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Moment estimates for actual component tensor phases

Lax342547.ComponentMoments · concepts/Lax342547/ComponentMoments.lean · lax-342547

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    Natural Language Statement

    Lemma

    The finite high moment and Markov tail are bounded by the actual summed component rank under independent uniform component channels, with arbitrary bounded unit-side signs.

    Concept map
    21 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 component_even_moment_bound proven

    2 component_phase_tail proven

    Lean source view on GitHub

    1import Lax342547.TensorCharacters
    2import Lax342547.MomentTails
    3
    4/-!
    5---
    6title: Moment estimates for actual component tensor phases
    7type: lemma
    8---
    9The finite high moment and Markov tail are bounded by the actual summed component rank under independent uniform component channels, with arbitrary bounded unit-side signs.
    10-/
    11
    12namespace Lax342547.ComponentMoments
    13
    14open Lax342547.MomentSpace Lax342547.RelativeEntropy Lax342547.FiniteSampling Lax342547.RetainedImages
    15open Lax342547.ChannelCharacters Lax342547.TensorCharacters
    16open scoped BigOperators
    17
    18axiom component_even_moment_bound {e I J Ω : Type} [Fintype e] [Fintype I] [Fintype J] [Fintype Ω]
    19 [DecidableEq e] [DecidableEq I] [DecidableEq J]
    20 (μ : Ω → ℝ) (A : Ω → e → Matrix I J Binary) (ψ : Ω → ℝ) (h t : ℕ)
    21 (hμ : ∀ x, 0 ≤ μ x) (hψ : ∀ x, |ψ x| ≤ 1) :
    22 (∑ c, productLaw (fun _ : e × Fin h => channelLaw) c*
    23 |∑ x, μ x*ψ x*tensorCharacter h (A x) c|^(2*t)) ≤
    24 ∑ sample : Fin (2*t) → Ω, productLaw (fun _ : Fin (2*t) => μ) sample/
    25 (2 : ℝ)^(h*∑ a : e, (∑ i, A (sample i) a).rank)
    26
    27axiom component_phase_tail {e I J Ω : Type} [Fintype e] [Fintype I] [Fintype J] [Fintype Ω]
    28 [DecidableEq e] [DecidableEq I] [DecidableEq J]
    29 (μ : Ω → ℝ) (A : Ω → e → Matrix I J Binary) (ψ : Ω → ℝ) (h t : ℕ) (θ : ℝ)
    30 (hμ : ∀ x, 0 ≤ μ x) (hψ : ∀ x, |ψ x| ≤ 1) (hθ : 0 < θ) :
    31 cellMass (productLaw (fun _ : e × Fin h => channelLaw))
    32 (fun c => θ ≤ |∑ x, μ x*ψ x*tensorCharacter h (A x) c|) ≤
    33 (∑ sample : Fin (2*t) → Ω, productLaw (fun _ : Fin (2*t) => μ) sample/
    34 (2 : ℝ)^(h*∑ a : e, (∑ i, A (sample i) a).rank))/θ^(2*t)
    35
    36end Lax342547.ComponentMoments
    37
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