Ordered products with global outcome multipliers (lem:qld-global-products) #
The global measurement of lem:qld-global-pvm is consistent, on average over a uniform padded
point u = (x, z, α, β, w), with the sandwich combination
P_u(c) = ∑_{αa + βb = c} Z_b X_a Z_b of Bob's expanded point measurements. Stage 4b needs the
same statement for the two ordered products ∑_{αa + βb = c} Z_b X_a and
∑_{αa + βb = c} X_a Z_b, in the state-distance form: with v_g = (G_g ⊗ 1) Φ,
∑_g E_u ‖v_g - (1 ⊗ B_u(g(u))) v_g‖² ≤ 2δ + 2κ, where δ is the consistency error and κ the
average commutator weight E_{x,z} ∑_{a,b} ‖(1 ⊗ [X_a, Z_b]) Φ‖² of Bob's two measurements.
The argument #
Two steps. (1 ⊗ (1 - P)) v_g is controlled by the consistency itself, because 0 ≤ P ≤ 1 gives
(1 - P)² ≤ 1 - P and so ∑_g E ‖(G_g ⊗ (1 - P_u(g(u)))) Φ‖² ≤ ∑_g E ⟨G_g ⊗ (1 - P_u(g(u)))⟩ ≤ δ.
(1 ⊗ (P - B)) v_g is controlled by the commutators: grouping the outcomes by their value
c = g(u) and dropping the projector ∑_{g(u) = c} G_g ≤ 1 leaves
∑_c ‖(1 ⊗ (P_u(c) - B_u(c))) Φ‖²,
a fibre-summed family of the deviations sand - ord; these sum to zero over (a, b), so Parseval
over F_q (BipartiteModel.sum_avg_stateSqNorm_fibre_eq) turns the average over (α, β) of the
fibre sums into (1 - 1/q) ∑_{a,b} ‖(1 ⊗ (sand - ord)(a, b)) Φ‖² with no loss of a factor q, and
each deviation is a contraction of a commutator: Z X Z - Z X = Z [X, Z] and
Z X Z - X Z = -(1 - Z) [X, Z].
Everything here is stated for abstract projective families X x, Z z of the second player,
with the consistency and the commutator weight as hypotheses; lem:qld-global-pvm and
lem:qld-combined-points supply them for the padded strategy.
Stated in a bipartite model #
Stated in a bipartite model (Phase 5 of planning/mipco-track.md). The state is that of a
bipartite model K (the model in which the low individual degree test is applied, padModel,
or any other), the global measurement G is a projective POVM in the first player's algebra, and
the two families X x, Z z are projective measurements in the second player's. A Kronecker
product G_g ⊗ D is K.πA (G.op g) * K.πB D, its state norm is K.snorm, and the second
player's squared norm ‖(1 ⊗ D) Φ‖² is K.swap.stateSqNorm D. The ordered products, the
commutator and the combinations are defined in any ring. The step (1 - P)² ≤ 1 - P is taken on
the represented operators, where the functional calculus lives (mul_self_le_self_of_le_one),
and Parseval is the model's (BipartiteModel.sum_avg_stateSqNorm_fibre_eq in K.swap).
An element of a projective measurement is at most one, in a star-ordered ring.
Q² ≤ Q for 0 ≤ Q ≤ 1, read through a Born probability: against a nonnegative element
of the first player, ⟨S ⊗ Q²⟩ ≤ ⟨S ⊗ Q⟩. The inequality Q² ≤ Q is taken on the represented
operator (Op.mul_self_le_self), where the functional calculus lives; the second player's
algebra need not have one.
The Born probability is monotone in the first player's operator when the second player's is nonnegative.
⟨ψ| 1 ⊗ D⋆ D |ψ⟩ = ‖(1 ⊗ D) ψ‖²: the second player's squared state norm.
The deviation of a projective outcome family from a family of the second player, grouped by
a value map: the first player's projection ∑_{f g = c} G_g ≤ 1 is dropped.
The two ordered products and the commutator #
The ordered product, X then Z (applied right to left: Z_b X_a).
Equations
- MIPRE.QLD.ordZX X Z p = Z p.2 * X p.1
Instances For
The deviation of the sandwich from the order Z_b X_a is at most the commutator, in the
second player's state norm.
The deviation of the sandwich from the order X_a Z_b is at most the commutator, in the
second player's state norm.
The averaged fibre bound #
The fibre sums of sand - ord are controlled by the commutators, on average over the
combining coefficients and without a factor q, for either order.
The uniform padded point as its blocks and combining coefficients #
Setting the two combining coordinates of a padded point.
Equations
- MIPRE.QLD.setAB u α β = Function.update (Function.update u (MIPRE.QLD.aIdx m) α) (MIPRE.QLD.bIdx m) β
Instances For
The involution of (point, α, β) exchanging the combining coordinates with (α, β).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Reading a padded point's blocks and combining coordinates is reading its blocks and a fresh uniform pair.
The uniform padded point, read through its blocks and combining coordinates: the average
over u ∈ F^{4m} of a function of (xBlk u, zBlk u, alph u, bet u) is the average over
independent uniform x, z ∈ F^m and (α, β) ∈ F².
The lemma #
The sandwich combination at a padded point, P_u(c) = ∑_{αa + βb = c} Z_b X_a Z_b, for
abstract point measurements X x, Z z in a ring.
Equations
- MIPRE.QLD.sandComb X Z u c = MIPRE.QLD.ptComb (MIPRE.sand (X (MIPRE.QLD.xBlk u)) (Z (MIPRE.QLD.zBlk u))) (MIPRE.QLD.alph u) (MIPRE.QLD.bet u) c
Instances For
An ordered-product combination at a padded point, ∑_{αa + βb = c} ord(a, b).
Equations
- MIPRE.QLD.ordComb ord X Z u c = MIPRE.QLD.ptComb (ord (X (MIPRE.QLD.xBlk u)) (Z (MIPRE.QLD.zBlk u))) (MIPRE.QLD.alph u) (MIPRE.QLD.bet u) c
Instances For
lem:qld-global-products, for an abstract ordered product. Let G be a projective
measurement of the first player with outcomes polynomials on F^{4m}, consistent with error δ,
on average over a uniform padded point, with the sandwich combination of two projective families
X x, Z z of the second player whose average commutator weight is κ; and let ord be an
ordered product of the two families summing to one and deviating from the sandwich by a
contraction of the commutator. Then ∑_g E_u ‖(G_g ⊗ (1 - ∑_{αa + βb = g(u)} ord(a, b))) Φ‖² ≤ 2δ + 2κ.
lem:qld-global-products for the order Z_b X_a.
lem:qld-global-products for the order X_a Z_b.