trureturing — Discovering truth

bibkey: designolle2023symmetricbell authors: Sébastien Designolle, Tamás Vértesi, Sebastian Pokutta year: 2023 title: “Symmetric multipartite Bell inequalities via Frank-Wolfe algorithms” doi: 10.1103/PhysRevA.109.022205 url: https://arxiv.org/abs/2310.20677v3 claim: “Section V conjectures that the symmetrised local polytope is affinely equivalent to the cross-polytope in dimension ceil(m/2).” strata_touched:

• D5/S3/Quantum/Entanglement/SymmetricBellPolytopeNotCrossPolytope
• D5/S3/Quantum/Entanglement/SymmetricBellPolytopeSupport license: citation-only triage: anchor

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DOI: 10.1103/PhysRevA.109.022205

Source: https://arxiv.org/abs/2310.20677v3

Journal: Physical Review A 109, 022205 (2024).

Source statement

Section V, PDF p. 4, states:

For all these examples, the number of facets is equal to $2^{\lceil m/2\rceil }$. This seems to indicate that the symmetric polytope is affinely equivalent to the cross-polytope in dimension $\lceil m/2\rceil$. Although we considered the orbits giving rise to the extreme points of this polytope and tried to infer a generalisable pattern, we could not establish this fact. We conjecture, however, that it holds in general, and hope that further research will identify these general extreme points.

The deterministic strategies and their local polytope are defined in Eqs. (1)–(2):

$$d_{x_1\dots x_N}^{{\vec{a}}^{(1)}\dots {\vec{a}}^{(N)}}:=\prod _{n=1}^N{a}_{x_n}^{(n)},{\mathcal{L}}_N^{(m)}:=\mathrm{conv}\{{\mathbf{d}}^{{\vec{a}}^{(1)}\dots {\vec{a}}^{(N)}}\}.$$

Equation (25) gives the four signed permutation generators: swap the first two parties, cycle the parties, shift their first two input indices in opposite directions, and reflect every input index. The antiperiodic rule supplies the signs. Equations (26)–(27) define the Reynolds operator:

$$\Gamma (\mathbf{p}):=\frac{1}{|G|}\sum _{g\in G}g\cdot \mathbf{p}.$$

Equation (27) describes its fixed subspace by the signed residue of the total input index and reflection about half the number of inputs. Table II counts distinct projected deterministic tensors; its entries do not count only extreme points of their convex hull.

Encoding and result

Inputs are read as $0,\dots ,m-1$, with $e_{x+m}=-e_x$. Integer quotient and remainder use floor division and nonnegative remainder. The Lean definition Gamma uses orthogonal projection onto the common fixed subspace, the convention specified in issue #15186. Equality to the finite-group Reynolds average is not a separately kernel-checked theorem in these modules.

The conjecture is refuted at $N=3$, $m=17$. Nine single-frequency strategies and their negatives remain uniquely exposed vertices. Writing $q_r=289(\Gamma 317d{)}_{[r,0,0]}$, a mixed strategy has the displayed scaled coordinates $(-141,133,-109,69,-13,-43,91,-123,139)$. In these scaled coordinates it is separated from their convex hull by $(-15,27,-21,8,59,24,-5,-6,10)$: the respective values are $8421$ and at most $8417$. Consequently there are at least nineteen extreme points, exceeding the eighteen possible in a nine-dimensional cross-polytope. The source’s other Bell inequalities and computational results are not consequences of this refutation alone.

Computed neighbours and open questions

The experiment entry docs/reports/designolle-vertesi-pokutta-2023-symmetric-polytope-cross-polytope/check.py in the-omega-institute/trureturing-experiments, revision 7ee14723c2f0ce18f299a0f9870eb3d5330ee9e9, has SHA-256 9cda6109955a66f89d7510741043b065bc600caf20e96a717b7f0dbd87dab998. Its reported command is python3 check.py, exit 0, final line bad= 0. These are computations, separate from the Lean theorem: for odd $m=5,\dots ,13$ at $N=3$ the projected polytope is the indicated cross-polytope; Table II’s $N=3$, $m=3,\dots ,9$ counts are $10,10,60,100,640,1540,10032$.

The first failing number of inputs at $N=3$ is open here: $m=14,15,16$ have not been computed. Other numbers of parties and the source’s remark that symmetric facets are full local-polytope facets for odd $N$ and even $m$ remain open here.