Transport along a tracial extension #
The pair law is invariant under the spatial extension: U intertwines the
left and right actions and preserves inner products.
The alignment defect is invariant under the isometry U.
ℓ¹ toolkit #
Two probability laws on a finite product are at unhalved ℓ¹ distance at most 2.
ℓ¹ conversion of a branch decomposition q̂ = cm · q_z + r with nonnegative
remainder of mass 1 − cm against a probability law q:
‖q̂ − q‖₁ ≤ ‖q_z − q‖₁ + 2(1 − cm) (06_otqcs.tex, the display deriving eq
otqcs-main: "on a bad branch the unhalved ℓ¹ cost is at most 2").
Vector-to-ℓ¹ for pair laws (06_otqcs.tex, eq vector-to-l1): two unit
vectors' pair laws under one pair of full POVMs differ in unhalved ℓ¹ by at most
2‖z − w‖₂ — the ℓ¹-POVM estimate sum_abs_re_inner_effect_sub_le applied to
the commuting product effects L(A_s^a) R(B_t^b).
The finite bad bound at the chosen trial count #
The selected-state chain (eq selected-state-preopt) #
‖k♯‖₂² = b (06_otqcs.tex, eq rounded-y: the normalization of ỹ).
The normalization inequality for the rounded polar vector:
‖ỹ_t − y_t‖₂ ≤ 2 ‖k_t^♯ − k_t‖₂ (06_otqcs.tex, eq normalization-inequality
applied to ỹ = b^{−1/2} k♯ v, y = k v, with v a right isometry on the
relevant vectors — Rop_v_norm).
‖k − k^{cut}‖₂ ≤ √(ρ + L²) when the bands cover [L, H] and the high tail
beyond H is at most ρ (06_otqcs.tex, eq rounding-tails).
The selected-state chain (06_otqcs.tex, eq selected-state-preopt):
‖z_{st} − u_{st}‖₂ ≤ 2√Γ_{st} + 2((r − 1) + κ) + ‖y_t − u_{st}‖₂, where
r − 1 = α bounds the rounding ‖k♯ − k^{cut}‖ and κ the cut ‖k^{cut} − k‖.
Hypotheses of the grid slab read off a joint spectral package #
Averaging and the parameter arithmetic #
Jensen for the square root under a probability weight:
∑ πᵢ √gᵢ ≤ √(∑ πᵢ gᵢ) ("this is the sole square-root conversion in the proof").
The final constant bookkeeping in the nontrivial range δ + ξ ≤ 1/2, with
δ = Δ^{1/6}, ρ = ξ²/32, L = ξ/8, α = δ + ξ (06_otqcs.tex, eqs cutoff-choice,
alpha-choice, alpha-optimization, Dbar-Delta): the averaged per-pair bound is at
most a universal multiple of δ + ξ.
Averaging the per-pair bound (06_otqcs.tex, the final display of the proof
of thm otqcs): if every pair obeys the selected-state/bad-branch bound in terms of
its grid disagreement Γᵢ and its alignment Yᵢ = ‖y − u‖², and the π-averages
obey eqs common-shift and Dbar-Delta, then the π-average of the ℓ¹ errors is at
most a universal multiple of δ + ξ, δ = Δ^{1/6}.
The trivial resource (degenerate range) #
The trivial sampling resource of the degenerate range α₀ > 1/2
(06_otqcs.tex, proof of thm otqcs: "take explicitly N̂ = N, Ω = 1,
Â_s^{a₀} = B̂_t^{b₀} = 1 and set all other output effects to zero").
Equations
- One or more equations did not get rendered due to their size.