Non-Archimedean Welch Bounds and Non-Archimedean Zauner Conjecture

2022/08/28 by K. Mahesh Krishna, Krishna, K. Mahesh
Mathematics · Physics and Astronomy · #12J25 #46S10 #47S10 #Black Holes and Theoretical Physics #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Information Theory (cs.IT) #Number Theory (math.NT) #Tensor decomposition and applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2210.07062

openalex publication_date 2022/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathbbK be a non-Archimedean (complete) valued field satisfying |∑_j=1ⁿλ_j²|=max_{1≤ j ≤ n}|λ_j|², ∀ λ_j ∈ \mathbbK, 1≤ j ≤ n, ∀ n ∈ ℕ. For d∈ ℕ, let \mathbbK^d be the standard d-dimensional non-Archimedean Hilbert space. Let m ∈ ℕ and Sym^m(\mathbbK^d) be the non-Archimedean Hilbert space of symmetric m-tensors. We prove the following result. If \τ_j\_j=1ⁿ is a collection in \mathbbK^d satisfying ⟨ τ_j, τ_j⟩ =1 for all 1≤ j ≤ n and the operator Sym^m(\mathbbK^d)\ni x ↦ ∑_j=1ⁿ⟨ x, τ_j^{⊗ m}⟩ τ_j^{⊗ m} ∈ Sym^m(\mathbbK^d) is diagonalizable, then (1) max_{1≤ j,k ≤ n, j ≠ k}\|n|, |⟨ τ_j, τ_k⟩|^2m \≥ \frac|n|²|d+m-1 \choose m| . We call Inequality (1) as the non-Archimedean version of Welch bounds obtained by Welch [IEEE Transactions on Information Theory, 1974]. We formulate non-Archimedean Zauner conjecture.

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