Resource Theory of Aquaternionicity: Every Pure State Is a Replicable Resource for Exact Simulation of Arbitrary Quantum Operations

Abstract

An alchemical resource theory admits a nonfree state that can be exactly replicated by free operations and whose finitely many copies enable exact simulation of arbitrary quantum instruments. The only known nontrivial examples, imaginarity and parity asymmetry, exhaust the qubit case (arXiv:2609.12988) and exhibit an all-or-nothing property: any nonmaximal resource exactly simulates only free unitaries. We introduce aquaternionicity for even-dimensional qudits $d\ge4$, based on a quaternionic structure represented by an antiunitary $Θ$ with $Θ^2=-I$. It is extremal in two senses: every pure state is alchemical, and alchemicality is maximally abundant under mixing. Within this framework, for any nontrivial $d$-dimensional resource theory and pure alchemical state $g$, the admissible pure-mixing set satisfies $\dim\mathcal M_g\le 2d-4,$ which aquaternionicity saturates for every pure state. Conversely, these two extremal properties characterize the quaternionic structure. Because every pure state is alchemical, the initial resource can be chosen to match the known noise structure. Moreover, when the noisy resource states remain within a common isotropic subspace, the same free processor works exactly for all of them while returning the resource unchanged. This enables recalibration-free exact simulation, including universal quantum computation, and is realized for standard dephasing and dissipative noise, where aquaternionicity substantially outperforms imaginarity and parity asymmetry.