p-adic Welch Bounds and p-adic Zauner Conjecture

Let p be a prime. For d is an element of N, let Q(p)(d) be the standard d-dimensional p-adic Hilbert space. Let m is an element of N and Symm(Q(p)(d)) be the p-adic Hilbert space of symmetric m-tensors. We prove the following result. Let {tau j}nj=1 be a collection in Q(p)(d) satisfying (i) <tau j,tau j >=1 for all 1 <= j <= n and (ii) there exists b is an element of Qp satisfying & sum;(n)(j=1)< x,tau j >tau j=bx for all x is an element of Q(p)(d) . Then max(1 <= j,k <= n,j not equal k){|n|,|<tau j,tau k >|(2m)}>=|n|(2)divided by divided by(d+m-1 m)divided by divided by. We call Inequality (1) as the p-adic version of Welch bounds obtained by Welch [\textit{IEEE Transactions on Information Theory, 1974}]. Inequality (1) differs from the non-Archimedean Welch bound obtained recently by M. Krishna as one can not derive one from another. We formulate p-adic Zauner conjecture.