LLogical Methods in Computer ScienceVolume 15, Issue 2, 2019, pp. 3:1–3:51https://lmcs.episciences.org/ Submitted Apr. 25, 2017Published Apr. 26, 2019
GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS
STEFANO GOGIOSO AND WILLIAM ZENGQuantum Group, University of Oxford, UK e-mail address : [email protected] Computing, Berkeley, CA e-mail address : [email protected]
Abstract.
We broadly generalise Mermin-type arguments on GHZ states, and we provideexact group-theoretic conditions for non-locality to be achieved. Our results are of interestin quantum foundations, where they yield a new hierarchy of quantum-realisable All-vs-Nothing arguments. They are also of interest to quantum protocols, where they findimmediate application to a non-trivial extension of the hybrid quantum-classical secretsharing scheme of Hillery, Buˇzek and Berthiaume (HBB). Our proofs are carried out inthe graphical language of string diagrams for dagger compact categories, and their validityextends beyond quantum theory to any theory featuring the relevant algebraic structures.
Introduction
Non-locality is a defining feature of quantum mechanics, and its connection to the structure ofphase groups is a key foundational question. A particularly crisp example of this connection isgiven by Mermin’s argument for qubit GHZ states [Mer90], which finds practical applicationin the HBB quantum secret sharing protocol.In Mermin’s argument, N qubits are prepared in a GHZ state (with Pauli Z as compu-tational basis), then a controlled phase gate is applied to each, followed by measurement inthe Pauli X observable. Even though the N outcomes (each valued in Z ) are probabilistic,their parity turns out to satisfy certain deterministic equations. Mermin shows that theexistence of a local hidden variable model would imply a joint solution for the equations,which however form an inconsistent system. Mermin concludes that the scenario is non-local.Mermin’s argument has sparked a number of lines of enquiry, and this work is concernedwith two in particular: one leading to All-vs-Nothing arguments, and the other investigatingthe role played by the phase group. All-vs-Nothing arguments [ABK +
15] arise in the contextof the sheaf-theoretic framework for non-locality and contextuality [AB11], and generalisethe idea of a system of equations which is locally consistent but globally inconsistent. Thesecond line of research is brought forward within the framework of categorical quantum
Key words and phrases:
Mermin non-locality, categorical quantum mechanics, strongly complementaryobservables, All-vs-Nothing arguments, quantum secret sharing.An extended abstract for a small part of this work has appeared in the Proceedings of QPL 2015, and canbe found in EPTCS 195, 2015, pages 228-246, doi:10.4204/EPTCS.195.17.
LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.23638/LMCS-15(2:3)2019 c (cid:13)
S. Gogioso and W. Zeng CC (cid:13) Creative Commons :2 S. Gogioso and W. Zeng
Vol. 15:2 mechanics [AC09, CK15, CK17], and it focuses on the algebraic characterisation of phasegates and strongly complementary observables.A detailed analysis of Mermin’s argument shows that the special relationship betweenthe Pauli X and Pauli Z observables, known as strong complementarity, is key to itssuccess [CDKW12]. A pair of complementary observables corresponds to mutually unbiasedorthonormal bases: for example, both Pauli X and Pauli Y are complementary to Pauli Z .Strong complementarity [CD11, DD16] amounts to a strictly stronger requirement: if oneobservable is taken as the computational basis, the other must correspond to the Fourierbasis for some finite abelian group. Pauli X fits the bill, for the abelian group Z , but Pauli Y doesn’t (Pauli X is the only single-qubit observable strongly complementary to Pauli Z ).In [CDKW12], Mermin’s argument is completely reformulated in terms of stronglycomplementary observables (using † -Frobenius algebras) and phase gates. It can therefore betested on theories different from quantum mechanics, to better understand the connectionbetween non-locality and the structure of phase groups. A particularly insightful comparisonis given by qubit stabiliser quantum mechanics [CD11, Bac14] vs Spekkens’ toy model[Spe07, CE12]: both theories sport very similar operational and algebraic features, but thedifference in phase groups ( Z for the former vs Z × Z for the latter) results in the formerbeing non-local and the latter being local (both models have Z as group of measurementoutcomes, like Mermin’s original argument). The picture arising from comparing qubitstabiliser quantum mechanics and Spekkens’ toy model is iconic, and provides a first realglimpse into the connection between phase groups and non-locality [CES10].While presenting an extremely compelling case for stabiliser qubits and Spekkens’ toyqubits, the work of Refs. [CDKW12, CES10] does not treat the general case (i.e. beyond Z asgroup of measurement outcomes), nor does it provide a complete algebraic characterisation ofthe conditions guaranteeing non-locality. In this work, we fully generalise Mermin’s argumentfrom Z to arbitrary finite abelian groups, in arbitrary theories and for arbitrary phasegroups (we will refer to these as generalised Mermin-type arguments ). We also provideexact algebraic conditions for non-locality to be exhibited by our generalised Mermin-typearguments, thus bringing this line of investigation to a satisfactory conclusion.We proceed to make contact with the All-vs-Nothing line of enquiry [ABK + Synopsis of the paper.
In Section 1, we provide a brief recap of the technical backgroundfor this paper. We quickly review dagger compact categories, the CPM construction,Frobenius algebras, the CP* construction and the sheaf-theoretic framework for non-localityand contextuality. We introduce a new flavour of CP* categories, and prove some resultsconnecting them to the sheaf-theoretic framework for non-locality and contextuality.In Section 2, we present Mermin’s original argument in detail, deconstructing it in theinterest of our upcoming generalisation. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:3
In Sections 3 and 4, we introduce complementarity, strong complementarity and phasegroups, and elaborate on the relationship that ties them together. Some of the simplerresults are common knowledge in the field, and have appeared in similar form in other works(such as [CDKW12, Kis12, CK17]): they are re-proposed here to achieve a uniform, coherentpresentation of the material.In Section 5, we use strong complementarity and phase groups to formulate our gener-alised Mermin-type arguments, and we prove the exact algebraic conditions for the modelsto be non-local. In Section 6, we prove that our arguments are all quantum realisable. InSection 7, we connect with the All-vs-Nothing framework, showing that our argumentsalways result in All-vs-Nothing models whenever they are non-local.In Section 9, finally, we present an extension of the HBB quantum-classical secretsharing scheme to our newly generalised Mermin-type arguments, and we provide some noveldevice-independent guarantees of security.1.
Background
Categories for quantum theory.
The framework of dagger compact categories capturessome of the most fundamental structural features of pure-state quantum mechanics [AC09,CK15, CK17]: symmetric monoidal structure captures its characterisation as a theoryof processes, composing sequentially and in parallel; the dagger captures the state-effectduality induced by the inner product; the compact closed structure captures operator-stateduality. Dagger compact categories are commonplace in the practice of categorical quantummechanics, and we will assume the reader is familiar with them. We recommend [Sel09] fora detailed treatment of many subtle technicalities in the various constructions. The Hilbertspace model of (finite-dimensional) pure-state quantum mechanics corresponds to the daggercompact category fdHilb of (finite-dimensional) complex Hilbert spaces and linear mapsbetween them, while the operator model of mixed-state quantum mechanics corresponds tothe CPM category CPM[fdHilb].Categories of completely positive maps, also known as
CPM categories , can beconstructed for all dagger compact categories, in a process which mimics the way in whichthe operator model of mixed-state quantum mechanics is constructed from the Hilbertspace model of pure-state quantum mechanics [Sel07]. Given a dagger compact C , thecorresponding CPM category CPM[ C ] is defined as the subcategory of C having morphismsin the following form: ff ∗ EE ∗ B ∗ BA ∗ A (1.1)where f : A → B ⊗ E is a morphism of C , and f ∗ : A ∗ → E ∗ ⊗ B ∗ is its conjugate,obtained via the dagger compact structure. Processes in the CPM category are called completely positive (CP) maps . The system E in Diagram 1.1 is often interpreted asan environment which is operationally inaccessible, and hence must be “discarded” afterthe process has taken place. In the case of CPM[fdHilb], i.e. in the operator model ofmixed-state quantum mechanics, Diagram 1.1 can then be seen as an alternative formulationof Kraus decomposition. :4 S. Gogioso and W. Zeng
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Because diagrammatic reasoning about categories of completely positive maps ofteninvolves two distinct SMCs (the original category C and the CPM category CPM[ C ]), astylistic choice is adopted where systems and processes of the CPM category are denotedby thicker wires, boxes and decorations. For example, the “doubled” version f ⊗ f ∗ of aprocess f : A → B ⊗ E will be denoted as f with thicker wires and box: f ∗ f BB ∗ AA ∗ fA B := (1.2)The caps from compact closed structure play a particularly important role in the definitionof the CPM category, and are given their own decoration: A := AA ∗ (1.3)The CP map double [ f ] defined by Equation 1.2 is called the double of process f , whilethe CP map A defined by Equation 1.3 is called the discarding map on system A .In mixed-state quantum mechanics, the double of a linear map f is the rank-1 CP map double [ f ] : ρ (cid:55)→ f ◦ ρ ◦ f † , while the discarding map A sends a positive state ρ ∈ L [ A ] toits trace Tr[ ρ ] ∈ L [ C ] ∼ = R + .The discarding maps are also called an environment structure in the literature [CP10],and are tightly related to causality , an important feature arising at the interface betweenquantum theory and relativity [CL13, CK15, CDP11]. We say that a process is normalised (sometimes also called causal ) if performing it and then discarding the output is the sameas discarding the input: fA = AB (1.4)In particular, discarding normalised states results in the scalar 1.CPM categories CPM[ C ] are dagger compact, and the rules of diagrammatic reasoningfor dagger compact categories apply to them. The compact structure for CPM[ C ] is given bythe doubles of the cups and caps of C , while the adjoint of a process in the form of Diagram1.6 is given by first taking the adjoint in C , and then using the following equation for theadjoint of the discarding map: A = † A := AA ∗ A (1.5)Because the doubled processes double [ f ] and the discarding maps A are well-defined CPmaps, it is legitimate to rephrase the very definition of the CPM category by saying that itsprocesses are exactly those in the following form: fA B (1.6)This means that doubled processes and discarding maps are enough to express all CP maps,but to prove results about CP maps we need a graphical axiom relating a generic CPMcategory CPM[ C ] to the corresponding original category C . The required relationship is ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:5 encoded by the following CPM axiom , which characterises the action of discarding mapsin CPM[ C ] in terms of the dagger structure of C : fA = ⇔ gAf f † A A A g † Ag = in CPM[ C ]in C (1.7) Frobenius algebras.
Frobenius algebras are a fundamental ingredient of quantum theory,where they are intimately related to the notion of observable. A † -Frobenius algebra on an object A of a dagger symmetric monoidal category (henceforth † -SMC) is given bya monoid ( A, , ) on A (i.e. : A ⊗ A → A is associative and has : I → A as itsbilateral unit), and a corresponding comonoid ( A, , ), which are related by the followingFrobenius law:
AAAA multiplication unit A comultiplication AA A counit
AA AAA AA A A AAA = =Frobenius law (1.8)A † -Frobenius algebra is said to be special if the comultiplication is an isometry, and commutative if the monoid and comonoid are commutative: AA = AA = AA A AA A speciality commutativity (1.9)More in general, a quasi-special † -Frobenius algebra is one with comultiplication whichis an isometry up to a normalisation factor N , where N is in the form n † n for someinvertible scalar n : AA = AA quasi-speciality N (1.10)Because a several combinations of these properties will play a role in this work, we nowintroduce a number of short-hands for † -Frobenius algebras: † -Frobenius algebras commutative arbitraryspecial † -SCFA † -SFAquasi-special † -qSCFA † -qSFAarbitrary † -CFA † -FA :6 S. Gogioso and W. Zeng
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The importance of † -SCFAs in categorical quantum mechanics comes from the fact thatthey correspond to orthonormal bases, i.e. non-degenerate quantum observables. Key tothis correspondence is the notion of classical states for a † -FA , those states ψ which arecopied/transposed/deleted by in the following sense: ψ ψ ψψ = = ψ ψ † =copy transpose delete (1.11) Theorem 1.1 [CPV13] . We denote the set of classical states for a † -FA by K ( ) . In fdHilb , the classical states for a † -SCFA always form an orthonormal basis . Furthermore,any orthonormal basis arises this way for a unique † -SCFA. More in general, if is a † -qSCFA, with normalisation factor N , then the classical states for form an orthogonalbasis, each state having norm √ N . Furthermore, any orthogonal basis where all states havethe same norm √ N arises this way for a unique † -qSCFA. The concept of classical states forming a basis is generalised to arbitrary † -SMCs by thenotion of enough classical states. A † -FA on an object A is said to have enough classicalstates if its classical states separate morphisms from A , i.e. two morphisms f, g : A → B are equal whenever they satisfy f ◦ ψ = g ◦ ψ for all classical states ψ of . Because of thecopy condition, a † -FA with enough classical states is always commutative.This algebraic characterisation of quantum observables is not limited to the non-degenerate case of orthonormal bases, but can be extended to the more general case ofcomplete families of orthogonal projectors. To do so, one considers balanced-symmetric † -Frobenius algebras , i.e. those satisfying the following equation:= AA AA (1.12)
Theorem 1.2 [Vic11] . In fdHilb , balanced-symmetric † -SFAs are in bijective correspondencewith C*-algebras, and hence with complete families of orthogonal projectors. The correspondence between balanced-symmetric † -SFAs and C*-algebras will not play arole in this work, but it helps to frame the broad role played by † -Frobenius algebras incategorical quantum mechanics.Further to their correspondence with quantum observables, † -Frobenius algebras finddirect use as fundamental building blocks of quantum algorithms and protocols [Vic12,BH12, CK17, GK17]. When designing quantum protocols, classical data is often encodedinto quantum systems using orthonormal bases. In this context, the four processes in a † -SCFA can be seen as coherent versions of the basic data manipulation primitives:(a) the comultiplication = | ψ x (cid:105) (cid:55)→ | ψ x (cid:105) ⊗ | ψ x (cid:105) is the coherent copy of classical data;(b) the counit = | ψ x (cid:105) (cid:55)→ | ψ x (cid:105) ⊗ | ψ y (cid:105) (cid:55)→ δ xy | ψ x (cid:105) is the coherent matching of classical data;(d) the unit = (cid:80) x | ψ x (cid:105) is the coherent superposition of classical data (up to normalisation).In this sense, † -SCFAs in general † -SMC are often interpreted as modelling coherentcopy/delete/matching operations on some kind of classical data (usually modelled by their The copy and delete conditions are sufficient to characterise classical states in the case of fdHilb. Commutative † -FAs are a special case of balanced-symmetric † -FAs. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:7 classical states) . More in general, balanced-symmetric † -SFAs can be though of as coherentmanipulation of data which carries some residual entanglement after being copied .Because of their diagrammatic definition, † -FA in a dagger compact category C give riseto † -FA in the CPM category CPM[ C ] by doubling: the † -FA in CPM[ C ] that arise this wayare said to be canonical . In this work, we will only consider canonical † -FA when workingwith CPM categories. The CP* construction.
In the framework of mixed-state quantum mechanics, classicalsystems can be though of as quantum systems which are constantly undergoing decoherencein some basis. In CPM[fdHilb], the decoherence map dec in an orthonormal basis ( | x (cid:105) ) x is the one zeroing out all non-diagonal elements of a positive state:dec := ρ (cid:55)→ (cid:88) x | x (cid:105) ( (cid:104) x | ρ | x (cid:105) ) (cid:104) x | (1.13)The decoherence map can be written as follows in terms of the associated † -SCFA ::=dec (1.14)This means that a decohered quantum system can be though of as having undergone acoherent copy operation, with one copy lost in the environment. We take Equation 1.14 tobe the definition of decoherence maps for arbitrary canonical balanced-symmetric † -SFAs inarbitrary CPM categories.Starting from a CPM category CPM[ C ], we wish to construct a new category whichincludes some kind of classical systems, as defined by decoherence maps. This new categoryCP ∗ [ C ], known as the CP* category , can be defined as follows:(i) the objects are pairs (
A, a ) of an object A of CPM[ C ] and a normalised self-adjointidempotent process a : A → A taking either the form a := id A or the form a := decfor some balanced-symmetric † -SFA on A ;(ii) the morphisms ( A, a ) → ( B, b ) in CP* are the morphisms f : A → B in CPM[ C ] inCPM which are invariant under the specified idempotents on A and B , i.e. thosewhich satisfy b ◦ f ◦ a = f .Note that this definition is a hybrid of the perspective of Ref. [Sel08], based on decoherencemaps and the Karoubi envelope, and the perspective of Refs. [CHK14, CH15], based onC* algebras and quantum logic. Because we have picked processes to be invariant underself-adjoint idempotents, CP ∗ [ C ] is always dagger compact.The CP* category contains CPM[ C ] as the full subcategory associated with objects inthe form ( A, id A ): we refer to the latter as CPM objects , and we simply denote themby A for simplicity. We refer to the CP* objects in the form ( A, dec ) as super-selectedobjects , and often denote them by ( A, ) for simplicity.In the case of CP ∗ [fdHilb], a balanced-symmetric † -SFA on a finite-dimensional Hilbertspace A corresponds, by Theorem 1.2, to a complete family ( P j ) j of orthogonal projectors . This extends straightforwardly to † -qSCFA and their unnormalised classical states. In quantum mechanics, this is because non-demolition measurements in a degenerate observable onlybreaks entanglement between the subspaces associated with distinct projectors, but not within each subspace. We will keep using the doubled notation for them and morphisms between them. I.e. P i ◦ P j = δ ij P i and (cid:80) i P i = id A . :8 S. Gogioso and W. Zeng
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The associated decoherence map dec takes the following concrete form:dec = ρ (cid:55)→ (cid:88) j P j ρ P † j (1.15)This means that objects in CP ∗ [fdHilb] truly are super-selected quantum systems, withsuper-selection sectors given by the domains of the projectors ( P j ) j . In particular, super-selected objects associated to † -SCFAs (corresponding to non-degenerate observables, i.e.families of 1-dimensional projectors) behave as classical probabilistic systems.If is a balanced-symmetric canonical † -SFA on an object A , the decoherence mapdec is always a process dec : A → A in CP ∗ [ C ]. Because of idempotence, however, itis also a process A → ( A, ) and a process ( A, ) → A : we will refer to the former asthe measurement in , and the latter as the preparation in . The single and doublednotation distinguish between the different cases:( A, ) A measurement := : A → ( A, ):=( A, ) : ( A, ) → AA preparation := A : A → AA decoherence ( A, ) :=identity ( A, ) : ( A, ) → ( A, ) (1.16)In CP ∗ [fdHilb], preparations and measurements for a † -SCFA associated to an orthonormalbasis ( | x (cid:105) ) x ∈ X of a finite-dimensional Hilbert space A take the familiar form traditionallyadopted by the literature: (cid:55)→ (cid:16) (cid:104) x | ρ | x (cid:105) (cid:17) x ∈ K ( ) = ρ ( A, ) A ( A, ) A preparationmeasurement = x (cid:55)→ | x (cid:105)(cid:104) x | (1.17)Demolition measurements are traditionally thought to result in some kind of classical(probabilistic) data. Unfortunately, our framework does not yet allow us to conclude anythingof the sort, as we lack an appropriate definition of classical systems to work with. Followingthe footsteps of the sheaf-theoretic framework for non-locality and contextuality [AB11], wegeneralise probabilities from R + to some arbitrary commutative semiring R . For a fixedcommutative semiring R , we define our category of classical R -probabilistic systems tobe the category R -Mat of free, finite-dimensional R -semimodules and R -semilinear mapsbetween them:(a) the objects of R -Mat are in the form R X for all finite sets X ;(b) the morphisms R X → R Y in R -Mat are Y ⊗ X matrices with values in R (c) R -Mat is a SMC, with fSet (finite sets and functions) as a sub-SMC;(d) R -Mat inherits the discarding maps X := x (cid:55)→ (cid:63) of fSet;(e) R -Mat is enriched in itself, with morphisms R Y ⊗ X forming the free finite-dimensional R -semimodule R Y ⊗ X .If we generalise this definition to involutive commutative semirings R , i.e. those comingwith an involution † : R → R , the category R -Mat is in fact a dagger compact category. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:9 The traditional definition of classical probabilistic systems corresponds to workingin R + -Mat: normalised states are probability distributions over finite sets, and normalisedprocesses are stochastic maps (also, we always think of R + as coming with the trivial involu-tion id R + ). However, using arbitrary semirings opens the way to interesting generalisations:a prominent example is that of classical possibilistic systems , which are associated tothe semiring R = B of the booleans and play a large role in the sheaf-theoretic frameworkfor non-locality and contextuality. Definition 1.3.
We say that a SMC D is distributively CMon -enriched if the followingconditions hold:(1) the category is CMon-enriched, i.e. morphisms A → B form a commutative monoid (cid:16) Hom D [ A, B ] , + , (cid:17) for any fixed objects A, B ;(2) the tensor product ⊗ , associators and unitors are all linear.The definition can be extended to a † -SMC (or dagger compact category) D by asking thatthe dagger also be linear.The scalars of SMCs which are distributively CMon-enriched always form a commutativesemiring R (which is furthermore involutive in the case of † -SMCs), and all homsetsautomatically inherit the structure of R -semimodules. We use this observation to defineclassical systems within the context of CP* categories. Definition 1.4.
We say that a CP ∗ [ C ] is an R -probabilistic CP* category , or an R -probabilistic theory , if it satisfies the following conditions.(i) The dagger compact category CP ∗ [ C ] is distributively CMon-enriched, with R as itsinvolutive semiring of scalars .(ii) For each n ∈ N , there is some super-selected system ( A, ) in CP ∗ [ C ] such that:(a) is a † -SCFA with enough classical states;(b) the classical states of are mutually orthogonal.(c) has exactly n classical states;In this context, we refer to super-selected systems ( A, ) satisfying conditions (a) and (b)above as classical systems . Hence requirement (ii) above can be rephrased to say that forevery n ∈ N there is a classical system with n classical states. Theorem 1.5.
The full sub-SMC of an R -probabilistic CP* category spanned by the classicalsystems is equivalent to R -Mat .Proof. All we really need to show is that processes ( A, ) → ( B, ) between two classicalsystems in the CP* category form an R -module which is isomorphic to the R -module ofprocesses R K ( ) → R K ( ) in the category R -Mat of classical R -probabilistic systems. Firstly,every process f : ( A, ) → ( B, ) is determined by the R -valued matrix obtained by testingagainst classical states of the two † -SCFAs: ( B, ) =( A, ) (cid:80) x ∈ K ( ) (cid:80) y ∈ K ( ) f fx y yx ( A, ) ( B, ) (1.18) Equivalently, we can ask for CPM[ C ] to be enriched, as the two categories mutually inherit enrichment,scalars and discarding maps. :10 S. Gogioso and W. Zeng
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Secondly, every matrix (cid:0) F yx (cid:1) y ∈ K ( ) x ∈ K ( ) corresponds to a unique process ( A, ) → ( B, ): (cid:80) x ∈ K ( ) (cid:80) y ∈ K ( ) yx ( A, ) ( B, ) F yx (1.19)Because we have required that for each n ∈ N there be a classical system with n classical states,the bijective correspondence above establishes the desired equivalence of categories.As a special case, classical systems in CP* categories with R + as their semiring of scalarsbehave exactly like classical probabilistic systems, and the entire toolbox of probabilitytheory becomes available: we will refer to these as probabilistic theories , and they willcome into play in the context of our very last result. Since the time this work was firstwritten, a stand-alone framework for probabilistic theories capturing the constructions abovehas been developed by one of the authors, and can be found in Ref. [GS17]. Examples ofnotable quantum-like theories captured by this framework can be found in Ref. [Gog17]. Non-locality and contextuality.
Consider the abstract setup of a Bell-type scenario:(i) N parties are given devices B , ..., B N which might share some global state ρ ;(ii) each device B j takes an input, the measurement choice , freely chosen by party j from some finite set M j ;(iii) upon receiving input m j ∈ M j , the device B j produces some output o j in some finiteset O j , the measurement outcome ;(iv) no signalling is possible between the devices from before the first input is given toafter the last outputs has been produced.The sheaf-theoretic framework for non-locality and contextuality [AB11] characterises thedistribution of joint outputs conditional to joint inputs from the point of view of sheaftheory, showing that non-locality and contextuality are related to the (non-) existence ofglobal sections for a particular presheaf. The framework does not rely on any concretedescription of the state ρ or the devices B , ..., B N , focusing instead on the distributionalproperties of joint outputs/measurement outcomes o j := ( o , ..., o N ) conditional to the choice M := ( m , ..., m N ) of joint inputs/measurement choices.The framework begins by identifying a finite set X of inputs, which in the Bell-typescenario setup above (the one used in this work) would be X = (cid:116) Nj =1 M j . The disjoint unionpreserves information about which party each measurement is associated to, so we will adoptthe notation m j for generic elements of X , where m is the measurement and j is the party.For each subset U ⊆ X , the family of all potential joint outcomes takes the followingform: E [ U ] := (cid:89) m j ∈ U O j (1.20)The powerset P ( X ) is a poset (hence a poset category) under inclusion V ⊆ U of subsets.We can define a functor E : P ( X ) op → Set, i.e. a presheaf , by setting:(i) if U ∈ P ( X ), then we define E [ U ] := (cid:81) m j ∈ U O j as above(ii) if V ⊆ U , then we define E [ V ⊆ U ] := res UV to be the restriction map U Set −→ V :res UV = s (cid:55)→ s | V (1.21) Not all subsets of measurements need be compatible in each concrete scenario: see below for the definitionof measurement contexts. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:11 A section s over U is a U -indexed family of outcomes in the following form: s = { ( m j , s ( m j )) | m j ∈ U } ∈ (cid:89) m j ∈ U O j (1.22)The restriction map then sends a section s over U to its restriction over V : s | V = { ( m j , s ( m j )) | m j ∈ V } ∈ (cid:89) m j ∈ V O j (1.23)The definition of the set of possible joint inputs requires further consideration: itis a fundamental feature of quantum mechanics that not all measurements on a systemare compatible, and we should not expect different measurement choices in each M j tohave a consistent assignment of outputs. Instead, the framework requires us to specify aset M of measurement contexts , subsets C ⊆ X of measurements which are mutuallycompatible (and therefore have a well-defined notion of joint outcome). Even though moregeneral setups are allowed, we will assume that our measurement contexts all take the form C = { m , ..., m N } for m j ∈ M j , which we will denote by m : each party chooses exactly oneinput for their device, but we allow the possibility that not all combinations of inputs mightbe allowed/interesting. The only requirement is that ∪ C ∈M C = X , i.e. that M be a globalcover of X (each measurement choice for each player appears in at least one measurementcontext), which we assume to be endowed with the discrete topology. One can also definethe local covers for any U ⊆ X as the families ( U i ) i ∈ I such that ∪ i ∈ I U i = U .The choice of the discrete topology on X makes P ( X ) its locale of open subsets, and onecan define a notion of sheaf on it. Because it is defined in terms of sections , the presheaf E is in fact a sheaf on the locale P ( X ), and we shall refer to it as the sheaf of events .The measurement cover and the sheaf of events are the two ingredients required to definea measurement scenario ( E , M ): the former gives the compatible joint measurementchoices, while the latter gives the joint measurement outcomes conditional on all possiblemeasurement choices.The next step in the framework sees the introduction of generalised notions of prob-abilities and distributions. In quantum mechanics, probabilities can be seen as takingvalues in the commutative semiring R = ( R + , + , , · ,
1) of the non-negative reals (in factthey fall within the interval [0 , R of the normalisationcondition requiring that probabilities add up to 1). In other circumstances, one may beinterested in the possibilities associated with events, living in the commutative semiring B = ( { , } , ∨ , , ∧ ,
1) of the booleans. In the sheaf-theoretic treatment of contextuality, oneworks with an arbitrary commutative semiring R = ( | R | , + , , · , U , an R -distribution on U is a function d : U → R which has finite support supp d := { s ∈ U | d ( s ) (cid:54) = 0 } and such that (cid:80) s ∈ supp d d ( s ) = 1. One can thendefine a functor D R : Set → Set as follows:(i) for any set U , define D R [ U ] to be the set of R -distributions of U (ii) for any function f : U → V , define D R [ f ] := d (cid:55)→ (cid:104) t (cid:55)→ (cid:80) f ( s )= t d ( s ) (cid:105) .Composing this functor with the sheaf of events yields the presheaf of distributions D R E : P ( X ) op → Set, which captures the structure of R -distributions on joint measurementoutcomes under marginalisation. The presheaf sends each set U of measurements (the Compatibility of local sections amounts to compatibility over the intersection of the domains, and hencecompatible local sections can always be glued together by taking their union as relations. :12
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Vol. 15:2 objects of the presheaf category P ( X )) to the set D R E [ U ] of R -distributions on U -sections , and sends any inclusion V ⊆ U (the morphisms of the presheaf category P ( X ))to the corresponding marginalisation of distributions: D R E [ V ⊆ U ] = d (cid:55)→ d | V := t (cid:55)→ (cid:88) s | V = t d ( s ) (1.24)We will refer to d | V as the marginal of d .In quantum mechanics, if C is a set of compatible measurements on some state | ψ (cid:105) , thenthere is a probability distribution d ∈ D R + E [ C ] on the joint outcomes of the measurements,and the typical contextuality argument involves showing that the probability distributions ondifferent contexts cannot be obtained, in a no-signalling scenario, as marginals of some non-contextual hidden variable. In the sheaf-theoretic framework, a (no-signalling) empiricalmodel is defined to be a compatible family of distributions ( ζ C ) C ∈M for the global cover M of measurement contexts; the usual no-signalling property is shown in [AB11] to be aspecial case of the compatibility condition. In other literature (usually treating probabilisticmodels), empirical models for Bell-type scenarios are usually given explicitly as conditional(probability) distributions, in a format akin to the following: ζ m (cid:0) o (cid:1) := P (cid:2) o (cid:12)(cid:12) m (cid:3) (1.25)where m = ( m , ..., m N ) ∈ M are the measurement contexts used by the scenario and o ∈ (cid:81) j O j are the joint outcomes. This is the format we will use in the last section ofthis work. In the probabilistic case, empirical models for a fixed scenario form a polytope.However, this need not be the same as the no-signalling polytope which usually studied inquantum information theory, because the set of measurement contexts need not include allpossible combinations of all possible measurements for each party (i.e. it need not be thecase that M = (cid:81) j M j , although it is the case that M ⊆ (cid:81) j M j ).A global section for an empirical model ( ζ C ) C ∈M is a distribution d ∈ D R E [ X ] overthe joint outcomes of all measurements which marginalises to the distributions specified bythe empirical model: d | C = ζ C for all C ∈ M (1.26)The fundamental observation behind the sheaf-theoretic framework is that the existence of aglobal section for an empirical model is equivalent to the existence of a non-contextualhidden variable model (also known as a local hidden variable model ). Concretely,the existence of a global section d means that there is a finite set Λ, an R -distribution q ( λ ) : Λ → R and a family of functions f λj : M j → O j such that: ζ m (cid:0) o (cid:1) = (cid:88) λ ∈ Λ q ( λ ) (cid:89) j δ f λj ( m j )= o j (1.27)We will say that an empirical model ( ζ C ) C ∈M is contextual (or non-local ) if it doesn’tadmit a global section. From now on, no-signalling is implicitly assumed. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:13
Strong contextuality.
Contextuality of probabilistic models is interesting in itself, butmore refined notions can be obtained by relating R + to two other semirings: the reals,modelling signed probabilities, and the booleans, modelling possibilities. Observe that theconstruction D R is functorial in R , so that for any morphism of semirings r : R → R (cid:48) wecan define the following: D r [ U ] = [ d : U → R ] (cid:55)→ (cid:2) r ◦ d : U → R (cid:48) (cid:3) (1.28)In particular, there is an injective morphism of semirings i + : R + (cid:44) → R sending x ∈ R + to + x ∈ R , as well as a surjective morphism of semirings p : R + → B sending 0 (cid:55)→ x (cid:54) = 0 (cid:55)→ R + ).If ( ζ C ) C ∈M is a probabilistic empirical model, i.e. one in the semiring R + , then ( ζ C ) C ∈M can be seen as an empirical model ( i + ◦ ζ C ) C ∈M in the semiring R : regardless of whether( ζ C ) C ∈M was contextual or not over R + , it can be shown [AB11] that over the reals it alwaysadmits a global section. On the other hand, any probabilistic empirical model ( ζ C ) C ∈M canbe assigned a corresponding possibilistic empirical model ( p ◦ ζ C ) C ∈M in the semiring B ofthe booleans (and each boolean function p ◦ ζ C can equivalently be seen as the characteristicfunction of the subset supp ζ C ⊆ E [ C ]).Note that contextuality is a contravariant property with respect to change of semiring:if ( ζ C ) C ∈M is an empirical model in a semiring R and r : R → R (cid:48) is a morphism of semiring,then contextuality of ( r ◦ ζ C ) C ∈M implies contextuality of ( ζ C ) C ∈M (because a global section d of the latter is mapped to a global section r ◦ d of the former). We will say that a probabilisticempirical model ( ζ C ) C ∈M is possibilistically contextual if the corresponding possibilisticmodel ( p ◦ ζ C ) C ∈M is contextual (as opposed to probabilistically contextual , which weuse to say that ( ζ C ) C ∈M is contextual over R + ). Because of contravariance, possibilisticcontextuality implies probabilistic contextuality, but the opposite is not true: the Bell modelgiven in [AB11] is probabilistically contextual but not possibilistically contextual.Seeing distributions d ∈ D B E [ U ] as indicator functions of the subsets supp d ⊆ E [ U ]endows them with a partial order: d (cid:48) (cid:22) d if and only if supp d (cid:48) ⊆ supp d (1.29)The existence of a global section d ∈ D B E [ U ] for a possibilistic empirical model ( ζ C ) C ∈M implies that: d | C (cid:22) ζ C for all C ∈ M (1.30)We say that a possibilistic empirical model ( ζ C ) C ∈M is strongly contextual if there is nodistribution d ∈ D B E [ X ] such that Equation 1.30 holds. In particular, the GHZ model givenin [AB11], corresponding to Mermin’s original non-locality argument, is strongly contextual.Because of Equation 1.29, strong contextuality implies contextuality, but the opposite is nottrue: the possibilistic Hardy model give in [AB11] is contextual, but not strongly contextual.We will say that a probabilistic empirical model is strongly contextual if the associatedpossibilistic empirical model is strongly contextual, yielding the following strict hierarchy ofnotions of contextuality for probabilistic empirical models:probabilistically contextual ⇐ possibilistically contextual ⇐ strongly contextual(1.31) :14 S. Gogioso and W. Zeng
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Empirical models within the CP* construction.
The relevance of the sheaf-theoreticframework to this work stems from the following result: in any R -probabilistic CP* category,all Bell-type scenarios give rise to an empirical model. Definition 1.6.
Consider an R -probabilistic CP* category. A Bell-type scenario is aprocess Φ : ( A , ) ⊗ ... ⊗ ( A N , N ) → ( B , ) ⊗ ... ⊗ ( B N , N ), where all ( A j , j ) and all( B j , j ) are classical systems, which takes the following form, for some normalised state ρ and some normalised processes B , ..., B N : B B N ... R O R O N ... R M N R M ρ ... (1.32) Theorem 1.7.
Consider a Bell-type scenario Φ in the form given by Definition 1.6. Let M j be the finite set of classical states for j , and O j be the finite set of classical states for j . Then the process Φ gives rise to a no-signalling empirical model ( ζ m ) m ∈M as follows,for any cover M : B B N ... R O R O N ... m m N ρ ... := ∈ D R E [ { m , ..., m N } ] ζ m (1.33) Proof.
We need to show that the states in Equation 1.33 (indexed by the measurementcontexts m ∈ M ) satisfy no-signalling and are normalised (i.e. are R -distributions). Todo so, we (i) marginalise over party j , (ii) use the fact that the discarding map on theclassical systems ( B j , j ) can be written as ( B j , j ) = (cid:80) o j (cid:104) o j | , and (iii) use the fact thatthe measurement on j and the process B j are both normalised to show that the resultingstate is independent of m j : B B N ... R O R O N ... m m N ρ ... B j m j o j ... ...... (cid:80) o j ∈ O j ρ B ... R O N ... m j ... R O m B N = ...... m N ...= B j m j ... ... ρ R O B R O N m m N ...... B N ...... (1.34)Marginalising over all outputs leaves us with ◦ ρ , which equals 1 (independently of themeasurement context m ) because ρ is normalised. Hence the state ( ζ m ) m ∈M is also an R -distribution as desired, completing our proof. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:15 Mermin’s Original Argument
The parity argument.
In the original [Mer90], Mermin considers a 3-qubit GHZ state inthe computational basis, the basis of eigenstates for the single-qubit Pauli Z observable,together with the following four joint measurements :(a) the GHZ state is measured in the observable X ⊗ X ⊗ X ;(b) the GHZ state is measured in the observable Y ⊗ Y ⊗ X ;(c) the GHZ state is measured in the observable Y ⊗ X ⊗ Y ;(d) the GHZ state is measured in the observable X ⊗ Y ⊗ Y .We will denote the eigenstates of the Pauli Z observable by | z (cid:105) , | z (cid:105) , the eigenstates of thePauli X observable by |±(cid:105) := √ ( | z (cid:105) ± | z (cid:105) ) and the eigenstates of the Pauli Y observableby | ± i (cid:105) := √ ( | z (cid:105) ± i | z (cid:105) ). Mermin’s argument is a parity argument, where measurementoutcomes are valued in the abelian group Z = { , } according to the following bijections:(i) for the X observable, | + (cid:105) (cid:55)→ |−(cid:105) (cid:55)→ Y observable, | + i (cid:105) (cid:55)→ | − i (cid:105) (cid:55)→ Z sum of the three outcomes turns out to be deterministic, yielding the followingsystem of equations ( ⊕ here denotes the sum in Z ): X ⊕ X ⊕ X = 0 Y ⊕ Y ⊕ X = 1 Y ⊕ X ⊕ Y = 1 X ⊕ Y ⊕ Y = 1 (2.1)If there was a non-contextual assignment of outcomes for all measurements ( X , X , X , Y , Y and Y ), i.e. if there existed a non-contextual hidden variable model, then System 2.1 wouldhave a solution in Z , and in particular it would have to be consistent. However, the sum ofthe left hand sides yields 0 in Z :2 X ⊕ X ⊕ ... ⊕ Y = 0 X ⊕ ... ⊕ Y = 0 (2.2)while the sum of the right hand sides yields 0 ⊕ ⊕ ⊕ Z . This shows thesystem to be inconsistent. Equivalently, one could observe that the sum of the LHS fromEquation 2.2 can be written as 2( Y ⊕ Y ⊕ Y ), and that inconsistency of the system iswitnessed by the fact that the equation 2 y = 1 has no solution in Z .The first point of view, where contextuality is witnessed by an inconsistent systemwhere each equation individually admits a solution, is behind the generalisation of Mermin’sargument to All-vs-Nothing arguments, presented in [ABK + y = 1, will inspire thegeneralisation presented in this work. Where X j and Y j are the single-qubit Pauli X and Y observables on qubit j , for j = 1 , , :16 S. Gogioso and W. Zeng
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The role of phases.
To understand the role played by the equation 2 y = 1 in the originalMermin argument, we need to take a step back. First of all, we observe that the Pauli Y measurement can be equivalently obtained as a Pauli X measurement preceded by anappropriate unitary. A single-qubit phase gate , in the computational basis (the Pauli Z observable), is a unitary transformation in the following form: P α := (cid:18) e iα (cid:19) (2.3)where we eliminated global phases by setting the first diagonal element to 1. Measuring inthe single-qubit Y observable is equivalent to first applying the single-qubit phase gate P π ,and then measuring in the Pauli X observable.Because they pairwise commute, phase gates come with a natural abelian group structuregiven by composition, resulting in an isomorphism α (cid:55)→ P α between them and the abeliangroup R / (2 π Z ). Of all the phase gates, P (the identity element of the group) and P π stand out because of their well-defined action on the (unnormalised) eigenstates of the Pauli X observable: P = |±(cid:105) (cid:55)→ |±(cid:105) P π = |±(cid:105) (cid:55)→ |∓(cid:105) (2.4)If we see |±(cid:105) as the subgroup { , π } < R / (2 π Z ), then Equation 2.4 looks a lot like theregular action of { , π } on itself. This is not a coincidence. Each phase gate P α can be(faithfully) associated the unique phase state | α (cid:105) := | z (cid:105) + e iα | z (cid:105) obtained from its diagonal,and these phase states can be abstractly characterised in terms of the Pauli Z observable,with no reference to the phase gates they came from ( cf. section 4). The phase states inheritthe abelian group structure of the phase gates, and their regular action coincides with theaction of the group of phase gates on them. In particular, the phase gates P and P π haveorthogonal eigenstates of the Pauli X observable as their associated phase states | (cid:105) and | π (cid:105) , which coincide with √ | + (cid:105) and √ |−(cid:105) respectively: this endows the outcomes of Pauli X measurements with the natural Z abelian group structure arising from the inclusion { , π } < R / (2 π Z ). We will henceforth refer to the group of phase states as the group of Z -phase states , and to the subgroup { , π } as the subgroup of X -classical states ; thelatter will also be used to label the corresponding measurement outcomes.In order to pave the way to our generalisation, we now proceed to show how Mermin’soriginal argument can be re-constructed from the following statement: the equation y = π has no solution in the subgroup { , π } of X -classicalstates, but a solution y = π can be found in the larger group R / (2 π Z ) of Z -phase states. We begin by observing that tripartite qubit GHZ state used in Mermin’s argument has aspecial property when it comes to the application of phase gates followed by measurementsin the Pauli X observable. The abelian group R / (2 π Z ) is isomorphic to the circle group S . We prefer the former because of itsadditive notation, as opposed to the traditionally multiplicative notation of the latter (which is a subgroup ofthe non-zero multiplicative complex numbers C × ). Corresponding to {± } < S in the circle group. Natural because there is a unique isomorphism Z ∼ = { , π } . Corresponding to y = e i π = + i in the circle group S . ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:17 Lemma 2.1 [CDKW12] . If α j ∈ R / (2 π Z ) , denote by X α j j the measurement outcome on qubit j obtained by first applying phase gate P α j , and then measuring in the Pauli X observable.If α ⊕ α ⊕ α = 0 or π ( mod π ) , then X α ⊕ X α ⊕ X α = 0 or π ( mod π ) respectively. Now consider System 2.1 again, with values on the the RHS now obtained by applyingLemma 2.1 to X j := X j and Y j := X π j (and valued in { , π } instead of the original Z ): X ⊕ X ⊕ X = 0, the control X π ⊕ X π ⊕ X = π , the first variation X π ⊕ X ⊕ X π = π , the second variation X ⊕ X π ⊕ X π = π , the third variation (2.5)There are two complementary parts to the Mermin non-locality argument: (i) System 2.5above must be inconsistent, to rule out the existence of a non-contextual hidden variablemodel, and (ii) joint measurements yielding the individual equations must be possible (inquantum theory). For the first part, inconsistency of the system is witnessed by the factthat the equation 2 y = π has no solution in the subgroup of X -classical states. For thesecond part, notice that only measurements in the Y observable contribute to the sum foreach equation, as measurements in the X observable are associated with the group unit 0 ofthe group of Z -phase states. As a consequence, the existence of measurements implementingeach individual equation reduces to the existence of a Z -phase state | y (cid:105) satisfying equation2 y = π : the Y observable is chosen exactly because y = π/ Z -phase state.The following steps summarise the skeleton of the argument, and open the way to ourgeneralisation:1. consider a non-degenerate observable, call it Z , on an arbitrary quantum system;2. consider another non-degenerate observable, call it X , such that the X -classical statesare a subgroup (call it K ) of the abelian group of Z -phase states (call it P );3. consider an equation in the following form, generalising 2 y = π : n y ⊕ ... ⊕ n M y M = a (2.6)(here a ∈ K , n , ..., n M are integers , and ⊕ is the group addition in P );4. construct an appropriate system of equations, generalising System 2.5, with inconsis-tency witnessed by non-existence of solutions for Equation 2.6 in K , and consistencyof the individual equations witnessed by the existence of solutions in P ;5. a measurement scenario can be implemented if and only if a solution exists in P ;6. the measurement scenario is contextual if and only if no solutions exist in K .To give a first example of how such an appropriate system of equations might be constructed,we consider the simple generalisation of the argument from a 3-partite to an N -partite GHZstate, for appropriate values of N ≥
2. Our requirements are as follows:(i) we want the phases in the control to sum to 0, and hence we will take them all tobe 0 (i.e. measurements in the X observable), just as in the original argument;(ii) we also want the phases in each variation to sum to π , and hence we will take twomeasurements in each variation to be with phase π/ Y observable), and all the other ones to be with phase 0;(iii) we want an odd number V of variations, so that the RHSs will sum to 0 ⊕ V π = π ; This is a general equation in abelian groups, seen equivalently as Z -modules. :18 S. Gogioso and W. Zeng
Vol. 15:2 (iv) we want the LHSs to sum to an even multiple of X π ⊕ ... ⊕ X π N ;An appropriate choice is given by the following system of equations, where V := N and allvariations are cyclic permutations of the first one: X ⊕ X ⊕ X ⊕ ... ⊕ X N − ⊕ X N = 0, the control X π ⊕ X π ⊕ X ⊕ ... ⊕ X N − ⊕ X N = π , the 1 st variation X π ⊕ X ⊕ ... ⊕ X N − ⊕ X N − ⊕ X π N = π , the 2 nd variation... X ⊕ X π ⊕ X π ⊕ X ⊕ ... ⊕ X N = π , the N th variation (2.7)As long as N = 1 (mod k ), where k = 2 is the exponent of K , the RHSs will sum to π in K . Having chosen our variations by cyclic permutation also makes for the desired sum ofthe LHSs, since each X π/ j will be counted exactly twice: (cid:0) X ⊕ ... ⊕ X N (cid:1) ⊕ · (cid:0) X π ⊕ ... ⊕ X π N ) ⊕ ( N − · (cid:0) X ⊕ ... ⊕ X N )control X π j s from the variations X j s from the variationsWriting x for X ⊕ ... ⊕ X N and y for X π ⊕ ... ⊕ X π N , the sum above can be rearrangedto take the form ( N − x ⊕ y , which is equal to 2 y in K (since ( N −
1) = 0 (mod k )) .Hence summing all the LHSs and RHSs leaves us with the equation 2 y = π , which we knowto be unsatisfiable in K . 3. Strong Complementarity
Mermin’s parity argument is fundamentally group-theoretic, and it depends almost entirelyon the special relationship between the Pauli Z and Pauli X observables. Fixing theeigenstates of the Pauli Z observable as the computational basis, the requirement that the X -classical states are Z -phase states is satisfied by the Pauli X observable, but also bythe Pauli Y : in fact, the Z -phase states are exactly the unbiased states for the Pauli Z observables, the states lying on the equator of the Bloch sphere, and hence any observable complementary , or mutually unbiased , to Pauli Z would do the trick; because theireigenstates lie on the equator of the Bloch sphere, we will refer to observables complementaryto Pauli Z as equatorial observables . Definition 3.1 gives an algebraic/diagrammaticpresentation of complementarity using Hopf’s Law, and Lemma 3.3 shows that observableswhich are complementary observables under this definition are always mutually unbiased. Amore general result relating complementarity and mutual unbias in † -SMCs will be given byTheorem 4.10 in the next section. The smallest positive integer such that kx = 0 for all x ∈ K . In this specific case, it is also true that 2 = 0 (mod k ), but this is not key to the argument. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:19 Definition 3.1.
Two † -qSFAs and on the same object H of a † -SMC are said to be complementary if they satisfy the following Hopf ’s Law := = (3.1)where the antipode : H → H is the unitary defined as follows, which we require to beself-adjoint (or equivalently self-inverse) as part of the definition of complementarity::= = (3.2)
Definition 3.2.
Let be a † -qSFA in a dagger compact category. Then a state u is a -unbiased state if the following holds: = u (3.3)Equation 3.3 can be unfolded into the following more general definition, which holds in anarbitrary † -SMCs: = u † u (3.4)In fdHilb, the -unbiased states are those which, once normalised, yield the uniformdistribution upon measurement in the observable. Lemma 3.3.
Consider a complementary pair of † -qSFAs and . The -classical statesare -unbiased, and the -classical states are -unbiased.Proof. We prove that a -classical state χ is -unbiased: χ χ † == χ = = χ χ χ † (3.5)The first equality is by -classicality (delete condition), the second equality is Hopf’s law(together with the self-adjoint requirement for the antipode), the third equality is again by-classicality (copy condition for the bottom state, transpose condition for the top state),the last equality is by Frobenius law and unit law for . The proof for -classical states isthe same, with colours swapped.Complementarity is not sufficient for Mermin’s argument: Lemma 2.1 only holds if wemeasure the GHZ state in the Pauli X observable, not in any other equatorial observable.The algebraic relationship between the Pauli X , Y and Z observables is vividly captured bythe ZX calculus [CD11]: there, the special property relating the Pauli Z and X observablesis axiomatised under the name of strong complementarity , to distinguish it from thecomplementarity of Pauli Z and any other equatorial observable (such as Pauli Y ). Strongcomplementarity is behind the proof of Lemma 2.1, which lies at core of the fully diagrammatictreatment of Mermin’s original argument appearing in [CDKW12]. :20 S. Gogioso and W. Zeng
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Definition 3.4 gives an algebraic/diagrammatic presentation of strong complementarity,while Theorem 3.6 provides an exact correspondence in finite-dimensional Hilbert spacesbetween complementarity and the representation theory of finite abelian groups. A moregeneral characterisation of strong complementarity in † -SMCs will be given by Theorem 4.10in the next section. Definition 3.4.
Two † -qSFAs and on the same object H of a † -SMC are said to be strongly complementary if they are complementary and furthermore satisfy the followingequations : = == == = (3.6) Remark 3.5.
Technically speaking, the central equations of both rows are not necessary:indeed, they do not usually feature in the literature. The central equation on the top row of3.6 is in fact a consequence of Hopf ’s law and the other two equations of the top row: = = = == = (3.7)
Similarly, the central equation of the top row with colours swapped is a consequence of Hopf ’slaw, the rightmost equation of the top row, and the rightmost equation of the bottom row.The central equation of the bottom row can also be obtained using Hopf ’s law, self-adjointnessof the antipode, and the other equations.
Strong complementarity means that the eigenstates of the Pauli X observable are veryspecific equatorial states, given by the two multiplicative characters χ : Z → S of theabelian group Z : | χ (cid:105) := χ (0) | z (cid:105) + χ (1) | z (cid:105) = (cid:40) √ | + (cid:105) if χ is the trivial character χ ( j ) := +1 √ |−(cid:105) if χ is the alternating character χ ( j ) := ( − j (3.8)This group-theoretic characterisation of strong complementarity fully generalises to arbitraryfinite-dimensional Hilbert spaces and finite abelian groups. Theorem 3.6 [CDKW12, Kis12] . Let and be a † -SCFA and a † -qSCFA on the same finite-dimensional Hilbert space H .Then and are strongly complementary iff there exists an abelian group ( G, ⊕ , suchthat ( , ) endows the set of -classical states with the abelian group structure of G , i.e.iff we can label the -classical states as ( | g (cid:105) ) g ∈ G in a way such that: gh = g ⊕ h (3.9) The empty diagram on the RHS of the top right equation is the scalar 1. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:21
If and are strongly complementary, then the -classical are labelled by the multiplicativecharacters χ : G → S of the abelian group G , and take the following form: | χ (cid:105) := (cid:88) g ∈ G χ ( g ) | g (cid:105) (3.10) Furthermore, ( , ) turns the set of X -classical states into the finite abelian group ( G ∧ , · , ) of multiplicative characters of G : χχ (cid:48) g = χ gχ (cid:48) g = gχ · χ (cid:48) for all g ∈ G (3.11)The importance of strong complementarity for quantum algorithms [Vic12, GK17] mostlylies in the following observation: if the -classical states are taken to form the computationalbasis, then the -classical states form the (unnormalised) Fourier basis, and measuring inthe observable amounts to performing the quantum Fourier transform. The situation withMermin-type arguments, however, is different: the relevant facet of strong complementaritywill be the special relationship between -classical points and -phase states, explored indetail in the coming section. Remark 3.7.
Theorem 3.6 extend to the case where is a † -qSFA, i.e. not commutative,as long as the adjective abelian is dropped for the group G (which becomes a generic finitegroup); it should however be noted that the -classical states form a basis if and only if G isabelian (if and only if is commutative, i.e. a non-degenerate observable). The group G ∧ of multiplicative characters of a finite group G is always abelian, and Pontryagin duality necessarily fails when G is not abelian. Theorem 3.6 also extends to the case where isa † -qSCFA, in which case the -classical states form an orthogonal basis, rather than anorthonormal one. Most of Theorem 3.6 can be further extended to the case where is ageneric † -qSFA: it is still true that ( , ) endows the -classical states with the structureof a finite group, and that ( , ) endows the -classical states with the structure of a finitegroup, but Equation 3.10 need not hold, and the group structure given by ( , ) need notbe that of the group of multiplicative characters. The phase group
If strong complementarity is the fundamental algebraic property at work in Mermin’s argu-ment, phase gates and GHZ states are the operational components key to its implementation.Phase gates arise in the context of quantum-to-classical transitions, where they provide acharacterisation, in the spirit of groups and symmetries, of how much information is lost byperforming a (demolition) measurement in a non-degenerate observable. Multiplicative characters form an abelian group under pointwise multiplication (cid:0) χ · χ (cid:48) (cid:1) ( g ) := χ ( g ) · χ (cid:48) ( g )and with the trivial character := g (cid:55)→ Pontryagin dual G ∧ of G . The existence of a (canonical) isomorphism ( G ∧ ) ∧ ∼ = G . :22 S. Gogioso and W. Zeng
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Definition 4.1.
Let be a † -qSFA on an object H of a dagger compact category. Then the -phase gates are the unitaries U : H → H which are annihilated by the measurement:= U (4.1)Equation 4.1 can be unfolded into the following equivalent definition, which extends to anarbitrary † -SMC: = U † U (4.2) Remark 4.2.
A simpler algebraic characterisation of phase gates is given by the followingtwo equations, which are equivalent to Equation 4.2 (because U is assumed to be unitary): = UU (4.3) U = U (4.4) Both equations will play a pivotal role in this section: Equation 4.3 will features shortly inLemma 4.5, the result relating phase gates and GHZ states, while Equation 4.4 will featurelater on in Theorem 4.6, the result relating phase gates and unbiased states.
From Equation 4.1, it is not hard to see that -phase gates form a group: we willrefer to this as the -phase group , and we will denote it by P ( ). If is a † -SFA on afinite-dimensional Hilbert space H , associated with a direct sum decomposition H = ⊕ j H j ,then the phase group P ( ) is given by the corresponding direct sum of unitary groups,modulo a global phase: P ( ) = (cid:16) ⊕ j U ( H j ) (cid:17)(cid:14) S (4.5)In the special case where is a † -SCFA on H , i.e. when all H j subspaces are 1-dimensional,the phase group is abelian, the translation group of a torus: P ( ) = (cid:16) ⊕ dim H j =1 U (1) (cid:17)(cid:14) S ∼ = T dim H− (4.6)The connection between abelian phase groups and commutative Frobenius algebras generalisesfrom fdHilb to arbitrary dagger compact categories. The following result shows that thephase group of a commutative Frobenius algebra is always abelian, while the converse willbe proven later on in Corollary 4.9 (conditional to the existence of enough unbiased states) Lemma 4.3.
Let be a † -qSFA on an object H of a dagger compact category. If iscommutative, then the -phase group P ( ) is abelian. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:23 Proof.
VVU = U UV = =
VUUV = V = U = = UV = V U (4.7)The first equality is by unit law for ; the second equality is by Equation 4.3; the thirdequality is some topological manipulation; the fourth equality (top right to bottom left) is bycommutativity of ; the fifth equality is by Equation 4.3; the sixth equality is commutativityof ; the seventh and last equality is by Equation 4.3, followed by unit law for .Having defined the phase group and proven Lemma 4.3, we are now in a position tostate the first important result of this section. Lemma 4.5 characterises the sates that canbe obtained by application of phases gates to a GHZ state: in the context of our generalisedMermin-type arguments, it will play the same role that Lemma 2.1 played in Mermin’soriginal argument.
Definition 4.4.
If is a † -qSFA on an object H of a dagger compact category, the N -partite-GHZ state is the following state of H ⊗ N : ... N (4.8) Lemma 4.5.
Let be a † -qSCFA on an object H of a dagger compact category. Thenthe state obtained by applying -phase gates U , ..., U N to the N -partite -GHZ state onlydepends on the composition U · ... · U N of the phase gates: U U N ... = ... U U N . . . (4.9) Proof.
Each -phase gate is pushed down by using Equation 4.3 and commutativity of .Formally, the proof is by induction, with inductive step given by the following equality: U N = ... U U N − ... U U N = U N U N − ... U = U ... U N − U N U ...= U N − U N (4.10)We have remarked before that the phase gates in Mermin’s original argument areassociated to certain phase states, extracted from their diagonalisation, which are alsounbiased states for the relevant observable. As the following Theorem 4.6 shows, theconnection between -phase gates and -unbiased states holds true in full generality, and :24 S. Gogioso and W. Zeng
Vol. 15:2 as a consequence we will also refer to -unbiased states as -phase states . In the case offdHilb, the decomposition of a -phase gate U given by Equation 4.11 for a † -SCFA isequivalent to saying that U is diagonal in the orthonormal basis ( | x (cid:105) ) x associated with ,and has diagonal encoded by state | u (cid:105) as U xx = (cid:104) x | u (cid:105) . Theorem 4.6.
Let be a † -qSFA on an object H of a dagger compact category. Then the-phase gates are exactly the maps P u taking the following form for a -unbiased state u : u = P u (4.11) Proof.
First we prove that any phase gate U takes the form above, for some -unbiased state u . An appropriate state u can then be obtained by unit law for : U = UU = (4.12)By using Equation 4.2, we can prove that the state we obtained is -unbiased: U = U † = (4.13)Then we prove that any U in the form above with u a -unbiased state is a unitary:= u u † u † u = (4.14)Finally, we prove that any unitary U in the form above with u a -unbiased state is a -phasegate: = u u † u † u = (4.15)Because of the correspondence above, we will adopt a uniform notation for phase gatesand phase states, known in the literature as decorated spider notation [CD11, CK17]:phase gate P u u u phase state u (4.16) Corollary 4.7.
Let be a † -qSCFA on an object H of a dagger compact category. Thenthe state obtained by applying -phase gates P u , ..., P u N to the N -partite -GHZ state takes ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:25 the following form in terms of the corresponding -phase states u , ..., u N : u u N ... = ... u u N ... u = ... (4.17) That is, the states that can be obtained by applying -phase gates to the N -partite -GHZ stateare exactly those obtained by comultiplying N -times some -unbiased state u (specifically,above we have u = u · ... · u N , and all -unbiased states can be obtained this way).Proof. From Lemma 4.5, by re-writing each -phase gate in terms of the corresponding-phase state using Theorem 4.6, and then using associativity to group the -phase states.The group structure of phase gates transfers to unbiased states via the correspondencegiven by Theorem 4.6. Albeit not surprising, this result plays an important role in ourgeneralisation of Mermin-type arguments, where it connects the operational side of phasegates and GHZ states to the algebraic side of strong complementarity (in the group-theoreticcharacterisation given by Theorem 3.6 and Theorem 4.10 below).
Lemma 4.8.
Let be a † -qSFA on an object H of a dagger compact category. Then ( , ) endows the set of -unbiased states with the structure of P ( ) .Proof. The -phase gate corresponding to the -unbiased state is the identity, the unit of P ( ), so all we need to show is that composition of phase gates is the same as multiplicationunder of the corresponding -unbiased states:= v u vu (4.18)As a bonus, the correspondence between the -phase group and the group structure on-unbiased states can be used to prove a converse to Lemma 4.3. Corollary 4.9.
Let be a † -qSFA on an object H of a dagger compact category, and assumethat has enough unbiased states . Then is commutative iff P ( ) is abelian.Proof. We already know from Lemma 4.3 that if is commutative then the -phase group P ( ) must be abelian. Conversely, if P ( ) is abelian then so is the group structure induced by( , ) on the -unbiased states: in particular, this means that is commutative wheneverit is applied to -unbiased states, and the existence of enough unbiased states allows us toconclude that is always commutative.With Theorem 4.6 we have proven a general correspondence between phase gates andunbiased states, while with Lemma 4.5 and Corollary 4.7 we have characterised the statesthat can be obtained by applying phase gates to GHZ states. Phase gates and the GHZ statefor the Pauli Z observable are the key operational ingredients for Mermin’s original argument.However, just as important is the special algebraic standing of those phase gates derivedfrom the eigenstates of the Pauli X observable (an observable strongly complementary toPauli Z ), as opposed to the phase gates derived from other equatorial states (the eigenstatesof observables complementary to Pauli Z ). I.e. that two morphisms
F, G : H → K are equal whenever F ◦ u = G ◦ u for all -unbiased states u . :26 S. Gogioso and W. Zeng
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The last result of this section, Theorem 4.10, provides a general characterisation ofcomplementarity and strong complementarity in terms of the relation between classicalstates of one observable and unbiased states of the other. Together with Theorem 4.6 and4.7, it will form the basis for the formulation of our generalised Mermin-type arguments inthe next section.
Theorem 4.10.
Let and be † -qSFAs on an object H of a † -SMC. The following implications always hold: (i) if and are complementary, then the -classical states form a subset of the -unbiasedstates; (ii) if and are strongly complementary, then the -classical states form a subgroup ofthe -unbiased states.The converse implications hold if has enough classical states: (i) if the -classical states form a subset of the -unbiased states, then and arecomplementary; (ii) if the -classical states form a subgroup of the -unbiased states, then and arestrongly complementary.Note that the existence of enough -classical states implies the existence of enough -unbiasedstates when the former are a subset/subgroup of the latter.Proof. Implication (i) is the statement of Lemma 3.3: in its proof, Equation 3.5 is shownthat Hopf’s law is equivalent to the defining equation of a -unbiased state when it appliedto -classical state. χ χ † == χ = = χ χ χ † (4.19)Implication (ii) follows by applying the four defining equations of strong complementarity,together with Hopf’s law, to -classical states. The top row in 3.6 holds if and only if theunit is a -classical state: it is coherently copied, transposed and deleted by .= == (4.20)The bottom row in 3.6 holds applied to two -classical states if and only if the multiplicationunder of two -classical states is a -classical state. χχ (cid:48) = = χχ (cid:48) χχ (cid:48) χχ (cid:48) = χχ (cid:48) χχ (cid:48) (4.21)Conditional to ( , ) endowing the -classical states with the structure of a monoid, Hopf’slaw applied to a -classical state is equivalent to the antipode acting as group inverse on ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:27 -classical states. == χχ χ χ = (4.22)Implications (iii) and (iv) follow the same lines as implications (i) and (ii). Under theassumptions of (iii) we can conclude that Equation 4.19 holds, and under the assumptionof (iv) we can conclude that Equations 4.20, 4.21 and 4.22 hold: from the existence ofenough -classical states, we can conclude that the laws of complementarity and strongcomplementarity hold as desired.5. Generalised Mermin-type Arguments
Armed with the necessary results relating the classical and unbiased states of stronglycomplementary observables, we are now in a position to formulate our generalised Mermin-type arguments. To do so, we first review the ingredients of Mermin’s original parityargument for qubit GHZ states:(a) a 3-partite qubit GHZ state for the Pauli Z observable;(b) the abelian group P ( Z ) ∼ = R / (2 π Z ) of phase states for the Pauli Z observable;(c) the finite subgroup { , π } ∼ = Z given by the eigenstates of the Pauli X observable;(d) an equation 2 x = 1 with no solution in the subgroup { , π } given by the Pauli X eigenstates, but with a solution π/ R / (2 π Z ) of Pauli Z phase states;(e) measurements in the Pauli X observable.Similarly, our generalised Mermin-type arguments will involve the following ingredients:(a) an N -partite GHZ state for a † -qSCFA ;(b) the abelian group ( P ( ) , ⊕ ,
0) of -phase states ;(c) the subgroup ( K ( ) , ⊕ , † -qSFAwhich is strongly complementary to ;(d) a finite system of Z -module equations, together with a solution in the group P ( );(e) measurements in the observable.The non-existence of a solution in the subgroup K ( ) of -classical states is not part of ourgeneralised setup: it will be explicitly characterised as the necessary and sufficient conditionfor contextuality. Also, N will not be a free parameter, being instead determined by theexponent of the finite abelian group K ( ). Definition 5.1.
Consider an R -probabilistic CP* Category CP ∗ [ C ]. A generalised Mermin-type argument in CP ∗ [ C ] is specified by the following data:(i) a strongly complementary pair ( , ) of a canonical † -qSCFA and a canonical † -SCFA on some object H of C , such that has enough classical states; we furthermoreassume that the set K ( ) of -classical states is finite , and that | K ( ) | is invertibleas an element of the semiring R of scalars of C ; Isomorphic, by Theorem 4.6, to the -phase group, which we will denote by ( P ( ) , · , id ). This, together with commutativity of , means that ( K ( ) , ⊕ ,
0) is a finite abelian group. :28
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Vol. 15:2 (ii) a finite system of Z -module equations in the following form, with a , ..., a S ∈ K ( ): S = (cid:76) Mr =1 n r y r = a ... (cid:76) Mr =1 n Sr y r = a S (5.1)(iii) a given solution ( y r := β r ) Mr =1 in the abelian group P ( ) of -phase states;(iv) a positive integer N such that N ≥ (cid:80) Mr =1 n sr for all s = 1 , ..., S , and satisfyinggcd( N, exp[ K ( )]) = 1, where exp[ K ( )] is the exponent of K ( ).Therefore a generalised Mermin-type argument is specified by a quintuple ( , , S , β, N ).The quintuple ( , , S , β, N ) contains all the algebraic and operational ingredients weneed to formulate a measurement scenario, which sees N no-signalling parties sharingan N -partied -GHZ state. Each party makes a measurement choice m j ∈ { , , ..., M } ,applies the phase gate P β mj to her system, and then measures it in the observable (i.e.measurement outcomes are valued in the set K ( ) of -classical states).Not all combinations of measurement choices are needed for the argument, and themeasurement contexts will be determined by System 5.1. We begin by zero-padding thesystem as follows, so that exactly N phase states are involved in each equation: n y ⊕ y ... ⊕ y M = 0 n y ⊕ n y ... ⊕ n M y M = a ... n S y ⊕ n S y ... ⊕ n SM y M = a S (5.2)where we have defined a := 0, n s := N − (cid:80) Mr =1 n sr for all s = 1 , ..., S , n := N and n r := 0for all r = 1 , ..., M ; we will also extend the given solution by setting β := 0. The firstequation in System 5.2 (which we will refer to by the special value s = 0 of the parameter s )will contribute to a single measurement context, the control ; each further equation (i.e. foreach value s = 1 , ...., S of the parameter s ) will give rise to N measurement contexts, the variations , for a total of 1 + S · N measurement contexts involved in the scenario.In the control, all parties choose m j = 0, i.e. perform no phase gate before measuring.They obtain the following global state (where 1 / | K ( ) | N − is the normalisation factorrequired to obtain a R -distribution): ... O N O
00 = O N ... O ...1 | K ( ) | N − | K ( ) | N − (5.3)The first variation for each value s = 1 , ..., S is specified by the corresponding equation inSystem 5.2: the first n s parties choose m sj = 0, the next n s parties choose m sj = 1, the next I.e. equations with integer coefficients n sr ∈ Z and valued in abelian groups (aka Z -modules). The smallest positive integer e such that e · g = 0 for all g ∈ K ( ). ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:29 n s parties choose m sj = 2 and so on, until the last n sM parties choose m sj = M : m sj := the largest m ∈ { , ..., M } such that j ≥ m − (cid:88) r =0 n sr (5.4)They obtain the following global state, where the equality results from an application ofCorollary 4.7, using the relevant equation from System 5.2:... O N O β m s β m sN = O N a s ... O ... 1 | K ( ) | N − | K ( ) | N − (5.5)For each fixed value of s , the next N − j th party at the k th variation of a given s is m sj +( k − , wherethe sum j + ( k −
1) is taken modulo N :Parties: 1 2 ... N − N st variation for s m s m s ... m sN − m sN nd variation for s m s m s ... m sN m s rd variation for s m s m s ... m s m s ... ... ... ... ... N th variation for s m sN m s ... m sN − m sN − (5.6)Because is commutative, the global state obtained is the same as that for the first variationfor that value of s (shown on the RHS of Equation 5.5).By using strong complementarity and Theorem 4.10, we rewrite the global state obtainedby the N parties in the control and variations, obtaining an explicit R -distribution overthe set K ( ) N of joint measurement outcomes (from now on, the parameter s can take anyvalue in { , , ..., S } , unless otherwise specified). Lemma 5.2. a s ... O N O g N O N O ... | K ( ) | N − (cid:80) g ⊕ ... ⊕ g N = a s = g | K ( ) | N − (5.7) Proof.
Strong complementarity can be used to swap and , as shown in Corollary 4.1 of[CDKW12], and then a s can be pushed through because it is a -classical state (we haveleft normalisation aside, and we use a := 0 to treat control and variations uniformly):= O N a s ... O a s ... O N O = a s ... O N O (5.8)Using fact that has enough classical states, and recalling from Theorem 4.10 that ( , )acts as the group multiplication of K ( ) when restricted to the -classical states, we can :30 S. Gogioso and W. Zeng
Vol. 15:2 further decompose the state on the RHS of Equation 5.8 into an R -distribution over the set K ( ) N : = a s ... O N O | K ( ) | N − (cid:80) g ⊕ ... ⊕ g N = a s g g N O N O ... | K ( ) | N − (5.9)The joint outcome of measurements for the control is uniformly distributed over thesubgroup H (cid:69) K ( ) N specified by H := { ( g , ..., g N ) | g ⊕ ... ⊕ g N = 0 } , while the jointoutcome of any of the N variations for each specific value of s is uniformly distributed overthe coset H a s := ( a s , , ..., ⊕ H . For each s, s (cid:48) ∈ { , , ..., S } , the cosets H a s and H a s (cid:48) are disjoint whenever a s (cid:54) = a s (cid:48) . All in all, we get the following empirical model for thegeneralised Mermin-type argument: P [( g , ..., g N ) | control] = | K ( ) | N − if g ⊕ ... ⊕ g N = 00 otherwise (5.10) P [( g , ..., g N ) | k th variation for s ] = | K ( ) | N − if g ⊕ ... ⊕ g N = a s Z ). The proof of contextuality for our generalisedMermin-type arguments goes by similar lines, showing that the existence of a LHV model isequivalent to System 5.1 admitting solutions in the finite abelian group K ( ). Theorem 5.3.
Consider an R -probabilistic CP* category CP ∗ [ C ] , and let ( , , S , β, N ) bea generalised Mermin-type argument in it. If the associated empirical model is contextual,then the system S admits no solution in the finite abelian group K ( ) . Conversely, if thesystem S admits no solution in K ( ) and R is a positive semiring, then the empirical modelis contextual.Proof. The proof comes in two parts: ( ⇒ ) we show that any solution in K ( ) can be turnedinto a LHV model; ( ⇐ ) we show that, as long as R is a positive semiring, any LHV modelcan be turned into a solution in K ( ). Proof of ( ⇒ ). Assume that the system S (in the form of System 5.1) admits a solution( y r := b r ) Mr =1 , and define b := 0. A LHV model can be obtained as follows:(i) the uniform R -distribution on H (cid:69) K ( ) N is taken as a shared classical state amongstthe N parties: ... | K ( ) | N − O O N (5.12) ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:31 (ii) upon measurement choice m j ∈ { , , ..., M } for the j th party, a translation by b m j inthe group K ( ) is applied to the respective classical subsystem, independently of themeasurement choices of the other parties: ...1 | K ( ) | N − O b m s b m sN O N (5.13)All we need to show is that the procedure above produces the same R -distributions on K ( ) N as those given by the empirical model of Equations 5.10 and 5.11. To do so, wesimply observe that the global state obtained with the procedure above is the same as theglobal states obtained in the control 5.3 and in the variations 5.5 (which we treat uniformlyby considering s = 0 , , ..., S ), because b , b , ..., b N satisfy the same equations satisfied bythe phases β , β , ..., β N : = a s ... O N O | K ( ) | N − ...1 | K ( ) | N − O b m s b m sN O N (5.14) Proof of ( ⇐ ). Now assume that R is a positive semiring, and that the scenario admits aLHV model:(i) there is a some finite set Λ, the set of values for the hidden variable, coming with an R -distribution p : Λ → R ;(ii) for each possible measurement choice r = 0 , , ..., M that each party i = 1 , ..., N canmake, there is a family ( c i,λr ) λ ∈ Λ of -classical states, the deterministic local outcomesfor each value of the hidden variable;(iii) for each measurement context (either s = 0, k = 1 for the control, or ( s, k ) ∈{ , ..., S } × { , ..., N } for the N · S variations), a definite -classical outcome d i,λs,k isobtained by each party i = 1 , ..., N at each definite value λ ∈ Λ of the hidden variable: d i,λs,k := c i,λm si +( k − (5.15)(iv) if these definite -classical global states are weighted based on the R -distribution p onΛ, one obtains the same R -distribution on joint measurement outcomes that would beexpected from the measurement context:= a s ... O N O | K ( ) | N − ... (cid:80) λ ∈ Λ p ( λ ) O d ,λs,k d N,λs,k O N (5.16) :32 S. Gogioso and W. Zeng
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Given a LHV model, we can sum up all N outcomes of each side of Equation 5.16 in( K ( ) , ⊕ ,
0) to obtain an equation between R -distribution over K ( ):= a s ...1 | K ( ) | N − ... (cid:80) λ ∈ Λ p ( λ ) d ,λs,k d N,λs,k a s = (5.17)The last equation used the fact that was chosen to be special , and hence the normalisationfactor for the † -qSCFA is | K ( ) | (because has enough classical states) . Equation 5.17can be turned into the following conditions on the LHV: (cid:88) λ s.t. (cid:76) Ni =1 d i,λs,k = a s p ( λ ) = 1 (cid:88) λ s.t. (cid:76) Ni =1 d i,λs,k (cid:54) = a s p ( λ ) = 0 (5.18)Because R is a positive semiring, p ( λ ) = 0 for any λ such that ⊕ Ni =1 d i,λs,k (cid:54) = a s for some s .Conversely, picking any λ + such that p ( λ + ) > λ + exists, because p is an R -distribution) yields a family ( d i,λ + s,k ) s,k,i such that ⊕ Ni =1 d i,λ + s,k = a s for all s and k . Forthe control ( s = 0 and k = 1), we obtain the following equation: ⊕ Ni =1 c i,λ + = 0 (5.19)For each variation ( s, k ) ∈ { , ..., S } × { , ..., N } , we obtain the following equation: ⊕ Ni =1 c i,λ + m si +( k − = a s (5.20)If c i,λ + r was independent of the party i for all r = 1 , ..., M , this equation would yield asolution to system S in the form of b r := c i,λ + r for any i ; unfortunately, this need not be thecase. This is where our cyclic definition of the N variations for each value of s comes intoplay. For each fixed value of s , we add up the N equations for k = 1 , ..., N : ⊕ Nk =1 ⊕ Ni =1 c i,λ + m si +( k − = N a s (5.21)Because gcd( N, exp[ K ( )]) = 1, we can take the inverse of N modulo exp[ K ( )], and theequation above has solutions if and only if the equation below does: ⊕ Nk =1 ⊕ Ni =1 N − c i,λ + m si +( k − = a s (5.22)Now refer to the Table 5.6 defining the N variations for s , as well as to Equation 5.4 whichdefines the measurement choices,. The LHS of Equation 5.21 is a sum by rows of the N measurement choices in Table 5.6: each r = 0 , , ..., M appears n sr times in each row, butthe changing value of i along each row stops us from turning it into a solution to system S .However, we can switch the summations in Equation 5.21 to obtain a sum by columns of The special could have been replaced by a more general † -qSCFA, but at the price of an additionalnormalisation factor in all global states. The normalisation factor | K ( ) | refers to two wires: each additional wire is an additional copy of | K ( ) | ,for a total of | K ( ) | N − in the N -wire case here. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:33 the table, where each r = 0 , , ..., M still appears n sr times in each column (by the cyclicdefinition), but now i is constant along each column: ⊕ Ni =1 ⊕ Nk =1 c i,λ + m si +( k − = ⊕ Ni =1 ⊕ Mr =0 n sr c i,λ + r (5.23)We can then sum up all ( c i,λ + r ) Ni =1 for each r = 0 , , ..., M , and use Equation 5.21 (togetherwith Equation 5.19 to cancel out the contribution from r = 0) to finally obtain the desiredsolution ( b r ) Mr =1 to system S : ⊕ Mr =1 n sr (cid:0) ⊕ Ni =1 N − c i,λ + r (cid:1) = a s b r (5.24)6. Quantum realisability
In quantum theory, i.e. in the probabilistic CP* category CP ∗ [fdHilb], many of the require-ments of generalised Mermin-type arguments are automatically satisfied: canonical † -SCFAsin CPM[fdHilb] (i.e. † -SCFAs in fdHilb) always have enough classical states (and finitelymany so), the semiring R + of scalars is positive, and any non-zero integer is invertible in it.Hence, only strong complementarity is required in point (i) of the definition of generalisedMermin-type arguments, and Theorem 5.3 establishes an unconditional equivalence betweencontextuality of a generalised Mermin-type argument ( , , S , β, N ) and the existence ofsolutions to system S in the finite abelian group K ( ) of -classical states.The remarks above show that the correspondence between systems of equations in finiteabelian groups and generalised Mermin-type arguments is particularly tight in the case ofquantum theory, but an important question remains unanswered: which systems of Z -moduleequations lead to arguments which can be realised in quantum theory? As it turns out, allof them (but an obvious caveat applies). Theorem 6.1.
Let ( K, ⊕ , be a finite abelian group, and S be a finite system of Z -moduleequations in the following form, with a , ..., a S ∈ K : S = (cid:76) Mr =1 n r y r = a ... (cid:76) Mr =1 n Sr y r = a S (6.1) Assume that the system is consistent in the following sense, where by n s ∈ Z M we denotedthe row vectors of System 6.1: S (cid:77) s =1 c s · n s = Z M ⇒ S (cid:77) s =1 c s · a s = K , (6.2) Then for every | K | -dimensional quantum system H and every † -qSCFA on H with nor-malisation factor | K | , there exists a generalised Mermin-type argument ( , , S , β, N ) corre-sponding to System 6.1, i.e. we can always find: (i) a † -SCFA , strongly complementary to , such that ( K ( ) , , ) ∼ = ( K, ⊕ , ; (ii) a solution ( y r := β r ) Mr =1 to S in P ( ) ∼ = T | K |− ; :34 S. Gogioso and W. Zeng
Vol. 15:2 (iii) a positive integer N (infinitely many, in fact) such that N ≥ (cid:80) Mr =1 n sr for all s =1 , ..., S , and such that gcd( N, exp[ K ( )]) = 1 .Proof. Point (iii) is trivial: there are infinitely many N such that gcd( N, exp[ K ( )]) = 1,and hence we can always find one such that N ≥ (cid:80) Mr =1 n sr for all s = 1 , ..., S . Point (i) ismore interesting, and relies on Theorem 3.6 and Pontryagin duality for finite abelian groups.Point (ii) is perhaps the most interesting, and relies on the possibility of solving consistentsystems of Z -module equations in the torus T | K |− . Proof of point (i).
Because is a † -qSCFA with normalisation factor | K | on a | K | -dimensional Hilbert space H , it is associated with a basis of | K | vectors, each having norm (cid:112) | K | . Label the basis vectors by the | K | multiplicative characters χ ∈ K ∧ of the finiteabelian group K , and construct an orthonormal basis by using the multiplicative characters τ ∈ ( K ∧ ) ∧ of the finite abelian group K ∧ : | τ (cid:105) := 1 | K | (cid:88) χ ∈ K ∧ τ ( χ ) | χ (cid:105) (6.3)By Pontryagin duality, there is a canonical isomorphism ( K ∧ ) ∧ ∼ = K , so that the neworthonormal basis given by Equation 6.3 is canonically labelled by elements of K . Considerthe † -SCFA associated to the orthonormal basis thus defined to obtain the desired( K ( ) , , ) ∼ = ( K, ⊕ , Proof of point (ii).
The phase group P ( ) for a canonical † -qSCFA on a | K | -dimensionalHilbert space in CPM[fdHilb] is isomorphic to the ( | K | − y r := β r ) Mr =1 to System 6.1, we will show that one can alwaysfind solutions to arbitrary consistent systems of Z -module equations in a torus.While all K -valued systems with solutions in some super-group of K must necessarilybe consistent, the converse is not true in general: given a super-group P of K there may beconsistent systems with no solutions in P . Certainly if P is finite then at least one suchsystem exists (because of the finite exponent), and certainly if P = Q d then no such systemexists; in fact, every divisible torsion-free abelian group P is canonically a Q -vector space,and thus every consistent system of Z -modules equations (and, in fact, of Q -vector spaceequations) valued in a divisible torsion-free abelian group P has solutions in P (e.g. byGaussian elimination over the field Q ). Unfortunately, while tori are divisible, they are nottorsion-free, and in particular not Q -vector spaces: as a consequence, the reasoning abovedoes not apply.However, a more general argument can be used to show that any consistent system ofequations can be solved in any divisible abelian group, regardless of whether the group istorsion-free or not [Fuc15] (although uniqueness of solution need not hold for systems withlinearly independent row vectors). As tori are divisible abelian groups, all consistent systemsof Z -module equations can be solved in them, and in particular we can find our solution( y r := β r ) Mr =1 to System 6.1.7. All-vs-Nothing Arguments
Strong contextuality can be reformulated directly in terms of the supports of the distributions.The supports of the global sections, i.e. the d ∈ D B E [ X ] satisfying Equation 1.30, form a(possibly empty) lattice, and thus a probabilistic empirical model is strongly contextual iff ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:35 the following set is empty: S [ X ] := { s ∈ E [ X ] | s | C ∈ supp ζ C for all C ∈ M} (7.1)For a possibilistic (no-signalling) empirical model ( ζ C ) C ∈M , we can define [ABK +
15] a support subpresheaf S ⊆ E by setting: S [ U ] := { s ∈ E [ U ] | s | C ∩ U ∈ supp ζ C | U ∩ C for all C ∈ M} (7.2)Then a possibilistic empirical model is strongly contextual if and only if S [ X ] = ∅ .The fundamental observation behind the All-vs-Nothing arguments of [ABK +
15] isthat contextuality of Mermin’s original argument follows from the existence of the system of Z equations which has no global solution (corresponding to S [ X ] = ∅ in the sheaf-theoreticframework for contextuality [AB11]), but where each equation admits a solution (i.e. wehave S [ C ] (cid:54) = ∅ for the measurement context C associated to each equation). In this sectionwe summarise the basic framework of All-vs-Nothing arguments from [ABK + R be a commutative ring with unit: we will denote by + the addition in the ring R ,and by ⊕ the addition in R -modules. The ring R should not be confused with the semiring R over which the distributions are taken (i.e. the semiring of scalars of the enriched CPMcategory the arguments take place in). If G is some R -module, we will define an R -linearequation valued in G to be a triple φ = ( C, n, b ) where:(i) C is some finite set, and we define index( φ ) := C ;(ii) n : C → R is any function;(iii) b ∈ G is a given element of G .If φ = ( C, n, b ) is an R -linear equation valued in G , we will say that a function s : C → G (henceforth an assignment ) satisfies φ , written s | = φ , if and only if the following equationholds in G : (cid:77) m ∈ C n m s m = b (7.3)where we denoted n m := n ( m ) and s m := s ( m ). Any set W of assignments C → G can beassociated a corresponding set T R ( W ) of satisfied equations, which is itself an R -module : T R ( W ) := { φ | s | = φ for all s ∈ W } (7.4)Let ( ζ C ) C ∈M be a possibilistic empirical model for a measurement scenario ( E , M ), suchthat all measurements have the same R -module G as their set of outcomes (for examplewe had G = Z , a Z -module, for Mermin’s original argument). Let S ⊆ E be the supportsubpresheaf for the empirical model and define its R-linear theory to be: T R ( S ) := (cid:91) C ∈M T R ( S [ C ]) (7.5)We say that a possibilistic empirical model is All-vs-Nothing with respect to ring R and R -module G , written AvN R ,G , iff the R -linear theory admits no solution in G , i.e. iff thereexists no global assignment s : X → G such that: s | C | = φ for all C ∈ M and all φ ∈ T R ( S [ C ]) (7.6)To connect back with the notation in [ABK + R for AvN R , R .A straightforward generalisation (from rings to modules) of a result by [ABK +
15] provesthat any possibilistic empirical model which is AvN R ,G for some ring R and some R -module This gives rise to some interesting results on affine closures, see [ABK + :36 S. Gogioso and W. Zeng
Vol. 15:2 G is strongly contextual: if the model weren’t strongly contextual, then there would besome global section s ∈ S [ X ], and this would imply s | C ∈ S [ C ] for all C ∈ M , which in turnwould prove that global assignment s satisfies Equation 7.6 (by appealing to Equation 7.4).A result by [AB11] shows that a probabilistic empirical model is strongly contextualif and only if it is maximally contextual, i.e. if and only if it lies on a face of the no-signalling polytope with no local vertices. As a consequence, showing that our generalisedMermin-type arguments are AvN R ,G is a particularly neat way of proving that they aremaximally contextual, a highly desirable property for the device-independent security of thequantum-classical secret sharing protocol which we will present in the next section. Theorem 7.1.
Consider a R -probabilistic CP* category CP ∗ [ C ] , where R is a positivesemiring, and let ( , , S , β, N ) be a generalised Mermin-type argument in it. If the associatedempirical model is contextual, then it is AvN Z ,K .Proof. The associated probabilistic empirical model is given by Equations 5.10 and 5.11:the only scalars appearing are 0 and the invertible | K ( ) | , which are (necessarily) sent to0 and 1 respectively in the passage to the possibilistic empirical model. The possibilisticempirical model is as follows: P [( g , ..., g N ) | control] = (cid:40) g ⊕ ... ⊕ g N = 00 otherwise (7.7) P [( g , ..., g N ) | k th variation for s ] = (cid:40) g ⊕ ... ⊕ g N = a s S ⊆ E : S [control] = (cid:110) ( c im i ) Ni =1 ∈ K N (cid:12)(cid:12)(cid:12) ⊕ Ni =1 c im i = K (cid:111) (7.9) S [ k th variation for s ] = (cid:110) ( c im si +( k − ) Ni =1 ∈ K N (cid:12)(cid:12)(cid:12) ⊕ Ni =1 c im si +( k − = K a s (cid:111) (7.10)Amongst the (many) equations in T Z ( S ) we can find the following 1 + N · S equations: (cid:77) m s m = 0, satisfied by all s ∈ S [control] (7.11) (cid:77) m s m = a s , satisfied by all s ∈ S [ k th variation for s ] (7.12)Any global assignment satisfying all equations in T Z ( S ) would in particular satisfy the1 + N · S equations above, and hence provide a solution in K to the system S . By Theorem5.3, if the empirical model is contextual then no such solution exists: hence no globalassignment satisfying all equations in T Z ( S ) can exist, proving that the model is in particularAvN Z ,K . Corollary 7.2.
The generalised Mermin-type arguments provide an infinite family of quan-tum realisable
AvN Z ,K empirical models, indexed by all finite abelian groups K and all finiteconsistent systems S of Z -module equations valued in K which admit no solution in K .Furthermore, all AvN Z ,K arguments for some fixed K are equivalently AvN Z n ,K for anypositive integer n divisible by the exponent of K : as a consequence, there are generalisedMermin-type arguments providing quantum realisable AvN Z n models for positive integers n ≥ . ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:37 Proof.
The first part is a straightforward consequence of Theorems 5.3, 6.1 and 7.1. Thesecond part is a consequence of the fact that any Z -module equation valued in a finiteabelian group K is equivalent to a Z exp[ K ] -module equation (by taking remainders moduloexp[ K ] of all coefficients), and to a Z n -module equation for any n divisible by the exponentexp[ K ] (taking reminders modulo n of coefficients). The last part is the special case wherewe consider the finite abelian group K = Z n as a module over the ring R = Z n .One open question about All-vs-Nothing arguments asks whether all quantum realisableAvN Z models are in fact AvN Z . The following result answers the question negatively,showing that the infinite family of AvN Z models provided by the previous corollary form anon-collapsing hierarchy of AvN Z p models for all n ≥ Theorem 7.3.
For each n ≥ , there is a quantum realisable AvN Z n (and hence also AvN Z , Z n ) empirical model which is not AvN Z m ,K (cid:48) for any m ≥ coprime with n and anynon-trivial abelian group K (cid:48) with exponent dividing m ; in particular, it is not AvN Z m .Proof. The next Section fully works out the example of K := Z n with the system S consistingof a single Z -module equation ty = 1. If we pick a t ∈ { , ..., n − } which divides n , theequation cannot be satisfied for K = Z n , giving rise to a model which is both AvN Z , Z n andAvN Z n (because the equation can be replaced by an equivalent Z n -module equation). Nowconsider some m coprime with n , and some abelian group K (cid:48) with exponent dividing m .Then the equation has solutions in K (cid:48) , giving rise to a model which is not AvN Z ,K (cid:48) norAvN Z m ,K (cid:48) (nor AvN Z m , in the case K (cid:48) := Z m ). Indeed, we must have K (cid:48) ∼ = (cid:81) Ll =1 Z p ell forsome primes p l not dividing n and some exponents e l ≥
1, and the equation has solutions in Z p ell for all l (because t has the same prime factors of n , and hence no p l can divide t ).8. A fully worked-out example
In this Section, we fully work out a generalised Mermin-type argument, for the group K := Z d and the system S consisting of a single Z -module equation ty = 1 (i.e. we have S = M = 1);we take d ≥ t ∈ { , ..., d − } . This can equivalently be seen as a Z d -module equation ty = 1 (mod d ). Our presentation goes through four distinct Subsections, covering all aspectsfrom abstract definition of the argument to concrete realisation in quantum theory. In thefirst Subsection, we present the measurement scenario and empirical model, by writing downthe table of measurement choices (for the measurement scenario) and the table of outcomeprobabilities (for the empirical model). In the second Subsection, we characterise those casesin which the empirical model admits a local hidden variable model, which we describe in fulldetail. In the third Subsection, we write down the equations turning the empirical modelinto an All-vs-Nothing argument, in those cases in which a local hidden variable model isruled out. In the fourth and final Subsection, we give an explicit quantum realisation interms of GHZ states and phase gates on qudits (i.e. d -dimensional quantum systems). Whenrestricted to the special case d = 2 and t = 1, the setup we describe coincides with the oneoriginally proposed by Mermin [Mer90], as long as we take the observable in the quantumrealisation to correspond to Pauli Z, and the observable to correspond to Pauli X (recallthat performing the Pauli Z phase gate P e i π and then measuring in Pauli X is the same asmeasuring in Pauli Y). :38 S. Gogioso and W. Zeng
Vol. 15:2
Measurement scenario and empirical model.
Firstly, the exponent of Z d is d ,and we fix a number of parties N such that gcd( N, d ) = 1 (e.g. N := d + 1). Each party i = 1 , ..., N can make a measurement choice m i in the set { , } , and the measurementcontexts take the following form: in the control, all parties make measurement choice 0, whilethe variations take the form of N cyclic permutations, each one featuring N − t contiguousparties making measurement choice 0 and t parties making measurement choice 1. This issummarised by the table below:Party: 1 2 ... N − t − N − t N − t + 1 ... N − N control 0 0 ... ... st variation 0 0 ... ... nd variation 0 0 ... ... rd variation 0 0 ... ... N th variation 1 0 ... ... g , ..., g N ) for the N parties are valued in Z Nd , and thegeneralised Mermin-type argument is associated with the following empirical model: g ⊕ ... ⊕ g N = 0 g ⊕ ... ⊕ g N = 1 g ⊕ ... ⊕ g N (cid:54) = 0 , d N − st variation 0 d N − nd variation 0 d N − rd variation 0 d N − N th variation 0 d N − Local hidden variable models.
When t and d are coprime, the equation ty =1 (mod d ) has a (unique) solution y := t − (mod d ), and a local hidden variable model for theempirical model 8.2 can be obtained as follows. Consider the set Λ of all the ( g , ..., g N ) ∈ Z Nd such that g ⊕ ... ⊕ g N = 0, together with the uniform probability distribution p : Λ → R + on Λ (i.e. p ( g , ..., g N ) = d N − ). Also, consider deterministic local outcomes for each fixedvalue g ∈ Λ of the hidden variable such that, upon measurement choice m i for party i ,the measurement outcome is g i whenever m i = 0 and g i ⊕ t − whenever m i = 1. In thecontrol, all parties i = 1 , ..., N will choose m i = 0, and the joint measurement outcome willbe uniformly distributed over the subgroup Λ ⊂ Z Nd . In any variation, t parties will choose m i = 1 and N − t parties will choose m i = 0, and the joint measurement outcome will beuniformly distributed over the coset (1 , , ..., ⊕ Λ ⊂ Z Nd (using the fact that t · t − = 1in Z d ). Hence this really defines a local hidden variable model for the empirical model 8.2associated with the generalised Mermin-type argument.8.3. All-vs-Nothing arguments.
When t and d are not coprime, the equation ty =1 (mod d ) cannot have solutions in K = Z d (by a standard argument from number theory).The possibilistic empirical model associated with the argument has the following support ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:39 subpresheaf S ⊆ E (only the control and the first three variations are explicitly shown here,to exemplify the pattern): S [control] = the set of all ( g , g , ..., g N − t − , g N − t , g N − t +10 , ..., g N − , g N ) ∈ Z Nd such that N (cid:77) i =1 g i = 0 (8.3) S [1 st var’n] = the set of all ( g , g , ..., g N − t − , g N − t , g N − t +11 , ..., g N − , g N ) ∈ Z Nd such that (cid:16) N − t (cid:77) i =1 g i (cid:17) ⊕ (cid:16) N (cid:77) i = N − t +1 g i (cid:17) = 1 (8.4) S [2 nd var’n] = the set of all ( g , g , ..., g N − t − , g N − t , g N − t +11 , ..., g N − , g N ) ∈ Z Nd such that (cid:16) g N ⊕ N − t − (cid:77) i =1 g i (cid:17) ⊕ (cid:16) N − (cid:77) i = N − t g i (cid:17) = 1 (8.5) S [3 rd var’n] = the set of all ( g , g , ..., g N − t − , g N − t , g N − t +11 , ..., g N − , g N ) ∈ Z Nd such that (cid:16) g N − ⊕ g N ⊕ N − t − (cid:77) i =1 g i (cid:17) ⊕ (cid:16) N − (cid:77) i = N − t − g i (cid:17) = 1 (8.6)Amongst the (many) equations in T Z ( S ) we can find the N + 1 equations equations above,one for the control (Equation 8.3) and N for the variations (Equations 8.4, 8.5 and 8.6,corresponding to the first three variations, exemplify the pattern). Any global assignment( g ir ) i =1 ,...,Nr =0 , which satisfies all equations in T Z ( S ) would in particular satisfy those N + 1equations. However, adding up the N equations corresponding to the variations yields, aftera bit of rearranging, the following equation:( N − t ) (cid:16) N (cid:77) i =1 g i (cid:17) ⊕ t (cid:16) N (cid:77) i =1 g i (cid:17) = N (8.7)Taking this together with the equation (cid:76) Ni =1 g i = 0 associated with the control then resultsin the following equation (using gcd( N, d ) = 1 to obtain the inverse N − modulo d ): t (cid:16) N (cid:77) i =1 N − g i (cid:17) = 1 (8.8)But this means that setting y := (cid:76) Ni =1 g i would yield a solution to the equation ty = 1 in Z d , which we assumed not to exist. Hence we cannot have any global assignment satisfyingall equations in T Z ( S ), and the model is both AvN Z , Z d and AvN Z d . :40 S. Gogioso and W. Zeng
Vol. 15:2
Quantum realisation.
Now consider the d -dimensional quantum system C [ Z d ]. Pickto be the special commutative † -Frobenius algebra associated with the orthonormal basis | g (cid:105) g ∈ Z d , and to be the quasi-special commutative † -Frobenius algebra associated withthe orthogonal basis | χ (cid:105) χ ∈ Z ∧ d defined as follows in terms of the multiplicative characters χ : Z d → S of the finite abelian group Z d : | χ (cid:105) := (cid:88) g ∈ Z d χ ( g ) | g (cid:105) (8.9)Let H N be the subgroup of Z Nd defined by H := (cid:8) g ∈ Z Nd (cid:12)(cid:12) g ⊕ ... ⊕ g N = 0 (cid:9) . Then the(normalised) N -party GHZ state | GHZ (cid:105) in the observable takes the following form: | GHZ (cid:105) := 1 d N − (cid:88) g ∈ H | g (cid:105) ⊗ ... ⊗ | g N (cid:105) (8.10)Let P α , ..., P α N be phase gates for the observable, where each α i is a function Z ∧ d → S (not necessarily a group homomorphism): writing the phase gates explicitly we get P α i := d (cid:80) χ ∈ Z ∧ d α i ( χ ) | χ (cid:105)(cid:104) χ | . Applying the N phase gates P α , ..., P α N to the N subsystemsof the GHZ state is the same as applying their composition P α := P α ◦ ... ◦ P α N (where α := α · ... · α N ) to a single subsystem.Amongst the many functions α : Z ∧ d → S are the group homomorphisms, which formthe group (cid:0) Z ∧ d (cid:1) ∧ under pointwise product. By Pontryagin duality, there is a canonicalisomorphism Z d ∼ = (cid:0) Z ∧ d (cid:1) ∧ , given explicitly by h (cid:55)→ ( χ (cid:55)→ χ ( h )), and we can consider phasegates P h corresponding to all h ∈ Z d . The phase gate P h acts as P h | g (cid:105) = | g ⊕ h (cid:105) on thebasis | g (cid:105) g ∈ Z d , and hence applying P h to a single subsystem of the GHZ state results in thefollowing state, involving the coset H h = (cid:8) g ∈ Z Nd (cid:12)(cid:12) g ⊕ ... ⊕ g N = h (cid:9) : (cid:16) P h ⊗ id ⊗ ... ⊗ id (cid:17) | GHZ (cid:105) = 1 d N − (cid:88) g ∈ H Nh | g (cid:105) ⊗ ... ⊗ | g N (cid:105) (8.11)We wish to find a solution to ty = 1 in the group of phase gates for , i.e. we want some β : Z ∧ d → S such that ( P β ) t = P . To do so, first note that the multiplicative characters χ : Z d → S can equivalently be formulated in terms of elements k ∈ Z d , e.g. by considering χ k := g (cid:55)→ e i π k · gd , and then rewriting the equation ( P β ) t = P as follows:1 d (cid:88) k ∈ Z d β ( χ k ) t | χ k (cid:105)(cid:104) χ k | = 1 d (cid:88) k ∈ Z d e i π kd | χ k (cid:105)(cid:104) χ k | (8.12)It is now easy to see that a solution to ( P β ) t = P is given by β := χ k (cid:55)→ e i π t kd . Remark 8.1.
The explicit construction presented here for the case of cyclic groups Z d inordinary quantum theory is related to the recent work of Ref. [RLZL13] (as well as previouswork by Refs. [LLK06, CMP02, KZ02, ZK99] ), and the relationship between the dimension d of the quantum systems and the allowed numbers N of parties (i.e. gcd( N, d ) = 1 ) is thesame here and in Ref. [RLZL13] . Even when restricted to the special case of quantum theory,however, the work presented here is a significant generalisation of the work of Ref. [RLZL13] :in the latter, the authors focus on a specific family of M = 2 , S = 1 systems with values in afinite cyclic group Z d ; in this work, we provide necessary and sufficient algebraic conditionsfor arbitrary systems and arbitrary finite abelian groups. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:41 Quantum-classical Secret Sharing
In contrast to other information security protocols, classical secret sharing comes with theintrinsic assumption that some participants cannot, to some extent, be trusted. A dealer isinterested in sharing some secret with a number of players , with the caveat that the secret berevealed to the players only when all players agree to cooperate . Integrity and availabilityof communications is guaranteed by the existence of authenticated classical channels betweendealer and players, and the protocol is only concerned with confidentiality, defined as theimpossibility of recovering the secret unless all players cooperate.The quantum-classical scheme of Hillery, Buˇzek and Berthiaume [HBB99] introducesa new layer of security to secret sharing, employing entangled states and non-commutingobservables to detect eavesdropping. The HBB scheme is based on the same measurementcontexts of Mermin’s original parity argument: a dealer and N − N qubitsin a GHZ state (with respect to the computational basis associated with the Pauli Z observable), and randomly choose to measure their qubit in either of the mutually unbiasedPauli X or Pauli Y observables. It can be shown [Zam12] that confidentiality is an immediateconsequence of strong complementarity of the Pauli Z and X observables, while eavesdroppingdetection follows from mutual unbias of the Pauli X and Y observables.We extend the HBB scheme from Mermin’s original parity argument to our generalisedMermin-type arguments, and we use our result on contextuality to provide a number ofdevice-independent security guarantees. For the remainder of this section, we will consider ageneralised Mermin-type argument ( , , S , β, N ), on an object H of a R -probabilistic CP*category CP ∗ [ C ], where R is a positive semiring.Consider a dealer , call her Alice, who wishes to share a secret with N (cid:48) players , where2 ≤ N (cid:48) < N . As the owner of the secret, Alice is always a trusted party, the only trustedparty in the protocol. The secret is assumed to take the form of a string of elements of K ( ), the plaintext (at most one element of K ( ), the round plaintext , transmitted foreach round of the protocol). We wish to ensure that the plaintext can be decoded fromthe information Alice sends, the cyphertext , if and only if all players agree to cooperate(by which we mean that they all reveal their secret keys to some party in possession of thecyphertext). Alice and the players are given N devices (one per player, and N − N (cid:48) forAlice): at each round w , each device B j is fed an input m wj ∈ { , , ..., M } and returnsan output g wj ∈ K ( ) (we also refer to the outputs g w , ..., g wN (cid:48) as the secret keys of theplayers for round w ). We furthermore assume the following security conditions to hold.(i) Alice and the players share an authenticated classical channel, ensuring integrity andavailability of all classical communications involved in the protocol.(iia) Alice and the players are in possession of N secure independent classical sources of ran-domness, to generate independent inputs at each round which are uniformly distributedin { , , ..., M } .(iib) Alice is in possession of a secure classical source of randomness, independent from allother, to decide which rounds will be secret rounds (with probability (1 − τ ) >
0) andwhich rounds will be test rounds (with probability τ > . More in general, a minimum number of cooperating players can be specified. This can be achieved, for example, by ensuring the devices are operated in conditions controlled byAlice (trusted laboratories, synchronized time-stamp servers, etc). :42
S. Gogioso and W. Zeng
Vol. 15:2 β m w β m wN (cid:48) ...1 | K ( ) | N − β m wN β m w N (cid:48) +1 ... g w g wN (cid:48) +1 g wN (cid:48) g wN p w p w c w g wdealer ...... (cid:9) a s w Figure 1: Graphical presentation of a noiseless, trusted implementation.(iv) We will assume that in step 3 Alice is communicated the measurement choices faithfully Because tampering can only be determined after the protocol has ended and the entirety(or an otherwise significant portion) of the plaintext has been transmitted, we distinguishbetween the plaintext , the data that can be decoded using the secret keys, and the actual secret that Alice wants the players to share. Before the protocol begins, Alice will obtainthe plaintext by encrypting the secret with a secure symmetric encryption protocol , using afreshly generated ephemeral key which she will broadcast only if the protocol is successful. Ifthe protocol fails, the random key will not be broadcast and the secret will be unrecoverableeven if the plaintext is decoded.The quantum-classical secret sharing protocol then proceeds as follows for each round w = 1 , ..., W , until the entire secret has been transmitted. An individual round for a noiseless,trusted implementation is presented in Figure 1. Throughout the protocol, Alice keeps acount of occurrences of joint outputs g , ..., g N conditional to each joint input m , ..., m N that she observes in test rounds.1. Alice and the players share N subsystems of a state ρ : each player has an individualsubsystem and Alice keeps the remaining N − N (cid:48) subsystems. In a noiseless, trustedimplementation, ρ is the N -partite -GHZ state. For the purposes of a device-independentsecurity analysis, ρ can be potentially any state.2. Alice and the players each sample their classical source of randomness and obtaininputs m w , ..., m wN which are passed to the devices B , ..., B N and result in outputs g w , ..., g wN (cid:48) ∈ K ( ) for the players (the secret keys for the round) and g wN (cid:48) +1 , ..., g wN ∈ K ( ) This can be achieved by entrusting the laboratory setup with the communication of the randommeasurement choices to Alice, the player and the device. If the secret is in the form of a string of elements of K ( ), the natural choice for this protocol, thenthe plaintext can be obtained by generating a string of uniformly random k w elements of K ( ), obtainingthe round plaintext p w from the corresponding “round secret” q w as p w = q w ⊕ k w . Once the string ofrandom elements is broadcast, upon successful completion of the protocol, the secret can be recovered fromthe decoded plaintext as q w = p w (cid:9) k w . ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:43 for Alice. In a noiseless, trusted implementation, B j with input m wj applies the phasegate P β mwj to the subsystem j and then measures it in the observable.3. The inputs for the players are communicated to Alice. She checks that m w , ..., m wN define a valid measurement context (either the control ( s = 0) or a variation forsome s = 1 , ..., S ).4. Alice samples her source of randomness to decide whether the round will be a test roundor a secret round.4a. If the round is a test round, Alice requests all players to communicate their secret keys,and she increases the occurrence count for joint output ( g w , ..., g wN ) conditional to jointinput m w , ..., m wN .4b. If the round is a secret round, Alice computes g wdealer := (cid:76) Nj = N (cid:48) +1 g wj and broadcasts the round ciphertext c w := p w ⊕ g wdealer to the players, where the round plaintext p w is the next element of the plaintext to be sent. She also broadcasts the relevant value s w ∈ { , , ..., S } she obtained from the joint inputs m w , ..., m wN .5. Anyone in possession of the round ciphertext c w and all secret keys g w , ..., g wN (cid:48) can obtainthe round plaintext p w by computing p w = ( c w ⊕ g w ⊕ ... ⊕ g wN (cid:48) ) (cid:9) a s w , where s w is thevalue broadcast in Step 4.The chosen generalised Mermin-type argument determines the following promised con-ditional distribution P promised (cid:2) g (cid:12)(cid:12) m (cid:3) , the one which Alice and the players expect toobserve (asymptotically) in a trusted noiseless implementation (we use the more compactnotation g := ( g , ..., g N ) for the joint output and m := ( m , ..., m N ) for the joint input): P promised (cid:2) g (cid:12)(cid:12) m (cid:3) = | K ( ) | N − if g ⊕ ... ⊕ g N = βm ⊕ ... ⊕ βm N observed conditional distribution P observed (cid:2) g (cid:12)(cid:12) m (cid:3) (which need not beno-signalling). She then computes the noise parameter (cid:15) as follows: (cid:15) := 1 − | K ( ) | N − min (cid:110) P observed (cid:2) g (cid:12)(cid:12) m (cid:3)(cid:12)(cid:12)(cid:12) g ⊕ ... ⊕ g N = β m ⊕ ... ⊕ β m N (cid:111) (9.2)The error parameter as defined above is the smallest (cid:15) ∈ [0 ,
1] such that the observedconditional distribution can be decomposed as the following convex combination of promisedconditional distribution and some noise conditional distribution P noise (cid:2) g (cid:12)(cid:12) m (cid:3) : P observed (cid:2) g (cid:12)(cid:12) m (cid:3) = (1 − (cid:15) ) P promised (cid:2) g (cid:12)(cid:12) m (cid:3) + (cid:15) P noise (cid:2) g (cid:12)(cid:12) m (cid:3) (9.3)Before a run of the protocol begins, Alice sets a maximum (cid:15) max that she is going to acceptfor the noise parameter. Alice chooses as low an (cid:15) max as possible compatibly with thespecifications of the device provider (and any other beliefs she might have) on the amountof noise she should expect from the devices and states in the absence of any tampering fromEve. At the end of the protocol run, Alice compares the noise parameter (cid:15) she computedwith the maximum (cid:15) max she decided to accept: if (cid:15) ≤ (cid:15) max , she declares the protocol run asuccess and broadcasts the ephemeral key she used to encode the secret into the plaintext;if (cid:15) > (cid:15) max , she declares the protocol run a failure and she destroys the ephemeral key,rendering the secret unrecoverable even if the plaintext is at some point obtained by theplayers or by Eve. :44 S. Gogioso and W. Zeng
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The HBB quantum-classical secret sharing protocol comes with two security guarantees:(i) ignorance about any one secret key for a round denies knowledge about the plaintextfor that round; (ii) successful, undetected eavesdropping has low probability. It can beshown [Zam12] that in a noiseless and trusted implementation the first guarantee followsabstractly from strong complementarity of the Pauli Z and X observables, and the proofstraightforwardly transfers to the strongly complementary pairs ( , ) appearing in ourgeneralised protocol. Instead of treating eavesdropping directly, we will present a moregeneral, device-independent proof of security, based solely on contextuality of the generalisedMermin-type argument used by the protocol.Works on device-independent security (such as [BHK05, VV14] on quantum key distri-bution) usually posit Eve to be an adversary who can arbitrarily tamper with the sharedstate and measurement devices, and is only bound in her attempts by the physical theoryunder consideration and by the security conditions explicitly enforced by the protocol(including no-signalling). Examples of things that the Eve is allowed to do include:(i) the measurement outcomes broadcast at a test round can reveal to Eve informationabout measurement outcomes in previous secret rounds;(ii) Eve can keep a subsystem of the shared state to herself, which she can optimallymeasure, once all inputs and test round outputs have been broadcast, to obtaininformation about the secret keys.Our choice of a device-independent setting comes from the more modest desire to show thatthe security guarantees follow from contextuality of the generalised Mermin-type argument,regardless of the specific implementation; as a consequence, we will be content with a morerestricted model of attack. We assume that Alice and the players might be provided withnoisy or imperfect states and devices, which might give Eve a variety of security loopholesto exploit. However, we assume that the device provider shows no malice:(i) the devices are memoryless and operate independently at each round;(ii) the states used at different rounds are independent and identical;(iii) the states are not entangled with any additional system.However, Eve might possess classical information about the states which is unavailable tothe players (such as information leaked through noise or side channels, information acquiredvia eavesdropping, etc).Although not fully general, this setup subsumes a variety of more specialised securityscenarios that are of interest in classical and quantum cryptography:(i) Real-world implementations are unavoidably noisy, and one should consider any noiseas a potential source of cryptophthora. Our setup allows for the possibility that boththe shared state and the measurement devices be noisy, with no dependence on aspecific model of noise; it also allows for the possibility that what looks like randomnoise to Alice and the players might actually carry side-channel information to Eve.(ii) Eavesdropping detection is a typical desideratum in quantum cryptography, where Eveintercepts the local state of a player , measures it in some basis to obtain classicalinformation, and forwards the resulting collapsed state to the player. Our setup allows Eve is often assumed to be bound by the laws of quantum theory, but sometimes super-quantumattackers are also considered, bound only by causality and no-signalling. In our secret sharing protocol, a single player’s secret key is all that Eve needs to break confidentiality,as we may freely assume that the remaining players are colluding and asked Eve to help them. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:45 B B N (cid:48) ... B N B N (cid:48) +1 ... g w g wN (cid:48) +1 g wN (cid:48) g wN p w p w c w g wdealer ...... (cid:9) a s w m wN (cid:48) m wN m wN (cid:48) +1 m w ρ e w Figure 2: Graphical presentation of a generic, untrusted implementation at a single roundof the protocol. Eve might have some classical information e w about the stateswhich is unknown to Alice and the players. The classical side of the protocolis entirely in the hands of Alice and the players, and proceeds as in the trustednoiseless case.for the possibility of eavesdropping : the classical information that Eve possessesabout the state can be used to model the information she acquired by eavesdropping.Our security proof then has eavesdropping detection as a special case of protocolfailure.Figure 2 displays a single round w of the protocol in a generic, untrusted implementation. An N -partite state ρ is shared between Alice and the players at a given round of the protocol, withno additional subsystem accessible to Eve (who might however be in possession of classicalinformation e w about it). The measurement devices B , ..., B N operate independently ateach round, with no memory or shared resource other than the state ρ . At each round w , device B j takes measurement choice m wj as a classical input and returns measurementoutcome g wj as a classical output. The rest of the protocol is entirely in the hands of Aliceand the players, and proceeds as in the trusted noiseless case.Our first result shows that lack of contextuality implies the existence of a scenario inwhich a perfect undetectable attack may take place. In fact, the scenario is not particularlyremote: it might well happen happen that the device provider inadvertently chose phasesstates β , ..., β M which happen to be -classical states (maybe she did not notice, maybe shewas tricked by Eve into choosing them), and that the GHZ state decoheres (spontaneouslyor with a malicious helping hand) in the observable. In that case, Alice and the playerwill notice nothing wrong with their protocol, and Eve will obtain the entirety of the secretall by herself. However, it does not cover a more advanced attack in which Eve sends through a subsystem of anentangled state, keeping the rest of the state to herself and measuring it in the future to obtain moreinformation about the player’s outcome. :46
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Theorem 9.1.
Consider a quantum-classical secret sharing protocol based on a generalisedMermin-type argument ( , , S , β, N ) , in an R -probabilistic CP* category CP ∗ [ C ] , where R is a positive semiring. If the associated empirical model is non-contextual, then there areshared states ρ and measurement devices B , ..., B N such that test rounds will succeed withcertainty, and Eve will always know all the secret keys.Proof. By Theorem 5.3, if the empirical model is non-contextual then there exists a solution( y r := b r ) Mr =1 in K ( ) to the system S (which we take to be in the form of System 5.1). Foreach round w , Eve samples a random variable uniformly distributed over the following set: (cid:8) ( h w , ..., h wN ) ∈ K ( ) N (cid:12)(cid:12) h w ⊕ ... ⊕ h wN = 0 (cid:9) (9.4)Now assume that the separable pure state ρ w = | h (cid:105) ⊗ ... ⊗ | h N (cid:105) is given in input to themeasurement devices B , ..., B N , which are designed so that B j returns g wj := h wj ⊕ b m wj uponmeasurement choice m wj (i.e. applies a phase b m wj which happens to be -classical). Once themeasurement ( m wj ) N (cid:48) j =1 choices for the players are broadcast, Eve can compute all the secretkeys ( g wj ) N (cid:48) j =1 . Furthermore, since ( b r ) Mr =1 is a solution to S , the measurement outcomesobtained from this setup will have the same distribution as the ones from a noiseless trustedimplementation, and all test rounds will succeed with certainty.Our second result is restricted to probabilistic theories, i.e. distributively CMon-enrichedCPM categories having R + as their semiring of scalars. Consider the no-signalling polytopeassociated with the measurement scenario of a contextual generalised Mermin-type argument( , , S , β, N ), and let F be the face of the polytope specified by the support of the empiricalmodel (the one defined by Equation 9.1). For each vertex v ∈ F of that face, correspondingto empirical model P v (cid:2) g (cid:12)(cid:12) m (cid:3) , let H v be the average entropy across all measurement contexts: H v := 11 + N · S (cid:88) m ∈M H (cid:104) P v (cid:2) (cid:12)(cid:12) m (cid:3) (cid:105) (9.5)Let H ( min ) promised := min v ∈ F H v be the minimum average entropy across all vertices of theface: because the generalised Mermin-type argument is strongly contextual, the face cannotcontain any local vertices, and hence the minimum average entropy H ( min ) promised is alwaysstrictly positive; a tighter estimation of this quantity is left to future work. Call η := (cid:16) − H ( min ) promised | K ( ) | N − (cid:17) ∈ [0 ,
1) the information leakage fraction for the face: it is the maximumfraction of plaintexts that Eve can expect to decipher when the empirical model she sees lieson face F .We will now show that protocols based on contextual generalised Mermin-type argumentsalways provide a certain amount of security: for observed noise parameter (cid:15) small enough,the maximum expected fraction of plaintexts that Eve can expect to decipher is sharplypeaked somewhere between η and c · (cid:15) , where c is some constant depending on the geometryof the no-signalling polytope. In one extreme, we may have η = 0, i.e. all empirical model onthe face carry the same maximal amount of entropy. In this case, Eve’s chances of learningsome parts of the secret rely entirely on the noise parameter (cid:15) : in her best case scenario, sheobserves a deterministic empirical model for some fraction (cid:15) of rounds, in which case she cangain complete knowledge about the round plaintext. In the other extreme, we have η (cid:29) (cid:15) ,i.e. there are empirical models on the face F which might lead to more leakage of plaintext ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:47 information than any number of deterministic model which might be lurking in the noise (cid:15) .In this case, Eve’s best bet might just be to exploit the empirical models on the face F itself. Theorem 9.2.
Consider a quantum-classical secret sharing protocol based on a generalisedMermin-type argument ( , , S , β, N ) , in a probabilistic CP* category CP ∗ [ C ] (with R + asits positive semiring of scalars). Consider a run of the protocol with a large number W ofrounds, of which P secret rounds and T test rounds (with P → (1 − τ ) W and T → τ W almost certainly as W → ∞ ). Let (cid:15) be the noise parameter observed by Alice at the end (arandom variable), and let P Eve be maximum number of round plaintexts that Eve expects tosuccessfully decipher (another random variable). Then the maximum fraction of plaintexts P Eve /P that Eve expects to successfully decipher is sharply peaked around some value between η and O ( (cid:15) ) , with variance bounded above by O ( τ (1 − τ ) W ) almost certainly for W → ∞ (wherethe big- O notation hides a constant depending on the geometry of the polytope alone).Proof. As part of this proof, a number of different conditional distributions will be considered:(i) the no-signalling conditional distribution P true ( e ) (cid:2) g (cid:12)(cid:12) m (cid:3) determined by ρ and thedevices B , ..., B N conditional to Eve obtaining information e (this is the conditionaldistribution as seen from Eve’s vantage point);(ii) the no-signalling conditional distribution P true (cid:2) g (cid:12)(cid:12) m (cid:3) := (cid:80) e P [ e ] · P true ( e ) (cid:2) g (cid:12)(cid:12) m (cid:3) determined by ρ and the devices B , ..., B N , averaged over Eve’s information (this isthe true conditional distribution as seen from Alice’s vantage point, which hertests will estimate);(iii) the no-signalling conditional distribution P promised (cid:2) g (cid:12)(cid:12) m (cid:3) derived from the generalisedMermin-type argument (this is what Alice would expect to estimate in the absence ofany noise or tampering);(iv) the conditional distribution P observed (cid:2) g (cid:12)(cid:12) m (cid:3) estimated by Alice.Alice’s estimate of the true conditional distribution P true (cid:2) g (cid:12)(cid:12) m (cid:3) can be modelled by consid-ering the vector-valued random variables X w := (cid:0) X w ( g,m ) (cid:1) for all test rounds w , where X w ( g,m ) is the real-valued random variable defined as follows (note that g w is a random element of K ( ) N , and m w is a uniformly random element of the set of 1 + N S measurement contexts): X w ( g,m ) = (cid:40) g = g w and m = m w X w takes the value 1 over the joint input/joint output pair recorded by Alicefor round w , and 0 everywhere else: Alice’s estimate of the true conditional distribution isthen obtained from the average random variable T (cid:80) w test X w . By the central limit theorem,Alice’s estimate P observed (cid:2) g (cid:12)(cid:12) m (cid:3) will be normally distributed around the true conditionaldistribution, with variance O ( T ); because the noise parameter (cid:15) observed by Alice is obtainedfrom this estimate, it will similarly be distributed around the true noise parameter (cid:15) true defined below, with variance bounded above by O ( T ) (almost certainly for T → ∞ ).We define the true noise parameter (cid:15) true to be obtained from the conditional distri-bution P true (cid:2) g (cid:12)(cid:12) m (cid:3) in the same way that (cid:15) is obtained from the conditional distribution P observed (cid:2) g (cid:12)(cid:12) m (cid:3) . This means (cid:15) true is the largest such that P true (cid:2) g (cid:12)(cid:12) m (cid:3) decomposes asfollows, for some conditional distribution P true,noise (cid:2) g (cid:12)(cid:12) m (cid:3) :(1 − (cid:15) true ) P promised (cid:2) g (cid:12)(cid:12) m (cid:3) + ( (cid:15) true ) P true,noise (cid:2) g (cid:12)(cid:12) m (cid:3) (9.7) :48 S. Gogioso and W. Zeng
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For each value e ∈ E that Eve’s information can take, we define the parameter ξ ( e ) ∈ [0 , P true ( e ) (cid:2) g (cid:12)(cid:12) m (cid:3) decomposesas follows: (1 − ξ ( e )) P F ( e ) (cid:2) g (cid:12)(cid:12) m (cid:3) + ξ ( e ) P F,noise ( e ) (cid:2) g (cid:12)(cid:12) m (cid:3) (9.8)for some distribution P F ( e ) (cid:2) g (cid:12)(cid:12) m (cid:3) lying on the face F and some distribution P F,noise ( e ) (cid:2) g (cid:12)(cid:12) m (cid:3) lying outside of face F . To Eve, in possession of information e , the conditional distribution P true ( e ) (cid:2) g (cid:12)(cid:12) m (cid:3) looks like a biased coin deciding between the two following scenarios:(a) with probability (1 − ξ ( e )), she observes a distribution P F ( e ) (cid:2) g (cid:12)(cid:12) m (cid:3) lying on face F ,which means that the fraction of the round plaintext that she expects to learn is boundedabove by η ;(b) with probability ξ ( e ), she observes some other distribution P F,noise ( e ) (cid:2) g (cid:12)(cid:12) m (cid:3) , which inthe best case scenario could give her full knowledge of the round plaintext.Because marginalising over Eve’s knowledge must result in the distribution P true (cid:2) g (cid:12)(cid:12) m (cid:3) ,the geometry of the polytope implies that the convex combination (cid:80) e P [ e ] ξ ( e ) must go tozero as O ( (cid:15) true ) (i.e. there must be some constant c > (cid:80) e P [ e ] ξ ( e ) ≤ c · (cid:15) true ).It should be noted that the information e obtained by Eve is random to Eve herself:sometimes she will obtain information giving her better guessing probability, sometimes shewill obtain information giving her worse guessing probability. When the distribution of e is taken into account, the fraction of round plaintexts that Eve can expect to decipher isbounded above by the following value, falling somewhere between η and O ( (cid:15) true ): (cid:88) e P [ e ] (cid:16) (1 − ξ ( e )) η + ξ ( e ) (cid:17) (9.9)Again by central limit theorem, the maximum fraction P Eve /P of round plaintexts thatEve expects to successfully decipher is normally distributed around the value above, withvariance O ( P ) (almost certainly for P → ∞ ).Finally, because P Eve /P is sharply peaked around some value between η and O ( (cid:15) true ),with variance O ( P ), and because (cid:15) is sharply peaked around (cid:15) true , with variance boundedabove by O ( T ), we can conclude that P Eve /P is sharply peaked around some value between η and O ( (cid:15) ) , with variance bounded above by O ( T + P ) (which tends to O ( τ (1 − τ ) W ) almostcertainly as W → ∞ ). Conclusions and future work
Conclusions.
Using phase groups and strongly complementary observables, we have fullygeneralised Mermin-type non-locality arguments, and we have provided the exact group-theoretic conditions required for non-locality to arise. In this sense, our results completethe line of enquiry on the connection between phase groups and non-locality started inRefs. [CES10, CDKW12]. We have shown that all generalised Mermin-type argument canbe realised in quantum mechanics, using GHZ states and appropriate sets of phase gates.Furthermore, the abstract, diagrammatic nature of our proofs makes our results immediatelyapplicable to a much larger class of quantum-like theories, such as real quantum theory, I.e. taking the convex combination of the conditional distributions P true ( e ) (cid:2) g (cid:12)(cid:12) m (cid:3) with respect to theprobability distribution P [ e ] of Eve’s side-channel information. ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:49 relational quantum theory, hyperbolic quantum theory, p-adic quantum theory, and modalquantum theory [Gog17].We have proceeded to investigate the empirical models arising from our generalisedarguments, using the sheaf-theoretic framework for non-locality and contextuality [AB11].We have shown the models to provide a new infinite family of inequivalent quantum-realisableAll-vs-Nothing arguments [ABK + Future work.
In the process of pursuing the connection between phase groups and non-locality, this work has left several questions unanswered.Firstly, our generalised arguments are restricted to finite abelian group algebras, whileit is known that strongly complementary pairs extend to all finite group algebras (where oneFrobenius algebra is commutative, and one is potentially non-commutative), and even furtherto certain finite-dimensional compact quantum groups (where both Frobenius algebras arepotentially non-commutative). From a quantum perspective, this corresponds to usingpossibly degenerate observables in place of the non-degenerate ones employed in this work.In the future, we will be interested in investigating the impact that the introduction ofdegenerate observables—in the form of non-abelian group algebras and compact quantumgroups—might have on the connection between non-locality and phase groups that wasestablished here in the abelian group case.Secondly, we have shown that our generalised Mermin-type arguments are All-vs-Nothing,but the converse need not hold in general: there are many examples of All-vs-Nothingarguments which do not come in the format of Mermin-type arguments [ABK + Acknowledgements
The authors would like to thank Samson Abramsky, Bob Coecke, Aleks Kissinger and RuiSoares Barbosa for comments, suggestions and useful discussions, as well as Sukrita Chatterjiand Nicol`o Chiappori for their support. The authors would especially like to thank SamsonAbramsky for pointing out a mistake in a previous version of this work. Funding from :50
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EPSRC (OUCL/2013/SG) and Trinity College (Williams Scholarship) for the first author isgratefully acknowledged.
References [AB11] Samson Abramsky and Adam Brandenburger. The sheaf-theoretic structure of non-locality andcontextuality.
New Journal of Physics , 13, 2011.[ABK +
15] Samson Abramsky, Rui Soares Barbosa, Kohei Kishida, Raymond Lal, and Shane Mansfield.Contextuality, Cohomology and Paradox. , pages 211–228, 2015.[AC09] Samson Abramsky and Bob Coecke. Categorical Quantum Mechanics.
Handbook of QuantumLogic and Quantum Structures , pages 261–323, 2009.[Bac14] Miriam Backens. The ZX-calculus is complete for stabilizer quantum mechanics.
New Journal ofPhysics , 16(9), 2014.[BH12] Sergio Boixo and Chris Heunen. Entangled and sequential quantum protocols with dephasing.
Physical Review Letters , 108(12):1–5, 2012.[BHK05] Jonathan Barrett, Lucien Hardy, and Adrian Kent. No signaling and quantum key distribution.
Physical Review Letters , 95(1):1–4, 2005.[CMP02] Nicolas J. Cerf, Serge Massar, and Stefano Pironio. Greenberger-Horne-Zeilinger Paradoxes forMany Qudits.
Physical Review Letters , 89:080402, 2002.[CD11] Bob Coecke and Ross Duncan. Interacting quantum observables: Categorical algebra anddiagrammatics.
New Journal of Physics , 13, 2011.[CDKW12] Bob Coecke, Ross Duncan, Aleks Kissinger, and Quanlong Wang. Strong complementarity andnon-locality in categorical quantum mechanics.
Proceedings of the 2012 27th Annual ACM/IEEESymposium on Logic in Computer Science, LICS 2012 , pages 245–254, 2012.[CDP11] Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti. Informational derivation ofquantum theory.
Physical Review A - Atomic, Molecular, and Optical Physics , 84(1):1–39, 2011.[CE12] Bob Coecke and Bill Edwards. Spekkens’s toy theory as a category of processes.
Proceedings ofSymposia in Applied Mathematics , 71, 2012.[CES10] Bob Coecke, Bill Edwards, and Robert W. Spekkens. Phase groups and the origin of non-localityfor qubits.
Electronic Notes in Theoretical Computer Science , 270(2):15–36, 2010.[CH15] Oscar Cunningham and Chris Heunen. Axiomatizing complete positivity.
Electronic Proceedingsin Theoretical Computer Science , (Qpl 2015):148–157, 2015.[CHK14] Bob Coecke, Chris Heunen, and Aleks Kissinger. Categories of quantum and classical channels.
Quantum Information Processing , pages 1–31, 2014.[CK15] Bob Coecke and Aleks Kissinger. Categorical Quantum Mechanics I: Causal Quantum Processes,2015.[CK17] Bob Coecke and Aleks Kissinger.
Picturing Quantum Processes . Cambridge University Press,2017.[CL13] Bob Coecke and Raymond Lal. Causal Categories: Relativistically Interacting Processes.
Foun-dations of Physics , 43(4):458–501, 2013.[Coe12] Bob Coecke. The logic of quantum mechanics - Take II.
Logic and Algebraic Structures inQuantum Computing , 2012.[CP10] Bob Coecke and Simon Perdrix. Environment and classical channels in categorical quantummechanics.
Logical Methods in Computer Science , 8(4:14):1–24, 2010.[CPV13] Bob Coecke, Dusko Pavlovic, and Jamie Vicary. A new description of orthogonal bases.
Mathe-matical Structures in Computer Science , 23(03), 2013.[DD16] Ross Duncan and Kevin Dunne. Interacting Frobenius Algebras are Hopf.
Proceedings of the31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS ’16 , 2016.[Fuc15] L´aszl´o Fuchs.
Abelian groups . Springer, 2015.[Gog17] Stefano Gogioso. Fantastic Quantum Theories and Where to Find Them, 2017.[GK17] Stefano Gogioso and Aleks Kissinger. Fully graphical treatment of the quantum algorithm forthe Hidden Subgroup Problem, 2017[GS17] Stefano Gogioso and Carlo Maria Scandolo. Categorical Probabilistic Theories, 2017 ol. 15:2 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS 3:51 [HBB99] Mark Hillery, Vladim´ır Buˇzek, and Andr´e Berthiaume. Quantum secret sharing.
Physical ReviewA , 59(3):1829–1834, 1999.[KZ02] Dagomir Kaszlikowski and Marek Zukowski. Greenberger-Horne-Zeilinger paradoxes for N N-dimensional systems.
Physical Review A , 66:042107, 2002.[Kis12] Aleks Kissinger.
Pictures of Processes: Automated Graph Rewriting for Monoidal Categories andApplications to Quantum Computing . PhD thesis, 2012.[LLK06] Jinhyoung Lee, Seung-Woo Lee, and M. S. Kim. Greenberger-Horne-Zeilinger nonlocality inarbitrary even dimensions.
Physical Review A , 73:032316, 2006.[Mer90] David Mermin. Quantum mysteries revisited.
American Journal of Physics , 58(8):731–734, 1990.[RTHH16] Ravishankar Ramanathan, Jan Tuziemski, Micha(cid:32)l Horodecki, and Pawe(cid:32)l Horodecki. No QuantumRealization of Extremal No-Signaling Boxes.
Physical Review Letters , 117(5):050401, 2016.[RLZL13] Junghee Ryu, Changhyoup Lee, Marek Zukowski, and Jinhyoung Lee. Greenberger-Horne-Zeilinger theorem for N qudits.
Physical Review A , 88:042101, 2013.[Sel07] Peter Selinger. Dagger Compact Closed Categories and Completely Positive Maps.
ElectronicNotes in Theoretical Computer Science , 170:139–163, 2007.[Sel08] Peter Selinger. Idempotents in dagger categories ( extended abstract ).
Electronic Notes inTheoretical Computer Science , 210:107–122, 2008.[Sel09] Peter Selinger. A survey of graphical languages for monoidal categories.
Lecture Notes in Physics ,813:289–355, 2009.[Spe07] Robert W. Spekkens. Evidence for the epistemic view of quantum states: A toy theory.
PhysicalReview A , 75(3):032110, 2007.[Vic11] Jamie Vicary. Categorical Formulation of Finite-Dimensional Quantum Algebras.
Communica-tions in Mathematical Physics , 304(3):765–796, 2011.[Vic12] Jamie Vicary. Topological structure of quantum algorithms. In
Proceedings of the 2013 28thAnnual ACM/IEEE Symposium on Logic in Computer Science , 2012.[VV14] Umesh Vazirani and Thomas Vidick. Fully Device-Independent Quantum Key Distribution.
Physical Review Letters , 113(14):140501, sep 2014.[Zam12] Vladimir Nikolaev Zamdzhiev. An Abstract Approach towards Quantum Secret Sharing, 2012.[ZK99] Marek Zukowski and Dagomir Kaszlikowski. Greenberger-Horne-Zeilinger paradoxes with sym-metric multiport beam splitters.
Physical Review A , 59:3200, 1999.
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