## Jeremy Côté

Bits, ink, particles, and words.

# All in the Corners

As a quantum theorist, I spend a lot of time thinking about high-dimensional spaces. These are the playgrounds for quantum many-body 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 high-dimensional space that describe physically-relevant 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 entanglement1 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.

1. 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.

# The Curse of Dimensionality

If there’s one field of mathematics that everyone encounters in their daily life, I would argue that it’s combinatorics (with perhaps geometry being the other one). The rules of combinatorics cast a shadow over our lives. They affect how we make decisions and form the scaffolding for how options in our lives are displayed to us.

In this essay, I want to explore the idea which is known as the curse of dimensionality.

# Physics On A Cube

One of my favourite mathematical pieces of writing is Flatland, by Edwin Abbott Abbott (the book is in the public domain, so you can download it from Wikipedia). Published over a century ago, it’s a story1 involving residents (Flatlanders) who live in a two-dimensional world. Without giving too much of the story away (because you should seriously read it!), the inhabitants find themselves shocked when a strange shape dips into their world. That other “shape” is a sphere, which we know lives in a three-dimensional space. This confuses the residents to no end, and only a brave soul dares to push their mind further to explore the possibility of there being another dimension available.

1. Which, of course, is also a product of its time.