What is the meaning of measuring a Bell state with Pauli operators?

Question

There will be a certain value of getting the probability when measuring any Bell's state with Pauli operators such as observable X, Y, or Z. What is the meaning behind all this measurement? the result of the measurement will show or prove what? Are they any theory or explanation behind all the measurements?

Answer 1

Postulates of Quantum Mechanics:

1. The state of quantum mechanical system, including all the information you can know about it. It is represented by a normalized ket $|\psi\rangle$
2. A physical observable is represented mathematically by an operator $A$ that acts on kets.
3. The only possible result of a measurement of an observable is one of the eigenvalues $\lambda_n$ of the corresponding operator $A$.
4. The probability of obtaining the eigenvalue $\lambda_n$ in a measurement of the observable $A$ on the system in the state $|\psi \rangle$ is $|\langle \lambda_n | \psi \rangle|^2 $ where $|a_n\rangle$ is the normalized eigenvector of $A$ corresponding to the eigenvalue $a_n$.
5. After measurement of $A$ that yields the result of $a_n$, the quantum system is in a new state that is normalized projection of the original system ket onto the ket (or kets) corresponding to the result of the measurement: $$|\psi_{post} \rangle = \dfrac{P_n |\psi \rangle}{ \sqrt{ \langle \psi | P_n | \psi \rangle} } $$
6. (for completemess, I will add this postulate here as well) The time evolution of a quantum system is determined by the Hamiltonian or total energy operator $H(t)$ through the Schrodinger equation $$ i\hbar \dfrac{d}{dt} |\psi(t) \rangle = H(t)|\psi(t) \rangle $$

The postulates of quantum mechanics dictates how one treat a quantum mechanical system mathematically and how to interpret the mathematics to learn about the physical system in question. These postulate cannot be proven, but hey have successfully tested by many experiments, and so we accept them as an accurate way to describe quantum mechanical systems.

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Reference: Quantum Mechanics by David H. McIntyre.