Abstract

Let K be a non-Archimedean (complete) valued field satisfying n j=1 λ 2 j = max 1≤j≤n |λ j | 2 , ∀λ j ∈ K, 1 ≤ j ≤ n, ∀n ∈ N. For d ∈ N, let K d be the standard d-dimensional non-Archimedean Hilbert space. Let m ∈ N and Sym m (K d) be the non-Archimedean Hilbert space of symmetric m-tensors. We prove the following result. If {τ j } n j=1 is a collection in K d satisfying τ j , τ j = 1 for all 1 ≤ j ≤ n and the operator Sym m (K d) x → n j=1 x, τ ⊗m j τ ⊗m j ∈ Sym m (K d) is diagonalizable, then max 1≤j,k≤n,j =k {|n|, ||τ j , τ k | 2m } ≥ |n| 2 d+m−1 m. (1) 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.