## Outer Products in Quantum Probability

### Example 1: Quantum Logic

1. Start with logic. Something is either true 1 or false 0. No in-betweeners or outsiders today.
2. Now add probability. It has A% chance of being true 1 and B% chance of being false 0. A+B=100%
3. Actually, make that quantum probability. A and B are complex numbers (“amplitudes”) and still sum to 1.
4. So you have a vector with two possibilities True and False, both with probatilities in the unit disk that pair nicely. It represents a quantum state S.
$\dpi{300} \bg_white \begin{matrix} \mathbf{S} = \left( \! \begin{smallmatrix} A \\ B \end{smallmatrix} \! \right), \quad A,B \in \mathbf{D} \subset \mathbf{C}, \quad \Small{|\mathbf{S}|=1} \end{matrix}$
5. Now add probability again. Thought it was already in there from step 2? That was just the uncertainty principle telling us that the most fundamental state of matter has quantum-probability amplitudes to it.

Here I’m looking for uncertainty among quantum states. In other words it could be in a quantum state that’s 25% True and 75% False, call that state X. Or it could be in a quantum state that’s 0% True and 100% False, call that state Y.* What’s the chance of X being the case and the chance of Y being the case?
6. To find out use the outer product. For some state 2-vector S, takes its outer product with itself SSᵀ, and you get a 2⨯2 matrix. If there were seven possibilities you would have a 7⨯7.

Now you get to represent the probability of a couple different quantum-superposition states. Let’s say superposition state X has 10% chance, superposition state Y has 60% chance, and superposition state Z has 30% chance.
$\dpi{300} \bg_white 10\% \cdot \mid \!\mathbf{X} \rangle + 60\% \cdot \mid \!\mathbf{Y} \rangle + 30\% \cdot \mid \!\mathbf{Z} \rangle$
If you average together {the outer product of each with itself}, averaging by weight, you get a sensible matrix representation of the whole phenomenon I described above.

.1 XXᵀ  + .6 YYᵀ  + .3 ZZ= good, useful matrix

And instead of taking 10 paragraphs to describe, it just fills up a square.

* I thought you weren’t doing in-betweener truth values! I started with regular logic’s True and False only. I could have started with True, False, and Unsure. Or I could have started with True, False, Unsure, and N/A. Or I could have started with True, False, Maybe, Kinda, Almost Totally, I’m Not Sure, and N/A

That last case has 7 options so my basic quantum states from step 3 would have comprised 7-vectors. A, B, C, D, E, F, G in the state vector S and |S|=1. And that would be just ONE truth value of ONE entity.

Back to what’s above, state X and state Y are quantum superpositions of regular truth values T and F.

(Source: scottaaronson.com)

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