Periodic discretized continuous observables are neither continuous nor discrete

TL;DR

This paper demonstrates that discretized continuous quantum observables are neither fully continuous nor discrete, depending on outcome number, challenging traditional classifications and showing their complex nature through theoretical analysis and optical experiments.

Contribution

It introduces a novel perspective on discretized continuous observables, revealing their non-classifiable nature using mutual unbiasedness and experimental validation.

Findings

Discretized observables can behave as neither purely continuous nor discrete.

Experimental confirmation of four mutually unbiased measurements with three outcomes.

Surpasses previous limits on mutually unbiased measurements in continuous systems.

Abstract

Most of the fundamental characteristics of quantum mechanics, such as non-locality and contextuality, are manifest in discrete, finite-dimensional systems. However, many quantum information tasks that exploit these properties cannot be directly adapted to continuous-variable systems. To access these quantum features, continuous quantum variables can be made discrete by binning together their different values, resulting in observables with a finite number "$d$" of outcomes. While direct measurement indeed confirms their manifestly discrete character, here we employ a salient feature of quantum physics known as mutual unbiasedness to show that such coarse-grained observables are in a sense neither continuous nor discrete. Depending on $d$, the observables can reproduce either the discrete or the continuous behavior, or neither. To illustrate these results, we present an example for the construction of such measurements and employ it in an optical experiment confirming the existence of four mutually unbiased measurements with $d = 3$ outcomes in a continuous variable system, surpassing the number of mutually unbiased continuous variable observables.