### Example 1: Quantum Logic

- Start with logic. Something is either true
**1**or false**0**. No in-betweeners or outsiders today. - Now add probability. It has
**A%**chance of being true**1**and**B%**chance of being false**0**.**A**+**B**=**100%** - Actually, make that quantum probability.
**A**and**B**are complex numbers (“amplitudes”) and still sum to 1. - 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**.

- 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? - 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**YY****ᵀ +**.3**ZZ****ᵀ**=**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)