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Independent binary channel characters

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

proven

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    Natural Language Statement

    Lemma

    Actual independent uniform channel pairs have the exact rank character mean; multiplication over channels and sampled tensors gives the mean of their actual matrix sum.

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

    This concept declares 6 statements. Each proof establishes one of them relative to its assumptions.

    1 channel_probability proven

    2 character_add proven

    3 character_average proven

    4 character_sum proven

    5 product_channel_characters proven

    6 repeated_character_average proven

    Lean source view on GitHub

    1import Lax342547.BilinearMean
    2import Lax342547.FiniteMoments
    3import Mathlib.Algebra.BigOperators.Field
    4
    5/-!
    6---
    7title: Independent binary channel characters
    8type: lemma
    9---
    10Actual independent uniform channel pairs have the exact rank character mean; multiplication over channels and sampled tensors gives the mean of their actual matrix sum.
    11-/
    12
    13namespace Lax342547.ChannelCharacters
    14
    15open Lax342547.MomentSpace Lax342547.Walsh Lax342547.RelativeEntropy Lax342547.FiniteSampling
    16open scoped BigOperators
    17
    18noncomputable def channelLaw {I J : Type} [Fintype I] [Fintype J]
    19 (_ : (I → Binary) × (J → Binary)) : ℝ :=
    20 1/((2 : ℝ)^Fintype.card I*2^Fintype.card J)
    21
    22noncomputable def character {I J : Type} [Fintype I] [Fintype J]
    23 (A : Matrix I J Binary) (c : (I → Binary) × (J → Binary)) : ℝ := phase c.1 (A.mulVec c.2)
    24
    25noncomputable def channelCharacter {I J : Type} [Fintype I] [Fintype J] (h : ℕ)
    26 (A : Matrix I J Binary) (c : Fin h → (I → Binary) × (J → Binary)) : ℝ :=
    27 ∏ i, character A (c i)
    28
    29axiom channel_probability {I J : Type} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] :
    30 Probability (channelLaw (I := I) (J := J))
    31
    32axiom character_average {I J : Type} [Fintype I] [Fintype J]
    33 [DecidableEq I] [DecidableEq J] (A : Matrix I J Binary) :
    34 (∑ c, channelLaw c*character A c) = 1/(2 : ℝ)^A.rank
    35
    36axiom character_add {I J : Type} [Fintype I] [Fintype J]
    37 (A B : Matrix I J Binary) (c : (I → Binary) × (J → Binary)) :
    38 character (A+B) c = character A c*character B c
    39
    40axiom character_sum {I J ι : Type} [Fintype I] [Fintype J] [Fintype ι]
    41 (A : ι → Matrix I J Binary) (c : (I → Binary) × (J → Binary)) :
    42 character (∑ i, A i) c = ∏ i, character (A i) c
    43
    44axiom repeated_character_average {I J : Type} [Fintype I] [Fintype J]
    45 [DecidableEq I] [DecidableEq J] (A : Matrix I J Binary) (h : ℕ) :
    46 (∑ c, productLaw (fun _ : Fin h => channelLaw) c*channelCharacter h A c) =
    47 1/(2 : ℝ)^(h*A.rank)
    48
    49axiom product_channel_characters {I J ι : Type} [Fintype I] [Fintype J] [Fintype ι]
    50 (A : ι → Matrix I J Binary) (h : ℕ) (c : Fin h → (I → Binary) × (J → Binary)) :
    51 (∏ i, channelCharacter h (A i) c) = channelCharacter h (∑ i, A i) c
    52
    53end Lax342547.ChannelCharacters
    54
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