# Generalized Born Rule for partial measurements

I'm looking for the mathematical formulation of this trait (from an introductory Quantum Computing course):

How can I generalize this into higher dimensions? Or other observables?

It does not seem provable from the Born rule, which only gives information about the probability amplitudes of $B$.

• What is $A\downarrow 0$ supposed to denote? And, although one can guess it, what are $\lvert 0\rangle$ and $\lvert 1 \rangle$? Is there an implicit tensor product in the notation $\lvert 0 \rangle\lvert 1 \rangle$? Oct 28 '15 at 21:35
• 1) It denotes "A collapses to the spin-down state". 2) They are the computational basis vectors corresponding to the spin-down and spin-up states. 3) Yes.
– tba
Oct 28 '15 at 22:06

When performing a measurement based on a predicate $P$, such as the first qubit $P(\left|xy\right\rangle) = x$ or the second qubit $P(\left|xy\right\rangle) = x$ or some combination like $P(\left|xy\right\rangle) = x \text{ and } y$, you generally do the same thing:
1. Let $S$ be the set of states that match P.
2. Let $Z$ be the set of states that don't match P.
3. Let $t$ be the sum of the squared magnitudes of states in $S$.
4. Let $f = 1 - t$ be the sum of the squared magnitudes of states in $Z$.
5. The predicate measurement tells you if the quantum state is in $S$ or $Z$. It returns "S" with probability $t$, and "Z" with probability $f$.
6. If the measurement returned "S", then any states in "Z" now have amplitude 0 and any states in "S" had their amplitude scaled by $\frac{1}{t}$.
7. If the measurement returned "Z", then any states in "S" now have amplitude 0 and any states in "Z" had their amplitude scaled by $\frac{1}{f}$.