\end{matrix}\right)$$ as the observable.

How to calculate probability on the first qubit actually? What is the state after measurement?

The matrix $$\sigma_Y$$ has two eigenvalues $$\pm 1$$, so the one-qubit projectors for the measurement are $$\frac12(I\pm\sigma_Y).$$ Since you're measuring a two-qubit state, you need to express the measurement projects on both qubits. Since you're doing nothing to the second qubit, that's just the identity matrix: $$P_{\pm}=\frac12(I\pm\sigma_Y)\otimes I.$$ You can check that $$P_++P_-=I\otimes I$$.
Now measurement follows the standard formalism. The probabilities of getting either outcome are $$p_{\pm}=\langle\psi|P_{\pm}|\psi\rangle$$ and the possible states after the measurement are $$|\psi_{\pm}\rangle=P_{\pm}|\psi\rangle/\sqrt{p_{\pm}}.$$