I have learned a few things recently in my quantum reading group. First is that there is a difference between correlation and entanglement. In retrospect this is obvious, but I hadn’t thought about it before. Classically, it is easy to make correlated random variables, but quantum entanglement is something more. I need to study this a bit more because I don’t know how to tell the two types of states apart by looking at the form of the density matrix.

Second I learned that for every mixed state, there is a pure state of some larger quantum system such that the mixed state is the restriction of the pure state.

Thirdly there was a sentence or two on how computation requires no energy. I noticed before that it does require energy to set the state of registers. Now that I think about it more, you will probably want to set (or reset) all the registers that you are going to use in your computation before you use them. This means in practice the energy requirements are going to be proportional to your space requirements. Your power requirements are also proportional to your temperature, but this is the case for every process. Any process that “consumes energy” by producing heat can be made more efficient by operating at a lower temperature, because you could always run a heat engine between your low temperature environment and your waste heat.

Fourthly, entanglement can be distilled or diluted. Somewhat entangled states can be distilled into fewer more entangled states, and entangled states can be diluted into more less entangled states. Apparently the entanglement entropy of pure states is well understood, but the entanglement entropy of mixed states isn’t so well understood yet.

I also learned something at today’s Brouwer institute seminar. Apparently bar induction contradicts the Church-Turing axiom. Bar induction asserts that every barred tree is well-founded, which seems like a nice axiom. I think it has a nice functional interpretation. The Church-Turing axiom states that every function of type ℕ ⇒ ℕ is a recursive function. It also has a reasonable functional interpretation that allows you to inspect the code that defines a given function. I don’t see how these two axioms can be incompatible. I should study this more too.

Oh, and apparently point-free topology is the same thing as constructive logic.