In the quantum world, things are often weird. One of the classic sources of weirdness is entanglement, the “spooky action at a distance” that connects particles no matter how far apart they are. But there’s a lesser-known, yet equally fascinating, type of weirdness called “quantum nonlocality without entanglement.” This phenomenon occurs when you have a set of unentangled (product) quantum states that are, surprisingly, impossible to tell apart perfectly if you’re restricted to local operations.

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: