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

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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 $B$.

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  • $\begingroup$ 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 $\lvert 0 \rangle\lvert 1 \rangle$? $\endgroup$
    – ACuriousMind ♦
    Oct 28, 2015 at 21:35
  • $\begingroup$ 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. $\endgroup$
    – tba
    Oct 28, 2015 at 22:06

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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 $\frac{1}{f}$.

In other words: the total squared magnitude of states matching each measurement result tell you how likely that measurement result is. After you know the result, you renormalize the surviving amplitudes so that the total squared magnitude is 100% again.

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