A Structure Theorem for 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, only the ideal Gottesman-Kitaev-Preskill family; in finite dimension, 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.

32+11, no figures. Extended to DV QECCs, among many other things

arXiv:2607.02164 [quant-ph]

(or arXiv:2607.02164v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.02164 Focus to learn more arXiv-issued DOI via DataCite

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