bibkey: singhnechita2022evolution
authors: Satvik Singh; Ion Nechita
year: 2022
title: "Diagonal unitary and orthogonal symmetries in quantum theory II: Evolution operators"
doi: 10.1088/1751-8121/ac7017
url: https://arxiv.org/abs/2112.11123v1
claim: "The paper studies bipartite unitary and orthogonal evolution operators with diagonal symmetries and their entangling power; for the real case it introduces the Hadamardness measure Tr[(CC*)^2] of a sign matrix C, shows it is at least d^3 with equality exactly for Hadamard matrices, tabulates its minimum for d = 3, 5, 6, and conjectures that for every odd d the minimum is d^3 + d(d-1)."
strata_touched:
  - D5/S3/Quantum/Matrix/SinghNechitaOddHadamardnessRefutation
license: citation-only
triage: anchor

Singh and Nechita, diagonal unitary and orthogonal symmetries in quantum theory II

S. Singh and I. Nechita, arXiv:2112.11123v1 [quant-ph] (21 December 2021); J. Phys. A: Math. Theor. 55 (2022) 255302. Quotations are from the v1 TeX source, with the macros \overbar, \iden written as $\overline{\cdot }$, $I$ and cross-references omitted.

The Hadamardness measure of a sign matrix $C\in {\mathcal{M}}_d(\{\pm 1\})$:

$\mathfrak{h}(C)=\mathrm{Tr}[(C{C}^{†}{)}^2]={\sum }_{i,j=1}^d|⟨C_{i,\cdot },C_{j,\cdot }⟩{|}^2={\sum }_{i,j=1}^d{\left|{\sum }_{k=1}^d{C}_{ik}\overline{C_{jk}}\right|}^2$,

with the lemma

For a sign matrix $C\in {\mathcal{M}}_d(\{\pm 1\})$, we have $\mathfrak{h}(C)\ge d^3$, with equality being attained if and only if $C$ is a Hadamard matrix ($C{C}^{†}=d{I}_d$).

A table lists the minima $33$, $145$, $264$ for $d=3,5,6$ obtained by enumerating sign matrices. The conjecture:

We note that for the odd values of $d$ considered above, $\min \mathfrak{h}(C)=d^3+d(d-1)$, obtained for matrices $C$ with the property that all the scalar products between different works have absolute value $1$. We conjecture that this is the case for all odd integers $d$.

(“works” is the source’s wording for rows.)

Verified locator

Source: https://arxiv.org/abs/2112.11123v1

• DOI: https://doi.org/10.1088/1751-8121/ac7017 (J. Phys. A 55, 255302 (2022); author list, journal, volume and article number from Crossref).
• URL: https://arxiv.org/abs/2112.11123v1 (the e-print is a gzipped tar archive; member main.tex, 1078 lines, md5 d14ae7d4973cb1e432691fa8f590bb01):
  • the Hadamardness measure, Eq. eq:Hada_measure, l. 780–783;
  • the lemma $\mathfrak{h}(C)\ge d^3$, l. 785–792;
  • the table of minima, l. 796–846;
  • the conjecture, l. 848.
• The author list and the arXiv date are from the arXiv metadata.