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.Q:

Quantum probability difference between Born Rule and Born Sum Rule

If you know the probabilities for the measurement outcomes of two measurements on a system, the Born Rule tells you that the sum is the probability for the first measurement and the difference is the probability for the second measurement.
$$ P_m = \sum_{n} P_{m n} $$
The Born Sum Rule tells me that
$$\sqrt{P_{m n}} = \sum_{n} |P_{m n}| $$
Question:
If I had e.g. two measurements with outcomes $+1$ and $-1$, then there are only two possible outcomes. Then according to the Born Rule the probability for that measurement is given by $\sqrt{1 \cdot 0.5 + 0.5 \cdot 1} = 0.5$, but according to the Born Sum Rule it is $1$. Do you know any rules explaining that a double sum leads to the correct result?

A:

The reason for the difference between the two expressions is that they are dealing with different sets of probabilities.
The Born rule is based on the notion of being uncertain about the state of the system. Suppose the system is initially in a mixed state
$$\rho = \frac{1}{2} \left( |\psi_+\rangle \langle \psi_+| + |\psi_-\rangle \langle \psi_-| \right),$$
where $|\psi_{\pm}\rangle$ are eigenvectors of the operator that you measure, say the Pauli spin matrix $\sigma_z$. When you measure $\sigma_z$ with eigenvalue $+1$ (with probability $1/2$), you acquire information about the state of the system. The system is now in the state $|\psi_+\rangle$, and so the state of the system after

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