Topological proofs of contextuality in quantum mechanics Cihan Okay1, Sam Roberts2, Stephen D. Bartlett2, and Robert Raussendorf3 1Department of Mathematics, University of Western Ontario, London, Ontario, Canada 2Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW, Australia 3Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada 7 1 0 2 January 10, 2017 n a J Abstract 7 Weprovideacohomologicalframeworkforcontextualityofquantummechanicsthatissuited ] h to describing contextuality as a resource in measurement-based quantum computation. This p framework applies to the parity proofs first discussed by Mermin, as well as a different type of - contextuality proofs based on symmetry transformations. The topological arguments presented t n can be used in the state-dependent and the state-independent case. a u q 1 Introduction [ 1 Contextuality [1]-[5] is a feature that distinguishes quantum mechanics from classical physics. To v 8 describe it, let’s consider the question of whether it is possible to assign “pre-existing” outcomes 8 to measurements of quantum observables which are merely revealed by measurement. If this were 8 possible, it would amount to a description of quantum mechanics in terms of classical statistical 1 0 mechanics. Assuming such a model, for any two different sets A and B of mutually compatible 1. observablescontainingagivenobservableA, itisreasonabletorequirethatthevalueλ(A)attached 0 totheobservableAisapropertyofAalone, andthusagreesinAandB. AandB aremeasurement 7 contexts for A, and the constraint on λ(A) just described is called “context independence”. Can 1 : context-independent pre-assigned outcomes λ, or probabilistic combinations thereof, describe all of v quantum mechanics?—This turns out not to be the case [1], [2], a fact which is often referred to as i X contextuality of quantum mechanics. r For quantum computation, contextuality is a resource. In quantum computation with magic a states [6] and in measurement-based quantum computation (MBQC) [7], no quantum speedup can occur without it [8]–[10]; [11]–[13]. Forthepresentwork,thelinkbetweencontextualityandquantumcomputationisthemotivation to investigate the mathematical structures underlying contextuality. In this regard, Abramsky and coworkers have provided a sheaf-theoretic description of contextuality [4]. They have further identified cohomological obstructions to the existence of the classical models described above, so- callednon-contextualhiddenvariablemodels[14], [15]. Thesemethods, basedonCˇechcohomology, have a wide range of applicability, covering the Bell inequalities [2], Hardy’s model [16], and the Greenberger-Horne-Zeillinger setting [17]. Here, we provide a different cohomological framework for contextuality, involving group co- homology. It is designed to describe the form of contextuality required for the functioning of 1 measurement-based quantum computation. The connection between contextuality and MBQC was first observed in the example of Mermin’s star [11], and subsequently extended to all MBQC on multi-qubit states [12], [13]. From the latter works it is known that all contextuality proofs relevant for MBQC are generalizations of Mermin’s star, in the sense that they invoke an algebraic contra- dictiontotheexistenceofevenasinglenon-contextualconsistentvalueassignment. Byitsintended scope, the present framework only needs to apply to such kinds of proofs. But then there is an additional requirement: the cohomological framework in question needs to reproduce the original parity proofs in a topological guise. The reason for this requirement is that both the parity proofs and the classical side-processing required in every MBQC are based on the same linear relations (See Appendix A for a summary on contextuality in MBQC; also see [18]). Next to the parity-based proofs of contextuality exemplified by Mermin’s square and star, we investigate a different, yet related type of contextuality proof which is based on symmetry. The central object in these proofs is the group of transformations that leave the set of observables involved in a parity-based contextuality proof invariant, up to phases. We show that nontrivial cohomology of the symmetry group implies contextuality. To summarize, we examine proofs of contextuality of quantum mechanics that have two at- tributes. They can either be parity-based or symmetry-based, and be state-independent or state- dependent. There are thus four combinations, and for each of these types of proofs we present a topological formulation. The parity-based contextuality proofs are discussed in Section 4 and the symmetry-based proofs in Section 5. Furthermore, the parity-based and the symmetry-based contextuality proofs are related. Every symmetry-based proof implies a parity-based proof; See Section 5.5. 2 First example To illustrate what “reproducing the original parity proofs in topological guise” means, we consider as a first example Mermin’s square [3] (also see [19]), one of the simplest proofs of contextuality of quantum mechanics. Mermin’s square, depicted in Fig. 1a, demonstrates that in Hilbert spaces of dimension ≥ 4 it is impossible to consistently assign pre-existing values to all quantum mechanical observables. Each row and each column of the square represents a measurement context, consisting of com- muting observables. Furthermore, the observables in each context multiply to ±I. For exam- ple, in the bottom row in Fig. 1a, we have (X Z )(Z X )(Y Y ) = +I, and in the right column 1 2 1 2 1 2 (X X )(Z Z )(Y Y ) = −I. Now assume the nine Pauli observables T in the square have pre- 1 2 1 2 1 2 a existing context-independent outcomes λ(Ta) = (−1)s(Ta), with s(Ta) ∈ Z2 (the eigenvalues of the Pauli observables are ±1). Then, the product relations among the observables translate into con- straintsamongtheconsistentvalueassignments. Continuingwiththeaboveexample,weobtainthe constraints λ(X Z )λ(Z X )λ(Y Y ) = 1, and λ(X X )λ(Z Z )λ(Y Y ) = −1. It is convenient to 1 2 1 2 1 2 1 2 1 2 1 2 express these relations in terms of the value assignments s(·) rather than the measured eigenvalues λ(·), which leads to a system of linear equations, s(X )+s(X )+s(X X ) mod 2 = 0, 1 2 1 2 s(Z )+s(Z )+s(Z Z ) mod 2 = 0, 2 1 1 2 s(X Z )+s(Z X )+(Y Y ) mod 2 = 0, 1 2 1 2 1 2 (1) s(X )+s(Z )+s(X Z ) mod 2 = 0, 1 2 1 2 s(Z )+(X )+s(Z X ) mod 2 = 0, 1 2 1 2 s(X X )+s(Z Z )+s(Y Y ) mod 2 = 1. 1 2 1 2 1 2 2 (a) (b) X1 X2 XX X X2 X 1X 2 1 β=1 Z2 Z Z1 Z2 1 ZZ Z2 Y1Y2 Z2 Z2 X1 Z1X 2 Z1 XZ ZX -YY X2 Figure 1: Mermin’s square [3]. (a) Each horizontal and vertical line corresponds to a measurement context. Each context is composed out of three commuting Pauli observables A,B,C which satisfy the constraint ABC = ±I. (b) Mermin’s square re-arranged on a surface. The 9 Pauli observables are now associated with the edges, and each measurement context is associated with the boundary of one of the six elementary faces. The exterior edges are identified as shown. No assignment s can satisfy these relations. To see this, add the above equations mod 2, and observe that each value s(T ) appears twice on the lhs. This results in the contradiction 0 = 1. a We now reproduce this contradiction in a topological fashion. For this purpose, the six ob- servables are regarded as labeling the edges in a tessellation of a surface; See Fig. 1b. The value assignment s is now a 1-cochain. Denote by f any of the six elementary faces of the surface, such that ∂f = a + b + c, for three edges a, b, c. Then there is a binary-valued function β defined on the faces f such that T = (−1)β(a,b)T T . As before, these product constraints among (com- c a b muting) observables induce constraints among the corresponding values, namely s(a)+s(b)+s(c) mod 2 = β(f). By dialing through the six faces f, we reproduce the six constraints of Eq. (1). These constraints have a topological interpretation. Namely, β is a 2-cochain, and, for any consistent context-independent value assignment s it holds that ds = β. (2) Therein, d the coboundary operator and the addition is mod 2. We can now show that for the present function β, which evaluates to 0 on 5 faces and to 1 on one face, no consistent value assignment s exists. To this end, we integrate over the whole surface F which is a 2-cycle, ∂F = 0. By Stokes’ theorem, (cid:90) (cid:90) (cid:73) (cid:73) 1 = β = ds = s = s = 0, F F ∂F 0 where all integration is mod 2. In chain/cochain notation, this reads 1 = β(F) = ds(F) = s(∂F) = s(0) = 0. This is the same contradiction as above in Eq. (1), but in cohomological form. As we show in Section 4 of this paper, all parity proofs consisting of a set of conflicting linear constraints of the form Eq. (1) can be given a cohomological interpretation. To conclude this section, we remark that the above topological version of Mermin’s square, in its mathematical structure, resembles a certain aspect of electromagnetism [20]. First, consider the vector calculus question of whether a given vector field B can be written as the curl of some vector potential A, i.e., B = ∇ × A. This possibility is ruled out by the existence of a closed (cid:82) surface F for which dF·B (cid:54)= 0. Here, A is a 1-cochain (1-form) and B is a 2-cochain (2-form). F They are the counterparts of the value assignment s and the function β, respectively. Now let B 3 (cid:82) be a magnetic field. The statement dF·B (cid:54)= 0 for some closed surface F—the counterpart of F a contextuality proof β(F) (cid:54)= 0—would indicate the presence of magnetic monopoles. However, in contrast to contextuality [21], magnetic monopoles—while being a theoretical possibility—have to date not been experimentally observed [22]. 3 Measurement and contextuality In this section we define our measurement setting and notion of contextuality. 3.1 Observables In this paper, we consider observables with a restriction on their eigenvalues. Specifically, the eigenvalues are all of the form ωk, where ω = e2πi/d, for some d ∈ Z, and k ∈ Z . For d > 2, such d observables are in general not Hermitian operators. However, that doesn’t matter. We may look at the measurement of these observables in two equivalent ways. (i) The observables are unitary, and their eigenvalues can thus be found by phase estimation. Further, due to the special form of the eigenvalues, phase estimation is exact. (ii) If O = (cid:80)iωsi|i(cid:105)(cid:104)i|, with all si ∈ Zd, one may instead measure O˜ = (cid:80) s |i(cid:105)(cid:104)i|, which is Hermitian and has the same eigenspaces as O. i i Out of the set of observables O, we identify an indexed set {T ,a ∈ E} over a set E. Every a observable O ∈ O is related to an element T from this indexed set by a phase ωk for some k. That a is, O is of the form O = {ωkT |a ∈ E,k ∈ Z }. (3) a d TheproductoftwooperatorsT andT belongstoO iftheycommute, [T ,T ] = 0. Forcommuting a b a b operators the product T T will correspond to an operator T up to a phase. We write c = a+b a b c for this unique element in E. The operators {T } satisfy the relation a a∈E T = ωβ(a,b)T T , ∀a,b ∈ E, s.th.[T ,T ] = 0. (4) a+b a b a b The function β takes values in Z . To see this, consider the simultaneous eigenvalues of the d operators Ta, Tb, Ta+b. With Eq. (4) it holds that ωka+b = ωβ(a,b)+ka+kb, and ka+b,ka,kb ∈ Zd. Thus β(a,b) ∈ Z , as stated. d For any triple {T ,T ,T } of observables satisfying the commutativity condition [T ,T ] = 0, a b a+b a b the simultaneous eigenvalues can be measured. While individually random, the measurement out- comesarestrictlycorrelated,λ(a+b)/λ(a)λ(b) = ωβ(a,b). Thesecorrelations,whicharepredictedby quantummechanicsandareverifiablebyexperiment, formthebasisofMermin’sstate-independent contextuality proofs [3]. The function β is thus a central object in present discussion, summing up the physical properties of O. 3.2 Definition of contextuality We now define the notion of a non-contextual hidden-variable model (ncHVM) with definite value assignments. First, a measurement context is a commuting set M ⊂ O. The set of all measurement contexts is denoted by M. Definition 1 Consider a quantum state ρ and a set O of observables grouping into contexts M ∈ M of simultaneously measurable observables. A non-contextual hidden variable model (S,q ,Λ) ρ consists of a probability distributon q over a set S of internal states and a set Λ = {λ } of ρ ν ν∈S value assignment functions λ : O → C that meet the following criteria. ν 4 (i) Each λ ∈ Λ is consistent with quantum mechanics: for any set M ∈ O of commuting ν observables there exists a quantum state |ψ(cid:105) such that A|ψ(cid:105) = λ (A)|ψ(cid:105), ∀A ∈ M. (5) ν (ii) The distribution q satisfies ρ (cid:88) tr(Aρ) = λ (A)q (ν), ∀A ∈ O (6) ν ρ ν∈S Condition (i) in Definition 1 means that for every internal state ν of the non-contextual HVM the corresponding value assignment λ is consistent across measurement contexts. ν We say that a physical setting (ρ,O) is contextual if it cannot be described by any ncHVM (S,q ,Λ). ρ Lemma 1 For any triple A,B,AB ∈ O of simultaneously measurable observables and any internal state ν ∈ S of an ncHVM (S,q ,Λ) it holds that ρ λ (AB) = λ (A)λ (B). (7) ν ν ν Therelation Eq.(7)was firstusedin [3]torule outtheexistence ofdeterministicvalueassignments for Mermin’s square and star. In the same capacity, it as also of central importance for the present discussion. Proof of Lemma 1. Consider a set M = {A,B,AB} ⊂ O of observables such that [A,B] = 0. This set qualifies as a possible M in the sense of point (i) of Def. 1. Therefore, for any ν ∈ Λ there exists a quantum state |ψ(cid:105) such that A|ψ(cid:105) = λ (A)|ψ(cid:105), B|ψ(cid:105) = λ (B)|ψ(cid:105), AB|ψ(cid:105) = λ (AB)|ψ(cid:105). ν ν ν Furthermore, (AB)|ψ(cid:105) = A(B|ψ(cid:105)) = λ (A)λ (B)|ψ(cid:105). By comparison, λ (AB) = λ (A)λ (B). (cid:3) ν ν ν ν ν 4 Parity-based contextuality proofs The example of Section 2 is not special. As we show here, every parity-based contextuality proof— consisting of a set of conflicting linear constraints on the value assignments as in Eq. (1)—can be given a cohomological formulation. The main result of this section is Theorem 1. 4.1 The chain complex C ∗ We have two assumptions on the set of operators O: 1. O is closed under products of commuting operators i.e., if [O ,O ] = 0 for O ,O ∈ O then 1 2 1 2 O O ∈ O. 1 2 2. O contains the identity operator. Let η : E → O denote the map η(a) = T , with E the index set introduced in Eq. (3). The a set E has more structure coming from Eq. (4). We say two elements a,b ∈ E commute if the corresponding operators commute [T ,T ] = 0. Given two commuting elements a,b ∈ E we define a b the sum a+b ∈ E to be the unique element which satisfies T = ωβ(a,b)T T , cf. Eq. (4). We a+b a b assume that there is an element in E denoted by 0 corresponding to the identity operator η(0) = I in O. Under this addition operation every maximal subset of commuting elements in E has the structure of an abelian group. 5 a b a+b face (b,c) c a + + b c b + c Figure 2: An elementary volume V ∈ C , bounded by four faces. 3 Let us define the chain complex C = C (E). A standard reference for chain complexes is [23]. ∗ ∗ It will suffice to describe this complex up to dimension three C = {C ,C ,C ,C }. The geometric ∗ 0 1 2 3 picture is as follows. The space we consider consists of a single vertex (0-cell). It has an edge (1-cell) for each element of the set E whose both boundary points attached to the single vertex. A face (2–cell) is attached for every product relation among commuting operators. The set of faces is thus given by F = {(a,b) ∈ E ×E| [T ,T ] = 0}. (8) a b Thus, every face (a,b) ∈ F is bounded by three edges, namely a, b and a+b. Volumes (3-cells) are constructed from triples of commuting observables T ,T ,T (see Fig. 2 a b c for an illustration). The set of volumes is V = {(a,b,c) ∈ E ×E ×E| [T ,T ] = [T ,T ] = [T ,T ] = 0}. (9) a b b c a c Now comes the description of the chains: 1. C = Z since there is a single vertex. 0 d 2. C = Z E i.e. its elements are linear combinations 1 d (cid:88) α [a] where α ∈ Z . a a d a∈E In other words C is freely generated as a Z -module by [a] where a ∈ E. 1 d 3. C is freely generated as a Z -module by the pairs [a|b] where (a,b) ∈ F. 2 d 4. C is freely generated as a Z -module by the triples [a|b|c] where (a,b,c) ∈ V. 3 d In summary C ,C ,C are freely generated by E,F,V as a Z -module. We stop at dimension 1 2 3 d three although the definition can be continued for higher dimensions analogously, see [25]. The differentials in the complex ∂ ∂ ∂ C → C → C → C 3 2 1 0 are defined by ∂[a] = 0, ∂[a|b] = [b]−[a+b]+[a], ∂[a|b|c] = [b|c]−[a+b|c]+[a|b+c]−[a|b]. The homology groups of C are defined by ∗ ker(∂) H (C ,Z ) = . n ∗ d im(∂) The dual notion of cochains C∗ gives a cochain complex C3 ←d C2 ←d C1 ←d C0 where Cn consists of Z -module maps φ : C → Z . The differential d : Cn → Cn+1 is defined by d n d dφ(α) = φ(∂α) where α ∈ C . n+1 6 4.2 β is a 2-cocycle We may now formally extend the function β introduced in Eq. (4) from F to all of C via the linear 2 relations β(u+v) = β(u)+β(v), β(ku) = kβ(u), for all u,v ∈ C , k ∈ Z . The function β is thus 2 d a 2-cochain, β ∈ C2. The function β is constrained in the following way. Consider three commuting elements a,b,c ∈ E, and expand the observable T in two ways, a+b+c T = T a+b+c (a+b)+c = ωβ(a+b,c)T T a+b c = ωβ(a+b,c)+β(a,b)T T T , a b c and T = T a+b+c a+(b+c) = ωβ(a,b+c)T T a b+c = ωβ(a,b+c)+β(b,c)T T T . a b c Comparing the two expressions, we find that β(a+b,c)+β(a,b)−β(a,b+c)−β(b,c) mod d = 0, (10) whenever [T ,T ] = 0, [T ,T ] = 0, and [T ,T ] = 0. a b a c b c The four faces (a,b), (a+b,c), (a,b+c), (b,c), with appropriate orientation (hence sign), bound a volume V, i.e., ∂V = (a+b,c)+(a,b)−(a,b+c)−(b,c). Geometrically, the situation looks as displayed in Fig. 2. We can follow the convention that (a,b) denotes a face in the geometric sense and [a|b] denotes an element of the chain complex. So ∂V = [a+b|c]+[a|b]−[a|b+c]−[b|c]. Therefore, with Eq. (10), dβ(V) = β(∂V) = β((a+b,c)+(a,b)−(a,b+c)−(b,c)) = β(a+b,c)+β(a,b)−β(a,b+c)−β(b,c) = 0. Applying this relation to all volumes V ∈ C , we obtain 3 dβ ≡ 0. (11) Finally,thereisanequivalencerelationamongthefunctionsβ. Toseethis,recallthemapη : E −→ O which is defined by a (cid:55)→ T . Note that there is a certain freedom in this definition which does a not affect the commutation relations of the operators. Consider the following re-parametrization η (·) = ωγ(·)η(·), (12) γ where γ : E −→ Z . Then [η(a),η(b)] = 0 if and only if [η (a),η (b)] = 0. From the perspective d γ γ of contextuality, it does not matter which map η we use to define the observables {T ,a ∈ E}. γ a Contextuality cannot be defined away by rephasing. However, the function β is affected by the transformation Eq. (12). Namely, changing from η = η to η results in 0 γ β(a,b) −→ β (a,b) = β(a,b)−γ(a)−γ(b)+γ(a+b) γ (13) = β(a,b)−dγ(a,b). Therein, all addition is mod d. The functions β are thus subject to a restriction Eq. (11) and an identification Eq. (13). The various possible functions β thus fall into equivalence classes [β] = {β+dγ,∀γ}, and hence [β] ∈ H2(C,Z ). d 7 (a) (b) (c) Y 3 X 2 X X 1 1 F1 X3 XXX XXX XYY YXY YYX X Y1 2 X X Y Y2 Y Y2 F4 Y F5 Y F2 Y2 X Y Y Y X 1 1 Y YXY 3 X3 Y3 F3 Y1 X Y2 X2 2 Figure 3: (a) The state-independent version of Mermin’s star [3]. (b) Two elementary three-sided faces combining to a four-sided surface. (c) Topological representation of Mermin’s star. The left and right edges and the top and bottom edges, respectively, are identified. 4.3 Cohomological formulation of parity-based contextuality proofs The function β relates to the question of existence of non-contextual HVMs. We have the following result. First, a non-contextual value assignment s : E −→ Z , is such that λ(T ) = ωs(a). Again, d a by linearity, we can extend the assignment from E to all of C1, and s is thus a 1-cochain. We have the following relation. Lemma 2 For every consistent non-contextual value assignment s : E −→ Z it holds that d ds = −β. (14) Proof of Lemma 2. Evaluating Eq. (14) on any given face (a,b) ∈ F reads s(a)+s(b)−s(a+b) = −β(a,b). (15) Now, Eq.(4)andDef.(1)leadtotheconsistencyconditionωsa+b = ωβ(a,b)ωsaωsb, whichisprecisely what Eq. (15) requires. (cid:3) Theorem 1 Given set O of observables, if H2(C,Z ) (cid:51) [β] (cid:54)= 0 then O exhibits state-independent d contextuality. Proof of Theorem 1. If there were a value assignment s it would satisfy ds = −β. This means that β is a boundary: β = d(−s). Hence [β] = 0. (cid:3) Example: Mermin’s star. In addition to Mermin’s square, which we already discussed in Sec- tion 2, we now provide Mermin’s star [3] as a further example. Mermin’s star comes both in a state-independent and a state-dependent version, and is thus best suited as a running example for all topological constructions presented in this paper. Hereweconsiderthestate-independentversion;SeeFig.3. DenotebyF thesurfacedisplayed star in Fig. 3c, consisting of the five smaller surfaces F ,..,F each corresponding to a measurement 1 5 context in Fig. 3a. Each of the surfaces F may be split up into two elementary faces; See Fig. 3b. i F := (cid:80)5 F satisfies ∂F = 0. Since (X X X )(X Y Y )(Y X Y )(Y Y X ) = −I, we have star i=1 i star 1 2 3 1 2 3 1 2 3 1 2 3 β(F ) = 1, and for the other four measurement contexts it holds that β(F ) = 0. Hence, β(F ) = 5 i star 1. If β = ds for some 1-cochain s, then β(F ) = ds(F ) = s(∂F ) = s(0) = 0. Since this is star star star not the case, it holds that [β] (cid:54)= 0. Then, by Theorem 1, Mermin’s star exhibits state-independent contextuality, in accordance with the original proof [3]. 8 (a) (b) X X2 X1X2 X 3 X X2 X1X2 1 1 XXX Y2 Y2 X Y Y1 Y1 X X X3 Y2 YY Z1Z2 YX Y2 Y2 Z1Z2 Y2 X X Y2 X Y2 1 2 X1 X1 YXY X Y Z1 Z Y Y 1Y 2 Y3 Y1X2 Y1 Y1X2 Y1 2 X2 X2 Figure 4: Equivalence between Mermin’s square and star. (a) Volume V ∈ C (3) of Eq. (16). (b) 3 Flipping between the surfaces F (left) and F (right), by adding the boundary ∂V. star square 4.4 Squaring the star It tuns out that, from the cohomological perspective developed above, Mermin’s square and star are equivalent contextuality proofs. Denote by C (3) the complex induced by the set O = P3, the ∗ Pauli observables on 3 qubits. Both Mermin’s square and star embed into it. The star provides a closed surface F ∈ C (3) and the square provides a closed surface F ∈ C (3), such that star 2 square 2 β(F ) = 1 and β(F ) = 1. Both facts thus equally demonstrate that β (cid:54)= 0 ∈ H2(C (3),Z ). star square ∗ 2 Furthermore, there is a volume V ∈ C (3) such that 3 F = F +∂V. (16) square star Thatis,thesurfacesF andF representingtherespectivecontextualityproofsareelementsof square star the same homology class in H (C (3),Z ); and β(F ) = β(F ), for any 2-cocycle β. Mermin’s 2 ∗ 2 square star square and star are thus equivalent. The volume V of Eq. (16) is depicted in Fig. 4a. The surfaces F and F are shown in star square Fig. 4b. They are obtained from another by adding the boundary ∂V. The Mermin star resulting from this procedure is locally rotated w.r.t. the standard convention, namely X1 X2 XX Y Y 1 YY 2 XY YX ZZ 4.5 State-dependent parity proofs Mermin’sstar—whosestate-independentversionwasdiscussedinSection4.3—alsoexistsinastate- dependent version [3]. We use it as an initial example, to illustrate the adaption of the topological argument to the state-dependent case and to motivate the definitions Eq. (17), Def. 2 below. The state-dependent Mermin star contains a special set S = {X X X ,X Y Y ,Y X Y ,Y Y X } of 1 2 3 1 2 3 1 2 3 1 2 3√ observables and a special state, the Greenberger-Horne-Zeilinger state |GHZ(cid:105) = (|000(cid:105)+|111(cid:105))/ 2. 9 The latter is a simultaneous eigenstate of the observables in S, with eigenvalues +1,−1,−1,−1, respectively. There is thus a value assignment s(XXX) = 0, s(XYY) = s(YXY) = s(YYX) = 1. From the perspective of non-contextual hidden variable models, the question is whether the value assignment s can be extended in a consistent fashion to the local observables X and Y . i i Adapting the topological state-independent argument, we now demonstrate that this is not the case. We choose the mapping η such that X X X ,X Y Y ,Y X Y ,Y Y X ,X ,Y ∈ η(E), and 1 2 3 1 2 3 1 2 3 1 2 3 i i consider the surface F = (cid:80)8 f displayed in Fig. 5b. For any consistent value assignment s we i=1 i thus have s(∂F) = s(XXX)+s(XYY)+s(YXY)+s(YYX) mod 2 = 1. On the other hand, β(f ) = 0, for i = 1,..,8. Thus, assuming the existence of a consistent value assignment s, with i ds = β (cf. Lemma 2) and with Stokes’ theorem, we arrive at the following contradiction (addition mod 2): (cid:90) (cid:90) (cid:90) 0 = β = ds = s = 1. F F ∂F Hence our assumption that a consistent value assignment exists must be wrong. We now turn to the general state-dependent scenario. Any state-dependent contextuality proof singles out a subset O ⊂ O of observables of which a special state |Ψ(cid:105) is an eigenstate. Namely, Ψ O := {O ∈ O| ∃s ∈ Z such that O|Ψ(cid:105) = ωsO|Ψ(cid:105)}. (17) Ψ O d The set O may or may not be a context. It is required of O that the observables therein have Ψ Ψ at least one joint eigenstate, |Ψ(cid:105), but it is not required of them that they commute. We want to integrate this extra bit of information into our topological description. By the definition of O and Eq. (17), the set O has an extra property that whenever [O ,O ] = 0 for Ψ 1 2 O ,O ∈ O the product O O also lies in O . We need this condition to be able to construct a 1 2 Ψ 1 2 Ψ subcomplex of C = C (E). The corresponding labels determine a subset E ⊂ E of edges and a ∗ ∗ Ψ subcomplex C (E ) whose definition is analogous to C . ∗ Ψ ∗ Let us define s : E → Z via Eq. (17), i.e., Ψ Ψ d T |Ψ(cid:105) = ωsΨ(a)|Ψ(cid:105), ∀a ∈ E . (18) a Ψ We can regard s as an element of C1(E ) by extending it linearly. A consistent value assignment Ψ Ψ in the state-dependent case has to be compatible with the eigenvalues on the given state. This suggests the following definition. Definition 2 A state-dependent consistent value assignment is a function s : E → Z such that d its restriction to E coincides with s and satisfies Ψ Ψ s(a)+s(b)−s(a+b) = β(a,b) (19) for all commuting (a,b) ∈/ E ×E . Ψ Ψ According to Eq. (19) only the commuting labels which are not contained in E matters. Geomet- Ψ rically we can get rid of the edges in E by contracting them. For example in the case of Mermin’s Ψ star this process is depicted in Fig. 5. On the algebraic side the chain complex of the contracted space is described by the relative complex defined by the quotient C (E,E ) = C (E)/C (E ). ∗ Ψ ∗ ∗ Ψ 10