All in the Corners
As a quantum theorist, I spend a lot of time thinking about highdimensional spaces. These are the playgrounds for quantum manybody systems, and they are vast. The technical name is a Hilbert space, and it’s the space of complex vectors with the additional structure of a way to put vectors together (called an inner product).
Hilbert space is big (see “The Curse of Dimensionality”), but the usable area for quantum theory is often much smaller. This means we are stuck finding corners of a highdimensional space that describe physicallyrelevant phenomena.
A very important tool in my field is entropy. It can characterize the amount of entanglement between quantum states, which lets us talk about systems that produce behaviour far away from our usual ideas.
Recently, my supervisor and I were preparing homework for a class I help teach, and we wanted students to investigate the amount of entanglement^{1} present in a random quantum state of N qubits. This was a numerical exercise, and to complete it they needed to produce random unit vectors (because quantum states are normalized). This got me thinking about how to actually choose random unit vectors.

The measure we were interested in is called the entanglement entropy. It works like this. If you have a quantum state over a composite system (like many particles), you first divide it into two parts. Then, you can compute the reduced density matrix on one of those states, which is just a way to figure out the quantum state on one part by itself. Then, if you calculate the von Neumann entropy of that state, you get the entanglement entropy of the original quantum state. It also turns out that once you separate the composite system into two parts, it doesn’t matter which one you use to calculate the entropy. The result will be the same. ↩