# Contextual bound states for qudit magic state distillation

## TL;DR

This paper reformulates qudit magic state distillation within discrete phase space, revealing the existence of bound states that are contextual but ineffective for distillation, thus refining understanding of the role of contextuality in quantum computing.

## Contribution

It demonstrates that all finite qudit magic state distillation routines include bound states outside the Wigner polytope, highlighting limitations of contextuality as a resource.

## Key findings

- Bound states exist outside the Wigner polytope.
- Such states are contextual but useless for distillation.
- Refines the understanding of contextuality's role in quantum computing.

## Abstract

Identifying necessary and sufficient conditions for universal quantum computing is a long-standing open problem for which contextuality is, perhaps, the only promising solution [Howard et al., Nature 510, 351 (2014)]. To justify this conjecture, Howard et al. showed that contextuality is equivalent to Wigner negativity for qudits, and is therefore necessary and possibly sufficient for qudit magic state distillation. Here, we reformulate magic state distillation in the language of discrete phase space, to show that any finite qudit magic state distillation routine contains bound states outside the Wigner polytope -- these states exhibit contextuality but are useless for magic state distillation.