Generalized Born Rule and Entanglement

Let’s explore the generalized Born rule and find the probabilities for my measurement results given that Alice has observed spin-up. After Alice’s observation, we have new information, and that means that the appropriate quantum state vector for the overall system is, from this equationEq_9_17, as follows:

$$\ket{S\text{ after }\uparrow_A} = \underbrace{\ket{\uparrow_A}}_{\text{Alice's state}} \otimes \underbrace{\frac{c\ket{\uparrow_B} + d\ket{\downarrow_B}}{\sqrt{c^2+d^2}}}_{\text{Bob's state}}.$$

In other words, Alice’s qubit is now described by the state $\ket{\uparrow_A}$ (the basis state associated with her measurement outcome), which we factored out in the above equation, and mine is described by the remaining part of that equation.

An important observation: After a measurement on one of the qubits, the system state is no longer an entangled state. For example, the system state in the above equation is a simple product state, not an entangled state. Measurement crushes entanglement!