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The generalized Born rule
The Generalized Bron rule is an extension of the standard Born rule in quantum computing, and it is particularly relevant in the context of quantum computation. It provides the probability distribution of obtaining outcomes from measurements on quantum states. In more complex quantum system, it describe how quantum state evolve and change. The generalized Born rule can apply to post-measuring states and systems interacting with environment.

Details
This rule comes into play when only a single qubit from a set of n+1qubits is measured using a standard 1-qubit measurement gate. To understand this, consider the general state of n+1qubits, which can be expressed as: $$ \quad |\alpha_0|^2 + |\alpha_1|^2 = 1 $$ Here, $$\alpha_0$$ and $$\alpha_1$$ are complex coefficients satisfying $$|\alpha_0|^2 + |\alpha_1|^2 = 1$$, ensuring the state is normalized. The generalized Born rule specifies that when a measurement is made on the singled-out qubit, the outcome will either be 0 or 1. The probability of each outcome is given by the squared magnitude of the corresponding coefficient, $$|\alpha_0|^2$$ or $$|\alpha_1| ^2$$. After measurement, the state of the entire system reduces to a product state reflecting the outcome:$$|x\rangle |\phi_x \rangle_n$$where $$x$$ is the result of the qubit measurement. $$ |\Psi_{n+1}\rangle = \sum_{x=0}^{2^{n+1}-1} \gamma(x) |x\rangle_{n+1}, \quad \sum_{x=0}^{2^{n+1}-1} |\gamma(x)|^2 = 1 $$ $$|\Phi_0\rangle_n \text{ and } |\Phi_1\rangle_n \text{ are given by} \Phi_1\rangle_n = \frac{1}{\alpha_1} \sum_{x=0}^{2^n-1} \gamma(2^n + x) |x\rangle_n $$ $$ \alpha_0^2 = \sum_{x=0}^{2^n-1} |\gamma(x)|^2, \quad \alpha_1^2 = \sum_{x=0}^{2^n-1} | \gamma(2^n + x)|^2 $$
 * \psi \rangle_{n+1} = \alpha_0 |0\rangle |\phi_0 \rangle_n + \alpha_1 |1\rangle |\phi_1 \rangle_n,
 * \Phi_0\rangle_n = \frac{1}{\alpha_0} \sum_{x=0}^{2^n-1} \gamma(x) |x\rangle_n, \quad |

This rule implies that if the measured qubit is initially unentangled with the remaining n qubits, the measurement are calculated by the ordinary Born rule and the unmeasured qubits remain in their original state. In cases where multiple qubits are measured sequentially using 1-qubit gates, the generalized Born rule applies independently to each measurement, simplifying the analysis of such process.

Example
Suppose we have a quantum system of two qubits initially in an entangled state given by $$|\Psi\rangle = r_0|00\rangle + r_1|01\rangle + r_2|10\rangle + r_3|11\rangle$$ $$r_0, r_1, r_2,$$ and $$r_3$$ are complex numbers representing the amplitudes of the qubits being in states $$|00\rangle, |01\rangle, |10\rangle,$$ and $$|11\rangle$$ respectively. The sum of the squares of the magnitudes of these coefficients equals 1, ensuring that the total state is normalized. For each subgroup, define a normalized state corresponding to the second qubit: $$|\Phi_0\rangle = \frac{1}{\alpha_0} (r_0|0\rangle + r_1|1\rangle), \quad \alpha_0 = \sqrt{| r_0|^2 + |r_1|^2}$$ $$|\Phi_1\rangle = \frac{1}{\alpha_1} (r_2|0\rangle + r_3|1\rangle), \quad \alpha_1 = \sqrt{| r_2|^2 + |r_3|^2}$$ $$|\Psi\rangle = \alpha_0|0\rangle|\Phi_0\rangle + \alpha_1|1\rangle|\Phi_1\rangle$$ $$\text{Pr(first qubit at 0)} = |\alpha_0|^2 = |r_0|^2 + |r_1|^2$$

Upon measurement, if the first qubit is found to be in state $$|0\rangle$$, the system collapses to: $$|\Psi\rangle \rightarrow |0\rangle|\Phi_0\rangle = \left(\frac{r_0|00\rangle + r_1|01\rangle} {\sqrt{|r_0|^2 + |r_1|^2}}\right)$$