The marginals track the point measurements (lem:qld-helper) #
Stage 5 uses the simultaneous pair measurement of lem:qld-simultaneous through two statements,
the paper's lem:qld-constructing-the-paulis-helper. For W ∈ {X, Z}, on average over a uniform
point u ∈ F_q^m, with S^W_a the evaluated W marginal of Alice's pair measurement and
M̂^{W,u}_a the expanded point measurement:
- cross-party:
∑_a ⟨S^W_a ⊗ M̂^{B,W,u}_a⟩ ≥ 1 - δ_S, the consistency read as an agreement; - same-party:
∑_a ‖(S^W_a (1 - M̂^{A,W,u}_a) ⊗ 1) Φ‖² ≤ 4 δ_S + 2 κ, in which both factors sit on Alice --- the form the Pauli construction needs, since it multiplies the marginal by the point measurement on one side.
κ is the self-consistency of the expanded point measurements (lem:qld-expanded-points,
172 ε), which is what lets the second statement cross from one party to the other.
The identity behind the same-party statement #
With T = S^W_a and A = M̂^{A,W,u}_a projectors on Alice, B = M̂^{B,W,u}_a a projector on
Bob, and Φ the state,
T ⊗ 1 · (1 - A ⊗ 1) = (1 - 1 ⊗ B)(T ⊗ 1 - 1 ⊗ B) - (T ⊗ 1)(A ⊗ 1 - 1 ⊗ B),
because the cross terms (T ⊗ 1)(1 ⊗ B) cancel and (1 - 1 ⊗ B)(1 ⊗ B) = 0 by projectivity of
B. Both factors in front are contractions, so the deviation of the same-party product is at most
the sum of the two cross-party deviations: the first is the consistency of the marginal with Bob's
point measurement, the second the self-consistency of the point measurements. No positivity of the
state is used beyond its being a unit vector.
Stated in a bipartite model #
Stated in a bipartite model (Phase 5 of planning/mipco-track.md). The pair measurement is a
SimulPair M S K ι δ of a projective strategy S of a bipartite model M: its measurements live
in the model K, and the expanded point measurements, POVMs of the register model
M.reg (Anc F m), are read in K through the embedding ι. The identity above is an identity in
K's algebra, T ⊗ 1 being K.πA T and 1 ⊗ B being K.πB B; the expanded self-consistency
of lem:qld-expanded-points is carried to K exactly along ι (SimulPair.xSqNorm_aOp). The
projectivity of the strategy, a hypothesis of the matrix statements, is a field of S.
The same-party deviation #
The key identity: the same-party product T (1 - A) is a combination of the two
cross-party deviations, T ⊗ 1 - 1 ⊗ B and A ⊗ 1 - 1 ⊗ B.
The same-party deviation is at most the two cross-party ones.
The squared form, at the usual cost of a factor two.
One minus the inconsistency is the agreement #
The inconsistency is one minus the agreement.
The self-consistency of the expanded point measurements, at a uniform point #
The verifier's W block is uniform: an average over contents of a function of the
content's W point is the average over a uniform point.
lem:qld-expanded-points, read at a uniform point rather than at the verifier's content: the
content's W block is uniform, and the hatted measurement at a content is the one at its point.
The two items of lem:qld-helper #
The elements of a hatted point measurement are self-adjoint.
lem:qld-helper, item 1: the evaluated marginal agrees with the opposite party's expanded
point measurement with probability at least 1 - δ_S, on average over a uniform point.
The cross-party deviation of the marginal from the point measurement, from item 1.
The cross-party deviation of the two parties' expanded point measurements, read in K:
lem:qld-expanded-points, carried along the embedding.
lem:qld-helper, item 2: the same-party product of the marginal with the complement of
Alice's own point measurement is small.