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Actual channel moments and phase tails

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

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    Natural Language Statement

    Lemma

    The checked moment expansion and exact channel character mean bound arbitrary bounded unit-side signs by the rank of the actual sampled matrix sum, and yield the explicit Markov phase tail.

    Concept map
    19 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 channel_even_moment_bound proven

    2 channel_phase_tail proven

    Lean source view on GitHub

    1import Lax342547.ChannelCharacters
    2import Lax342547.MomentTails
    3
    4/-!
    5---
    6title: Actual channel moments and phase tails
    7type: lemma
    8---
    9The checked moment expansion and exact channel character mean bound arbitrary bounded unit-side signs by the rank of the actual sampled matrix sum, and yield the explicit Markov phase tail.
    10-/
    11
    12namespace Lax342547.ChannelMoments
    13
    14open Lax342547.MomentSpace Lax342547.RelativeEntropy Lax342547.FiniteSampling Lax342547.RetainedImages
    15open Lax342547.ChannelCharacters
    16open scoped BigOperators
    17
    18axiom channel_even_moment_bound {I J Ω : Type} [Fintype I] [Fintype J] [Fintype Ω]
    19 [DecidableEq I] [DecidableEq J]
    20 (μ : Ω → ℝ) (A : Ω → 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 _ : Fin h => channelLaw) c*
    23 |∑ x, μ x*ψ x*channelCharacter h (A x) c|^(2*t)) ≤
    24 ∑ sample : Fin (2*t) → Ω, productLaw (fun _ : Fin (2*t) => μ) sample/
    25 (2 : ℝ)^(h*(∑ i, A (sample i)).rank)
    26
    27axiom channel_phase_tail {I J Ω : Type} [Fintype I] [Fintype J] [Fintype Ω]
    28 [DecidableEq I] [DecidableEq J]
    29 (μ : Ω → ℝ) (A : Ω → 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 _ : Fin h => channelLaw))
    32 (fun c => θ ≤ |∑ x, μ x*ψ x*channelCharacter h (A x) c|) ≤
    33 (∑ sample : Fin (2*t) → Ω, productLaw (fun _ : Fin (2*t) => μ) sample/
    34 (2 : ℝ)^(h*(∑ i, A (sample i)).rank))/θ^(2*t)
    35
    36end Lax342547.ChannelMoments
    37
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