Cazzandro @KodeGhinn@qoto.org

A large part of the mathematics and physics I've put on social media consists of notes to myself that I wish I'd read years ago. Things I didn't get when I first studied them.

Here's one from quantum mechanics:

Suppose you have a (normalised) two qubit state

ψ = a|00⟩+b|01⟩+c|10⟩+d|11⟩.

Now suppose you can do whatever you like to the first bit - apply any unitary transformation, make any observation, but the second bit is inaccessible (maybe it flew out the lab or fell through an event horizon).

Then I think I'm correct in saying that (the pure state) ψ is indistinguishable from the (mixed) "state" where you have

probability |a|²+|c|² of having state a|0⟩+c|1⟩ and

probability |b|²+|d|² of having state b|0⟩+d|1⟩.

Examples of where this is important:

1. When I was messing with quantum computer simulation code years ago people said that my code was incomplete because it didn't handle mixed states. Yet I could correctly compute outcomes for every (small enough) simulation. I think simulating mixed states is a (sometimes very large) convenience rather than a necessity. Given a mixed state you can add "ancilla" bits to embed it as a pure state.

2. When I read Wald's book on QFT on curved spacetimes I was confused at the final step that derives Unruh radiation. At one step in the working a pure state was replaced by a mixed state. It took me a long time to see why that's OK. (Having said that, I know physicists have debated over whether there "really is" an evolution from a pure to mixed state and that makes absolutely no sense to me.)

And the density matrix representation of states is cool because indistinguishable states get mapped (I think) to the same thing.

Useful discoveries made by allowing smoking in the lab #1