isomorphismes

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.
    
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.
   
   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 ZZᵀ = 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)