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 \(S = \{\ket{\psi_1}, \ldots, \ket{\psi_n}\} \subset \mathbb{C}^n\) 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.