bibkey: botteron2024nonlocalboxorbit authors: P. Botteron; A. Broadbent; R. Chhaibi; I. Nechita; C. Pellegrini year: 2024 title: “Algebra of Nonlocal Boxes and the Collapse of Communication Complexity” doi: null url: https://arxiv.org/abs/2312.00725v3 claim: “Conjecture 15 asks whether right multiplication maximizes CHSH over every parenthesization of copies of a box in the PR-SR-I triangle.” strata_touched:
- D5/S3/Quantum/Entanglement/NonlocalBoxOrbitRightProduct license: citation-only triage: anchor
Verified locator
Source: https://arxiv.org/abs/2312.00725v3
Quantum 8, 1402 (2024). The supplied DOI 10.22331/q-2024-07-11-1402 returned HTTP 404 from Crossref’s work endpoint; the pinned arXiv version is the verified source locator.
Definitions and question
The box product in Section 1, Equation (9), sums over the four internal bits, applying the input functions and the two output indicators of a wiring. The wiring W_BS is decoded in Section 4 as f1(x,a2) = x, g1(y,b2) = y, f2(x,a1) = x a1, g2(y,b1) = y b1, f3(x,a1,a2) = a1 XOR a2 and g3(y,b1,b2) = b1 XOR b2.
The source defines PR(a,b|x,y) = (1/2) 1[a XOR b = xy], P0 = 1[a=b=0], P1 = 1[a=b=1], I = 1/4 and SR = (P0+P1)/2. CHSH is one quarter of the sum of P(a,b|x,y) over a XOR b = xy. The set tilde A consists of boxes in Conv{PR,SR,I} with CHSH at least 3/4. The k-orbit includes every full binary parenthesization of k copies of P; the right product is the left-nested product, multiplying a fresh P on the right at each step.
Section 2.1, p. 12, defines the k-orbit by the words:
all possible products with k times the term P, using the multiplication .
Theorem 13, p. 16, gives maximality in the tilted orbit under its distillation hypothesis. Conjecture 15 (Dyck paths), p. 16, states verbatim:
We conjecture that the same result actually holds without the tilde, i.e. the right multiplication gives the highest CHSH-value of , as observed numerically.
The Lean encoding uses Fin 2, with addition and multiplication modulo two, and orders the arguments as P a b x y. The definition claim retains the source domain tilde A and k >= 2, and drops the distillation hypothesis. The theorem whole_triangle_max proves the bound for every box of the whole triangle and every orbit depth; result specializes it to claim. The value rightPow P 0 is P, while the zero-leaf orbit is empty.
What the proof shows
Proved in the module: the explicit polynomial coordinate product admits a closed invariant region. Its associator has a nonnegative second-coordinate gap, and multiplication on the right preserves second-coordinate order when the first coordinates agree. Structural induction straightens every tree to the left-nested right product without decreasing CHSH. The bound needs neither CHSH >= 3/4 nor the distillation hypothesis of Theorem 13.
Computed: the experiment entry
check.py,
sha256 6f31d58efed171fec57acefa680b3d08e2d37f6a97429e9ecdae433a9007e2fc,
was run as python3 check.py, exit 0, bad= 0. This is finite computational
evidence, separate from the universal Lean theorem.
Open: other wirings, boxes outside the triangle, and an explicit formula for the exact maximum CHSH value as a function of k. Maximality alone does not assert that every such box distills or collapses communication complexity.