Mutually unbiased bases for quantum states defined over p-adic numbers

TL;DR

This paper constructs mutually unbiased bases for quantum states over p-adic numbers, revealing a higher number of such bases than in the real case, and discusses reasons for this difference.

Contribution

It introduces a method to find multiple MUBs in p-adic quantum state spaces, expanding understanding beyond real Hilbert spaces.

Findings

For each prime p>2, at least p+1 MUBs exist.

Contrast with only 3 MUBs known over real numbers.

Discussion on reasons for differences in MUB counts.

Abstract

We describe sets of mutually unbiased bases (MUBs) for quantum states defined over the p-adic numbers Q_p, i.e. the states that can be described as elements of the (rigged) Hilbert space L2(Q_p). We find that for every prime p>2 there are at least p+1 MUBs, which is in contrast with the situation for quantum states defined over the real line R for which only 3 MUBs are known. We comment on the possible reason for the difference regarding MUBs between these two infinite dimensional Hilbert spaces.