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Characters of all independent tensor channels

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

proven

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    Natural Language Statement

    Lemma

    Multiplication over actual component channel pairs yields the exact exponent given by the sum of component ranks, including the actual sampled componentwise tensor sum.

    Concept map
    20 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 product_tensor_characters proven

    2 tensor_character_average proven

    Lean source view on GitHub

    1import Lax342547.ChannelMoments
    2
    3/-!
    4---
    5title: Characters of all independent tensor channels
    6type: lemma
    7---
    8Multiplication over actual component channel pairs yields the exact exponent given by the sum of component ranks, including the actual sampled componentwise tensor sum.
    9-/
    10
    11namespace Lax342547.TensorCharacters
    12
    13open Lax342547.MomentSpace Lax342547.ChannelCharacters Lax342547.FiniteSampling
    14open scoped BigOperators
    15
    16noncomputable def tensorCharacter {e I J : Type} [Fintype e] [Fintype I] [Fintype J] (h : ℕ)
    17 (A : e → Matrix I J Binary) (c : e × Fin h → (I → Binary) × (J → Binary)) : ℝ :=
    18 ∏ i, character (A i.1) (c i)
    19
    20axiom tensor_character_average {e I J : Type} [Fintype e] [Fintype I] [Fintype J]
    21 [DecidableEq e] [DecidableEq I] [DecidableEq J] (A : e → Matrix I J Binary) (h : ℕ) :
    22 (∑ c, productLaw (fun _ : e × Fin h => channelLaw) c*tensorCharacter h A c) =
    23 1/(2 : ℝ)^(h*∑ i, (A i).rank)
    24
    25axiom product_tensor_characters {e I J ι : Type} [Fintype e] [Fintype I] [Fintype J] [Fintype ι]
    26 (A : ι → e → Matrix I J Binary) (h : ℕ) (c : e × Fin h → (I → Binary) × (J → Binary)) :
    27 (∏ i, tensorCharacter h (A i) c) = tensorCharacter h (fun e => ∑ i, A i e) c
    28
    29end Lax342547.TensorCharacters
    30
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