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Phase-space representations of quantum error-correcting codes

Enrico Bozzetto1,2 and Jonte R. Hance2,*

  • *Contact author: jonte.hance@newcastle.ac.uk
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Phys. Rev. Research 8, 043076 – Published 21 October, 2026

DOI: https://doi.org/10.1103/7my9-zgfz

Abstract

We connect the structure theorem for quasiprobability representations to quantum error correction. For a code space invariant under a subgroup of an extended Weyl-Heisenberg group, the Brif-Mann construction gives a semifunctorial representation whenever its kernel satisfies the Stratonovich-Weyl axioms and composes on channels covering full error-correction cycles. With phase-space parity available, a kernel other than the ordinary Wigner one exists iff the code space is invariant under a lattice of displacements: Among compact-phase-space bosonic codes, this is only the ideal Gottesman-Kitaev-Preskill family; in finite dimension, this is every qudit stabilizer code, whose negative volume for odd prime dimension is the syndrome-averaged logical magic. For rotation-symmetric, one-photon, non-Pauli qudit and spin codes, only the ordinary kernel remains, its negativity reflecting the physical carrier, not logical content; a sector-graded representation built from recovery isometries restores a resource reading. We add a channel atlas, a closed-form magic lifetime under displacement noise, a Kirkwood-Dirac layer, and finite-energy corrections.

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Phys. Rev. Research 8, 043076– Published 21 October, 2026
  • Received 6 July 2026
  • Accepted 29 September 2026
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