Discrete phase-space structure of $n$-qubit mutually unbiased bases

TL;DR

This paper explores the phase-space structure of n-qubit systems using finite fields, classifying geometric structures that underpin mutually unbiased bases, and analyzing their properties in low dimensions.

Contribution

It introduces a phase-space framework for n-qubits, classifies discrete curves relevant to unbiased bases, and examines their transformations in specific dimensions.

Findings

Classified bundles of discrete curves compatible with unbiasedness.

Analyzed the structure in four- and eight-dimensional cases.

Studied effects of local transformations on these structures.

Abstract

We work out the phase-space structure for a system of $n$ qubits. We replace the field of real numbers that label the axes of the continuous phase space by the finite field $\Gal{2^n}$ and investigate the geometrical structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties. We provide a simple classification of such curves and study in detail the four- and eight-dimensional cases, analyzing also the effect of local transformations. In this way, we provide a comprehensive phase-space approach to the construction of mutually unbiased bases for $n$ qubits.