Nonlocality with less Complementarity

TL;DR

This paper explores the relationship between nonlocality and complementarity in quantum mechanics and toy theories, showing that nonlocality can occur without certain types of complementarity, challenging traditional links.

Contribution

It introduces two toy theories that exhibit nonlocality without the specific form of complementarity usually associated with quantum mechanics.

Findings

First toy theory maximally violates CHSH but has jointly measurable observables.

Second toy theory maximally violates CHSH with classical state space and nondisturbing measurements.

Both theories demonstrate nonlocality without the typical complementarity constraints.

Abstract

In quantum mechanics, nonlocality (a violation of a Bell inequality) is intimately linked to complementarity, by which we mean that consistently assigning values to different observables at the same time is not possible. Nonlocality can only occur when some of the relevant observables do not commute, and this noncommutativity makes the observables complementary. Beyond quantum mechanics, the concept of complementarity can be formalized in several distinct ways. Here we describe some of these possible formalizations and ask how they relate to nonlocality. We partially answer this question by describing two toy theories which display nonlocality and obey the no-signaling principle, although each of them does not display a certain kind of complementarity. The first toy theory has the property that it maximally violates the CHSH inequality, although the corresponding local observables are pairwise jointly measurable. The second toy theory also maximally violates the CHSH inequality, although its state space is classical and all measurements are mutually nondisturbing: if a measurement sequence contains some measurement twice with any number of other measurements in between, then these two measurements give the same outcome with certainty.