Hypergraph Contextuality

Mladen Pavičić

doi:10.3390/e21111107

Abstract

Quantum contextuality is a source of quantum computational power and a theoretical delimiter between classical and quantum structures. It has been substantiated by numerous experiments and prompted generation of state independent contextual sets, that is, sets of quantum observables capable of revealing quantum contextuality for any quantum state of a given dimension. There are two major classes of state-independent contextual sets—the Kochen-Specker ones and the operator-based ones. In this paper, we present a third, hypergraph-based class of contextual sets. Hypergraph inequalities serve as a measure of contextuality. We limit ourselves to qutrits and obtain thousands of 3-dim contextual sets. The simplest of them involves only 5 quantum observables, thus enabling a straightforward implementation. They also enable establishing new entropic contextualities.

Highlights

Quantum contextuality found applications in quantum communication [1,2], quantum computation [3,4], quantum nonlocality [5] and lattice theory [6,7]
That means that we consider classical models with predetermined binary values, which can be assigned to measurement outcomes of classical observables, which underlie the latter computation, versus quantum models that do not allow for such values and underlie quantum computation
As for the direct relevance of our results to quantum computation, we point out that the hypergraph presented in Figure 2 of Reference [3]—from which the contextual “magic” of quantum computation has been derived—is the kind of hypergraph contextual sets we present in this paper

Summary

Introduction

Quantum contextuality found applications in quantum communication [1,2], quantum computation [3,4], quantum nonlocality [5] and lattice theory [6,7]. Contextual theoretical models and experimental tests involve additional subtle issues, such as the possibility of classical noncontextual hidden variable models that can reproduce quantum mechanical predictions, up to arbitrary precision [30] or a generalization and redefinition of noncontextuality [31,32]. These elaborations are outside the scope of the present paper, though, since it is primarily focused on contextuality, which finds applications within quantum computation versus noncontextuality, which is inherent in the current classical binary computation. The hypergraph is from a 4-dim Hilbert space, so, we will not elaborate on it in this paper

Methods

Results

Conclusion