Hilbert space is big.

No, not big like the how the Earth is big compared to you. Rather, Hilbert space is astronomically big. Actually, that’s not quite right either. It’s bigger than that. I guess the best adverb I can use is that it’s mathematically big. In a Hilbert space, you tend to have a lot of room to maneuver. (To read more about that, check out my essay, “The Curse of Dimensionality”.)

In the last essay, we saw how to pick a random unit vector in a N dimensional space. This wasn’t for nothing, and now we are going to put that knowledge to use to ask the following question:

If I have a line of N qubits and randomly pick a state from the Hilbert space, what kind of entanglement will it have?

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.

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.

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.