Phase-Space Representations of Quantum Error-Correcting Codes
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 semi-functorial 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 stabiliser 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.
Versions
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BibTeX
@misc{bozzetto2026phasespace,
title = {{Phase-Space Representations of Quantum Error-Correcting Codes}},
author = {Enrico Bozzetto and Jonte R. Hance},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.02164},
eprint = {2607.02164},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.02164},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Phase-Space Representations of Quantum Error-Correcting Codes
AU - Enrico Bozzetto
AU - Jonte R. Hance
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2607.02164
UR - https://arxiv.org/abs/2607.02164
AB - 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 semi-functorial 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 stabiliser 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.
ER -