Mutually Unbiased Coarse-Grained Measurements of Two or More Phase-Space Variables

TL;DR

This paper explores how mutual unbiasedness of coarse-grained phase-space measurements can be achieved for multiple variables, extending previous work and demonstrating experimental realization with optics, bridging continuous and discrete quantum mechanics.

Contribution

It generalizes mutual unbiasedness to more than two phase-space variables with coarse-grained measurements and provides experimental validation using fractional Fourier transforms.

Findings

Mutual unbiasedness can be achieved for non-parallel phase-space operators.

Experimental demonstration using optics and fractional Fourier transform.

Differences between two and three mutually unbiased measurements are analyzed.

Abstract

Mutual unbiasedness of the eigenstates of phase-space operators-such as position and momentum, or their standard coarse grained versions-exists only in the limiting case of infinite squeezing. In [Phys. Rev. Lett. 120, 040403 (2018)] it was shown that mutual unbiasedness can be recovered for periodic coarse grainings of these two operators. Here we investigate mutual unbiasedness of coarse-grained measurements for more than two phase-space variables. We show that mutual unbiasedness can be recovered between periodic coarse graining of any two non-parallel phase-space operators. We illustrate these results through optics experiments using the fractional Fourier transform to prepare and measure mutually unbiased phase-space variables. The differences between two and three mutually unbiased measurements is discussed. Our results contribute to bridging the gap between continuous and discrete quantum mechanics and could be useful in quantum information protocols.