Realization of mutually unbiased bases for a qubit with only one wave plate: Theory and experiment

TL;DR

This paper demonstrates that a single wave plate with a specific phase shift can realize mutually unbiased bases for a qubit, reducing the number of optical elements needed and improving systematic error in quantum state tomography.

Contribution

It introduces a method to implement all three MUB for a qubit using only one wave plate with a specific phase shift, simplifying optical setups and reducing errors.

Findings

One wave plate can realize two MUB if phase shift is between 45° and 315°.

A third wave plate with 120° phase shift can realize all three MUB for a qubit.

Using a third wave plate reduces systematic error in quantum state tomography by 50%.

Abstract

We consider the problem of implementing mutually unbiased bases (MUB) for a polarization qubit with only one wave plate, the minimum number of wave plates. We show that one wave plate is sufficient to realize two MUB as long as its phase shift (modulo $360^\circ$) ranges between $45^\circ$ and $315^\circ$. {It can realize} three MUB (a complete set of MUB for a qubit) if the phase shift of the wave plate is within $[111.5^\circ, 141.7^\circ]$ or its symmetric range with respect to 180$^\circ$. The systematic error of the realized MUB using a third-wave plate (TWP) with $120^\circ$ phase is calculated to be a half of that using the combination of a quarter-wave plate (QWP) and a half-wave plate (HWP). As experimental applications, TWPs are used in single-qubit and two-qubit quantum state tomography experiments and the results show a systematic error reduction by $50\%$. This technique not only saves one wave plate but also reduces the systematic error, which can be applied to quantum state tomography and other applications involving MUB. The proposed TWP may become a useful instrument in optical experiments, replacing multiple elements like QWP and HWP.