Homotopical approach to quantum contextuality Cihan Okay1,2 and Robert Raussendorf1,2 1Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada 2Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, BC, Canada We consider the phenomenon of quan- on Čech cohomology by Abramsky and cowork- tum mechanical contextuality, and specif- ers [16], the geometric approach by Arkhipov [17], ically parity-based proofs thereof. Mer- and the approach based on cohomology of chain min’s square and star are representative complexes [18]. For other approaches to contex- examples. Part of the information invoked tuality see [19–23] and for refinements of the Čech in such contextuality proofs is the com- cohomology approach see [24–27]. mutativity structure among the pertain- In this paper we unify the approach of ing observables. We investigate to which Arkhipov and the approach based on chain com- extent this commutativity structure alone plexes, thereby generalizing one of Arkhipov’s determines the viability of a parity-based results [17]. Both [17] and [18] are concerned contextuality proof. We establish a topo- with so-called parity-based contextuality proofs, logical criterion for this, generalizing an of which Mermin’s square and star are represen- earlier result by Arkhipov. tative examples [28]. Generally, a parity-based contextuality proof is a set of linear constraints on hypothetical value assignments that has no so- 1 Introduction lution. Although concerned with the same type of contextuality proof, [17] and [18] use different Contextuality is a characteristic feature of quan- abstractions and address slightly different ques- tum mechanics [1,2] that distinguishes quantum tions. from classical physics. It rules out the descrip- Ref. [18] asks: Does a given set O of observ- tion of quantum phenomena by so-called non- ables produce a parity-based contextuality proof? contextual hidden variable models, in which each The first step in answering this question is in ex- observable has a pre-determined value which is tracting the essential information of the set O of merely revealed by the act of measurement. Such observables, in terms of a chain complex C that models are either internally inconsistent, or fail to captures the commutativity structure within O, reproduce the predictions of quantum mechanics. and a cocycle β that captures the phase infor- What leads to paradoxes under too strin- mation of product relations among commuting gent assumptions about physical reality be- elements in O. Then, a parity-based contextu- comes a commodity if these assumptions are not ality proof arises in the setting specified by O if made: contextuality is also a resource in quan- and only if the cocycle β is cohomologically non- tum computation. For two models of quantum trivial, [β] 6= 0. computation—quantum computation with magic Arkhipov [17] invokes a different abstraction arXiv:1905.03822v2 [quant-ph] 18 Dec 2019 states [3] and measurement-based quantum com- and addresses a different question. Rather than a putation [4]—it has been shown that a quantum chain complex, [17] defines an intersection graph speedup can occur only if contextuality is present G, with the edges labeling observables and the [5–12]. vertices labeling contexts. Again, G captures the At its core, contextuality is about the interplay commutativity structure among the observables between local consistency and global inconsis- involved. The counterpart to the cocycle β of tency [13]; and therefore topological methods are [18] is a sign function defined on the vertices of particularly suited to discuss it. Presently four G. Arkhipov’s question is, in a sense, the re- approaches to contextuality have been described verse of the above; namely: Given the intersec- that employ topology: the topos approach by tion graph G, does there exist a corresponding Döring and Isham [14, 15], the approach based set O of observables that produces a parity-based Accepted in Quantum 2019-12-09, click title to verify. Published under CC-BY 4.0. 1 contextuality proof? This question has a surpris- [17] all observables are constrained to have eigen- ingly simple—and topological—answer. Namely, values ±1, whereas the present treatment permits the intersection graph G can give rise to a parity- operators with eigenvalues e2πi/d, for any d ≥ 2. based contextuality proof if and only if G is non- The remainder of this paper is organized as fol- planar. lows. In Section2 we introduce magic arrange- In [17], rather than a set O of observables, the ments and explain the connection to the notion intersection graph G is the primary object. The of contextuality. Section3 we introduce the ba- broader significance of this is that a geometrical sic tools: topological realizations and path opera- object encoding commutativity relations among tors. Section4 contains our main result (Theorem observables emerges at the base of quantum me- 4.6). In Section4 we study topological realiza- chanical contextuality. In this paper, we seek gen- tions via the fundamental sequence of homotopy eralizations of Arkhipov’s results to a greater va- groups. Then in Section5 we move on to define riety of physical settings and to understand the a representation of the fundamental group of a structure of the emerging geometric objects. topological realization and use this to refine our main result (Corollary 5.6). Section6 is on the We now summarize how the two above- algorithmic approach [29, 30] to parity-based con- described approaches [17] and [18] are unified, textuality proofs and how it relates to the topo- how one of Arkhipov’s results is thereby general- logical approach presented in this paper. ized, and why such a generalization is desirable. First, we observe that the intersection graph G employed in [17] is defined only for settings in 2 Magic arrangements which each observable in O is part of no more than two contexts. This is a consequence of the Many proofs of contextuality of quantum me- simple fact that an edge in a graph has no more chanics employ fixed sets of observables to find than two end points. obstructions to the existence of non-contextual The extension of Arkhipov’s results to settings value assignments. Two key examples are Mer- with observables contributing to more than two min’s square and star [28]. However, the ex- contexts is, for example, relevant for the dis- istence of the obstruction does not depend on cussion of contextuality in measurement-based the precise choice of the observables; for exam- quantum computation (MBQC). In the stan- ple, conjugating each observable in the set by dard MBQC setting [4], applied to evaluation of the same unitary produces an equivalent proof. Boolean functions on m input bits [10], there are Rather, it depends only on which subsets of 2m measurement contexts overall. Each local ob- observables commute, and how commuting ob- servable measurable in the process, chosen from servables multiply. In Arkhipov’s formulation, a set of 2, appears in one half of these contexts. this information makes up the so-called “arrange- Therefore, each such observable appears in more ment", and the commutativity part of it the so- than two measurement contexts whenever m > 2. called “intersection graph" of the arrangement. Our main result is that one direction of The definition of the intersection graph is as Arkhipov’s result can be generalized in the for- follows. Given a fixed set of observables for a malism of [18] based on a chain complex C, with contextuality proof, such as Mermin’s square or no bounds on the number of contexts that any star, these observables become the edges of the given observable can appear in. Specifically, a intersection graph, and the sets of commuting complex C can yield a parity-based contextual- observables, i.e., the contexts, become the ver- ity proof only if the topological space XC corre- tices of the intersection graph. Thus, two edges sponding to C is not simply connected, i.e., only if have the same vertex as end point if and only if π1(XC) 6= 1. For example, in the case of Arkhipov the corresponding observables commute. To stick the complex C can be described by a closed sur- with the above examples, the intersection graphs face Σg of some genus g together with a cell struc- of Mermin’s square and star (see Fig. (2)) are the ture on it. The intersection graph G corresponds bipartite complete graph K3,3 and the complete to the dual cell structure of Σg. Vanishing of the graph K5, respectively. fundamental group coincides with the planarity Central in [17] is the notion of “magic" arrange- of the intersection graph in this case. Also, in ments. They appear in the context of nonlocal Accepted in Quantum 2019-12-09, click title to verify. Published under CC-BY 4.0. 2 games [31], cooperative strategies that demon- magic if it is quantum realizable but not classi- strate contextuality as a resource for winning cally realizable. with high probability. An arrangement is magic When the arrangement satisfies the property if the edges of its intersection graph can be popu- that each observable is contained in exacly two lated with Hermitian operators with eigenvalues contexts one can define the intersection graph. ±1, but not with numbers ±1. The significance Can it be judged from the intersection graph of of this notion is that every magic arrangement an arrangement alone whether the arrangement gives rise to a proof of contextuality. Formally, is magic? Arkipov answers this question in an a set of contraints among quantum observables is affirmative way, and provides the following char- specified, and a contextuality proof would be a acterization. demonstration of the failure to satisfy these con- Theorem A.