While quantum computers have continued to improve over the past few years, it’s a known issue that error rates cannot be made low enough simply by improvements to the hardware. One approach to this problem is the domain of quantum error correction (QEC) which promises fault-tolerant devices that properly deals with the issue of noise. While the theory of QEC is well-established, the implementation of the theory is still a while away from being physically realizable.

I created the toqito software project back in early 2020 as a pet project for delving into some research questions I wanted to pursue in quantum information but felt were lacking in the Python community. When I was a Ph.D. student, having software like toqito would have been indispensible for rapidly prototyping ideas and attacking problems numerically. I’ve been continuing to build my quantum information numerical arsenal that has been a real feather in my cap when probing into particularly thorny research questions and it’s also been a ton of fun to build and to watch grow.

Let \(n\) be an integer, let \(\mathcal{X}\) be a complex Euclidean space, let \(\rho_i \in \text{D}(\mathcal{X})\) be a pure quantum state represented as a density operator, and let $$ \eta = \left\{\left(\frac{1}{n}, \rho_1\right), \ldots, \left(\frac{1}{n}, \rho_n\right)\right\} \subset \mathcal{X} $$ be an ensemble of pure and mutually orthogonal quantum states. Define \(\eta^{\otimes 2}\) as the two-copy ensemble where $$ \eta^{\otimes 2} = \left\{\left(\frac{1}{n}, \rho_1 \otimes \rho_1\right), \ldots, \left(\frac{1}{n}, \rho_n \otimes \rho_n\right)\right\} \subset \mathcal{X} \otimes \mathcal{X}.

I’m going to discuss a conjecture that got me interested in the topic of “quantum state antidistinguishability”. Conjecture [Havlíček-Barret (2020)]: Let \(S = \{|\psi_1\rangle, \ldots, |\psi_n\rangle\} \subset \mathbb{C}^n\) be a set of \(n\) pure quantum states each of dimension \(n\). Then \(S\) is antidistinguishable when $$ |\langle \psi_i | \psi_j \rangle| \leq \frac{n-2}{n-1} $$ for all \(i \neq j\). Before unpacking this statement more formally, I want to mention some points that drew me to it:

I recently happened upon the textbook “Triangular Arrays with Applications” and decided to pick up a copy. I’m certainly no number theorist, but I still enjoy a good triangular arrangement of numbers as much as the next person. My typical strategy for understanding something mathy is to write some code and to see lots and lots of examples to develop intuition. I suppose I’m one of those people who need to burn their hands on the stove multiple times to realize that doing so is a bad idea.

There’s a scene in Cosmos (the good one with Carl Sagan) where he is at the public library in New York City. It’s a massive building filled with books. Sagan begins talking about how many books you could read if you read one book each week. He takes a few steps in front of some bookshelves he’s standing in front of and stops shortly after. The entirety of one’s reading life is contained within a few humble steps of an enormous library.