bibkey: grossgoedicke2025artisanal
authors: David Gross; Paulina Goedicke
year: 2025
title: "Thirty-six officers, artisanally entangled"
doi: 10.48550/arXiv.2504.15401
url: https://arxiv.org/abs/2504.15401v2
claim: "The paper constructs doubly perfect functions and the associated two-unitary complex Hadamard matrices, characterizes quadratic exponentials as doubly perfect exactly when N and N+J have trivial kernel, proves existence on the binary phase space Z_2^{2n} for every n>1 via trace-orthonormal bases, and reports from computer experiments that kite-shaped binary matrices A=B with A_ij=1 iff i+j<=n+1 give a solution for all n not congruent to 1 modulo 3."
strata_touched:
  - D5/S3/Quantum/Information/KiteShapedDoublyPerfectForm
license: citation-only
triage: anchor

Thirty-six officers, artisanally entangled

D. Gross and P. Goedicke, arXiv:2504.15401v2 [quant-ph, cross-listed math.OA]. Quotations are from the v2 TeX source.

The standard symplectic form:

To state them, introduce the \emph{standard symplectic form} $[\vec{a},\vec{b}]={\vec{a}}^{t}J\vec{b}$, $J=\left(\begin{array}{cc}{0}_{n\times n}& {\text{Id}}_{n\times n}\\ -{\text{Id}}_{n\times n}& 0_{n\times n}\end{array}\right)$ on $V$.

The quadratic-phase criterion:

If $N\in {\text{ZZ}}_d^{2n\times 2n}$, both auto-correlations reduce to a character sum, and we get the sought-for delta functions if and only if both $N$ and $N+J$ have trivial kernel.

As remarked after Eq.~(\ref{eqn:perfect_quadratic}), the quadratic exponential $\lambda (\vec{a})={\tau }^{\vec{a}N\vec{a}}$, $\vec{a}\in {\text{ZZ}}_d^{2n},N=N^t\in {\text{ZZ}}^{2n\times 2n}$ is doubly perfect if and only if both $N$ and $N+J$ have trivial kernel over ${\text{ZZ}}_d$.

The binary existence theorem:

There are doubly perfect functions on the phase space $V={\text{ZZ}}_2^{2n}$ for every $n>1$.

The kite-shaped family (Remarks after Theorem 5):

More generally, if $A,B\in {\text{ZZ}}_2^{n\times n}$ then \cite{silvester2000determinants} $\det \left(\begin{array}{cc}A& \text{Id}\\ \text{Id}& B\end{array}\right)=\det (AB-\text{Id})$. Therefore, if $A,B$ are a pair of non-singular symmetric matrices such that $AB-\text{Id}$ is also non-singular, one obtains a suitable quadratic form by taking $N$ to be the block-diagonal matrix with $A,B$ on the main diagonal. As a concrete example, computer experiments indicate that the matrices with ``kite-shaped support’’, $A_{ij}=B_{ij}=1$ iff $i+j\leq n+1$, give a solution for all $n$ not congruent to $1$ modulo $3$.

Verified locator

Source: https://arxiv.org/abs/2504.15401v2

• URL: https://arxiv.org/abs/2504.15401v2 (source artisanal.tex, md5 29b677f0287fbbb335cd79eb4f5f33d2):
  • the symplectic form, l. 349–358;
  • the quadratic-phase criterion, l. 1032 and l. 2225–2230;
  • the binary existence theorem, l. 2253–2255;
  • the kite-shaped family, l. 2296–2316.
• DOI: 10.48550/arXiv.2504.15401 (https://doi.org/10.48550/arXiv.2504.15401; arXiv lists v1 of 21 April 2025 and v2 of 20 June 2025).