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: