### Classical vs Quantum Computation (Week 5)

#### Posted by John Baez

Here are the notes for the latest installment of my course on Classical versus Quantum Computation:

- Week 5 (Nov. 2) - Theorem: evaluating the "name" of a morphism gives that morphism! The naturality of currying. A new "bubble" notation for currying and uncurrying. Popping bubbles to reveal the quantum world.

Last week’s notes are here; next week’s notes are here.

Last week I said that a monoidal category $C$ is **closed** if for each object $A$ there is a functor $\mathrm{hom}(A, -) : C \to C$
which is right adjoint to tensoring with $A$:
$- \otimes A : C \to C$
This means that there’s a natural isomorphism
$Hom(- \otimes A, --) \cong Hom(- , \mathrm{hom}(A, --))$
called **currying**.

Today, we actually used the naturality of this isomorphism to prove something interesting! We’ve already seen that every morphism
$f: A \to Y$
has a **name**:
$"f" : I \to \mathrm{hom}(A,Y).$
We also have an **evaluation** morphism:
$ev: \mathrm{hom}(A,Y) \otimes A \to Y$
Here we show that evaluating the name of $f$, as follows:
$A \cong I \otimes A \stackrel{"f" \otimes 1_A}{\to} \mathrm{hom}(A,Y) \otimes A \stackrel{ev}{\to} Y$
gives back $f$, as it should.

In preparing to give this proof, I realized that a slightly different string diagram notation for currying and uncurrying would be very nice. So, after giving the proof I explain this new notation. It’s copied after the usual notation for compact categories, where $\mathrm{hom}(A,Y) \cong Y \otimes A^*.$ But, it uses “bubbles” to remind us that certain parts of the picture shouldn’t be taken too literally. In the compact case we can “pop” these bubbles and our equations remain true.

The cute thing about this is that the compact case is applicable to *quantum* logic, while other sorts of monoidal closed categories (like cartesian ones) are applicable to *classical* or *intuitionistic* logic. So, our bubble diagrams make sense in all these forms of logic - but when we pop the bubbles, they only make sense in quantum logic.

So, we can “pop the bubbles to reveal the quantum world”! I’m not sure yet how important this is, or what it really means. But, it’s cool.

## Notation for currying.

You can really see the “zipper” on the top of page 7, not so well on the bottom of page 6.

It occurs to me that you don’t really need to draw either the bubbles

orthe zipper/ribbon. That is, the diagrams are unambiguous without these marks. On the other hand, drawing them in helps ensure that your diagrams are legitimate!; I suppose that this is why you keep them.