Metadata
bibkey: lapiana2025graphquantummagicsquares
authors: Francesca La Piana
year: 2025
title: Graph Quantum Magic Squares and Free Spectrahedra
doi: 10.48550/arXiv.2512.08797
url: https://arxiv.org/abs/2512.08797v3
claim: Question 1.3 asks whether graph quantum permutation matrices generate all graph quantum magic squares; Section 4.1 names C5 as a candidate for strict containment.
strata_touched:
- D5/S3/Quantum/LaPianaGraphQuantumMagicSquares
license: citation-only
triage: anchor
Verified locator
DOI: 10.48550/arXiv.2512.08797
Source: https://arxiv.org/abs/2512.08797v3
Statements
Question 1.3, page 3: “Given a graph Γ, does the matrix convex hull of its quantum permutation matrices P^(Γ) generate all graph quantum magic squares? That is, mconv(P^(Γ)) = M^(Γ)?”
Definition 2.1, page 4: “A quantum magic square of external size n and internal size s is a block matrix” with “Aij ∈ Psds(ℂ),” “such that, for every i, j,” the displayed row and column sums equal Is. The displayed relations are Σk=1ⁿ Aik = Is and Σk=1ⁿ Akj = Is.
Definition 2.3, page 5: “We say that R is matrix convex if” the displayed normalized blockwise sums belong to Rt. The normalization is Σi ViVi = It, and the new block matrix is Σi ViA^(i)Vi. “Here the conjugation acts blockwise” and “Equivalently, we say that R is closed under matrix-valued convex combinations.”
Definition 2.3, page 5: “For a subset S ⊆ Matn(Hers(ℂ)), the matrix convex hull of S, denoted as mconv(S), is the smallest matrix convex set containing S.”
Definition 3.1, page 8: “A quantum magic square X = (Xij)1≤i,j≤n ∈ Matn(Hers(ℂ)) is called a graph quantum magic square if” X(Is ⊗ AΓ) = (Is ⊗ AΓ)X. “This commutation condition can be written in terms of internal blocks as” Σk∼j Xik = Σk∼i Xkj “∀ i, j ∈ {1, . . . , n},” “where k ∼ j means that vertices k and j are adjacent in Γ.”
Definition 3.2, pages 8–9: “Let Γ be a finite graph with adjacency matrix AΓ, and let s ≥ 1. We define the following sets of graph quantum magic squares:” M^(Γ)s := { X ∈ M^(n)s | X(Is ⊗ AΓ) = (Is ⊗ AΓ)X } and P^(Γ)s := { X ∈ M^(Γ)s | Xij² = Xij = Xij* for all i, j }.
Section 4.1, page 20: “A natural class of candidates is vertex-transitive or symmetric graphs, such as the cycle C5, the Petersen graph, and, more generally, any k-regular graph with a large automorphism group.” The displayed conjecture is mconv(P^(Γ)) ⊊ M^(Γ) for various graphs Γ.
Encoding and boundary
The external indices are Fin n, the internal indices are Fin s, positivity is Matrix.PosSemidef, and the conjugate transpose is Matrix.conjTranspose. Graph adjacency is SimpleGraph.Adj. The smallest matrix convex family is expressed by membership in every matrix convex family containing the original one. Natural-number sizes include the unique empty size-zero level; the separating square has internal size two. The graph in the settling proposition is SimpleGraph.cycleGraph 5.
The C5 strict-containment statement is a repository-derived answer to the source’s question. It uses the block matrix with entries (1/25)[[5+a(i−j), b(i+j)], [b(i+j), 5−a(i−j)]], where a=(3,−3,0,0,0), b=(4,−4,0,0,0), and indices are modulo five. A decomposition into positive circulant and anti-circulant summands is obstructed by the distinct kernels of four rank-one blocks.
The universal odd-cycle extension, even cycles of size at least six, the Petersen graph and general vertex-transitive graphs are not concluded by this module. The source’s C4 counterexample and free-spectrahedral results remain separate results. Completeness of the literature exclusion is ASSUMED-UNVERIFIED.