Exact value and rigidity of the 2×3 magic rectangle

@misc{essayag2026exact,
  title = {{Exact value and rigidity of the $2\times3$ magic rectangle}},
  author = {Isaac Barouch Essayag and Aryeh Lev Zabokritskiy},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12001},
  eprint = {2610.12001},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12001},
}

TY  - UNPB
TI  - Exact value and rigidity of the 2×3 magic rectangle
AU  - Isaac Barouch Essayag
AU  - Aryeh Lev Zabokritskiy
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.12001
UR  - https://arxiv.org/abs/2610.12001
AB  - The 2×3 magic rectangle is a nonlocal game in which two players match entries subject to incompatible parity constraints. Its quantum value was suggested by numerical bounds and known quantum random access codes. We prove that this value is (1+√(2/3))/2 and classify all optimal strategies in arbitrary finite local dimensions, allowing general measurements: up to local isometries and ancillary systems, each contains a maximally entangled pair of four-dimensional systems with fixed measurements on the local state supports. The same value holds for commuting-operator strategies and for the almost-quantum relaxation, confirming the Adamson--Wallden conjecture for this rectangle. For finite-dimensional magic-square strategies, the largest score compatible with perfect prediction of Alice's complete first row by an adversary with quantum side information is one third plus two thirds of the rectangle value. We classify the strategies attaining this threshold with perfect prediction and give linear-order upper and lower bounds on one minus the best compatible guessing probability above it. The rectangle value is also the optimal average success probability of finite-dimensional three-bit random access codes that hide the full input parity, and we classify all optimal codes.
ER  -