Skip to content
DraconiticResearch
Search Sources API
190,851 studies31 sourcesUpdated 12 min ago
PreprintarXiv licencearXiv

The twisted convolution identity and ghost r-SICs from finite quantum dilogarithms

  • Marcus Appleby
  • Steven T. Flammia
  • Gene S. Kopp
Open at sourcePDFDOICite

Abstract

Radchenko and Wheeler (RW) recently proved a finite pentagon relation for real quadratic special values of the modular quantum dilogarithm and used it to establish the rank-$1$ twisted convolution identity conjectured by the current authors. RW gave an explicit argument in the principal case and remarked that their proof holds for all rank-$1$ admissible tuples. We extend their proof to all rank-r admissible tuples and provide an explicit dictionary between the modular quantum dilogarithm and the Shintani-Faddeev modular cocycle conventions in the respective papers. Thus, we establish that, if d,r are positive integers such that r<(d−1)/2 and (d2−1)/(r(d−r)) ∈ ℤ, then there exist ghost r-SICs: i.e., configurations of d2 rank-r subspaces in ℂd that satisfy a non-Hermitian equichordal condition. Under the Stark conjecture, these configurations are Galois conjugate to Hermitian equichordal configurations called r-SICs (or rank-r SIC-POVMs).

Related

  1. Real quadratic fields and finite quantum dilogarithms I
    Danylo Radchenko, Campbell Wheeler
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXivHAL
  2. Quantum gates from the middle convolution of twisted Burau representations
    Haru Negami
    arXivPreprintCC BY-NC-SA 4.0Full textAlgebra & number theoryMathematical physicsarXiv
  3. Approximate Uniformity in Finite Convolution Models
    Nilava Metya, Satyaki Mukherjee
    arXivPreprintCC BY-SA 4.0Full textCombinatorics & discrete mathematicsProbabilityarXiv
  4. Twisted Deligne modules and abelian varieties over finite fields
    Sergey Rybakov
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Classification of finite-dimensional Nichols algebras of rank two in twisted Yetter--Drinfeld categories
    Bowen Li
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{appleby2026twisted,
  title = {{The twisted convolution identity and ghost $r$-SICs from finite quantum dilogarithms}},
  author = {Marcus Appleby and Steven T. Flammia and Gene S. Kopp},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39192},
  eprint = {2609.39192},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39192},
}
RIS
TY  - UNPB
TI  - The twisted convolution identity and ghost r-SICs from finite quantum dilogarithms
AU  - Marcus Appleby
AU  - Steven T. Flammia
AU  - Gene S. Kopp
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39192
UR  - https://arxiv.org/abs/2609.39192
AB  - Radchenko and Wheeler (RW) recently proved a finite pentagon relation for real quadratic special values of the modular quantum dilogarithm and used it to establish the rank-$1$ twisted convolution identity conjectured by the current authors. RW gave an explicit argument in the principal case and remarked that their proof holds for all rank-$1$ admissible tuples. We extend their proof to all rank-r admissible tuples and provide an explicit dictionary between the modular quantum dilogarithm and the Shintani-Faddeev modular cocycle conventions in the respective papers. Thus, we establish that, if d,r are positive integers such that r<(d−1)/2 and (d^2−1)/(r(d−r)) ∈ ℤ, then there exist ghost r-SICs: i.e., configurations of d^2 rank-r subspaces in ℂ^d that satisfy a non-Hermitian equichordal condition. Under the Stark conjecture, these configurations are Galois conjugate to Hermitian equichordal configurations called r-SICs (or rank-r SIC-POVMs).
ER  -