The Magic Square item of the win implications #

Blueprint lem:qld-win, item 6 (the paper's enu:qld-ms in lem:qld-win-implications). It is the one item that is not a consistency relation: it says that on an anticommuting tuple ω the Pauli basis test's own Constraint/Variable measurements form a strategy for the Magic Square game of value 1 - O(ε). Composed with lem:ms-direct-anticomm --- which is what the Magic Square is in the test for --- it gives the anticommutator bound this file ends with, and that bound is what the expansion stage's sign cancellation runs on.

The conditional strategy #

msPOVM M ω is the strategy: at a Magic Square question x it is the Pauli test's measurement at the corresponding question msQ ω x, relabelled into the Magic Square's answer alphabet by msAns. The relabelling is where the paper's F_2^3 constraint answers meet the layout's full F_2^9 assignments: msAssign places the three bits at the constraint's cells (MIPRE.LCS.MagicSquare.cell) and 0 elsewhere, which is invisible to the Magic Square decider because it reads only the support.

sum_weight_bornProb_map is what makes the relabelling harmless: relabelling outcomes is data processing, so the relabelled strategy wins whenever the original answers would have been accepted. That is condFail_ms_le --- the Magic Square failure at an incidence is at most the Pauli test's failure at the corresponding question pair.

The error, and why it is a sum over the incidences #

msEps is the Magic Square game's own question-weighted sum of those Pauli-test failures, so one_sub_povmValue_ms_le is immediate off the support --- where the Magic Square distribution puts no mass, and the two deciders need not agree at all. On the support the pair is an oriented incidence, hence an edge of the Pauli test's type graph, and subtest_le applies. The average of msEps over the anticommuting tuples is then 86 ε, one factor of 86 and not thirty-six, because the Magic Square's own distribution is normalized.

In a bipartite model #

Stated in a bipartite model (Phase 5 of planning/mipco-track.md). The strategy is a model M : BipartiteModel 𝒞 𝒜 ℬ with ‖M.ψ‖ = 1 and two families of POVMs PA, PB in the players' ordered algebras; the conditional strategy msPOVM coarse-grains a family in any ordered ⋆-ring, the error msEps is a sum of the model's conditional failures M.condFail, the value is M.povmValue, and the second player's ‖(Id ⊗ Y)|ψ⟩‖² is M.swap.stateSqNorm Y. The relabelling and averaging steps need no projectivity. Item 6 does, on the constraint player's side: lem:ms-direct-anticomm is stated for a projective constraint player (MS.ms_direct_anticomm_avg, whose Naimark dilation is gone), so item_magicSquare asks the first player's measurements to be projective, as every strategy of the Pauli basis test is, and passes the projectivity of the conditional strategy on (isPVMIn_msPOVM). On a unit vector ψ : dA × dB → ℂ the statements are read at the tensor-product model BipartiteModel.tensor ψ with the families POVM.toIn.

The Magic Square item of the win implications #

The conditional strategy #

The error, and why it is a sum over the incidences #

In a bipartite model #

The conditional Magic Square strategy #

The assignment reads the three bits #

The Pauli test's Magic Square rule implies the Magic Square game's decider #

The failure transfers #

The point observables #

The anticommutator bound #