A histories perspective on characterising quantum non-locality

We introduce a framework for studying non-locality and contextuality inspired by the path integral formulation of quantum theory. We prove that the existence of a strongly positive joint quantum measure -- the quantum analogue of a joint probability measure -- on a set of experimental probabilities implies the Navascues-Pironio-Acin (NPA) condition $Q^1$ and is implied by the stronger NPA condition $Q^{1+AB}$. A related condition is shown to be equivalent to $Q^{1+AB}$.


Introduction
The phenomenon now commonly referred to as non-locality was, from the early days of quantum mechanics, central to debates on the adequacy of the standard formalism of quantum theory and other conceptual issues (see for example [1]). More recently, there has been interest in probing the exact quantitative extent of non-locality in quantum theory. For, while ordinary quantum models of experiments can violate Bellʼs local causality condition [2], they do not allow all sets of experimental probabilities or 'behaviours' [3][4][5] consistent with the weaker, operational condition, 'no superluminal signalling' [6]. A research program of 'characterizing quantum non-locality' has arisen with two closely related goals: to provide a method of determining whether a given experimental behaviour could have been produced by an ordinary quantum model, and to discover physical or information-theoretic principles that result in constraints on possible behaviours.
Building on the pioneering work of Tsirelson, Navascues, Pironio and Acin (NPA) made substantial progress towards the first goal. NPA introduced a sequence of conditions of increasing strength, satisfaction of all of which implies the existence of an ordinary quantum model for a behaviour. Most importantly, each condition can be decided-given unlimited computing power-using standard programming techniques [7,8]. As to the second goal, a number of proposals have been made for information-theoretic principles that result in bounds on behaviours, including 'non-triviality of information complexity' [9], 'no advantage for nonlocal computation' [10], 'information causality' [11], 'macroscopic locality' [12], and 'local orthogonality'/'consistent exclusivity' [13][14][15].
Progress on these issues could provide a more general understanding of the limitations on measures of success for quantum information processing tasks, along with new ways of bounding them. It could also throw light on the conceptual issues mentioned above. One view of the conflict between locality and quantum theory is that we should seek to uphold some essential feature of Bellʼs notion of local causality beyond mere no-signalling, while shedding some (at present unidentified) conceptual 'excess baggage'. Drawing an analogy with the apparent conflict between relativity of motion and constancy of the speed of light which resolved itself in the theory of relativity, the hope is that a significant gain in understanding might result from this rehabilitation of causal ideas in quantum theory: see, for example [16], for an approach to quantum theory influenced by such considerations. Developing an understanding of what restrictions on non-locality remain in quantum theory could aid this endeavour.
In the meantime, related questions have arisen more or less independently in work on the problem of quantum gravity in a path integral framework. A number of quantum gravity researchers have argued that, rather than trying to canonically quantize general relativity (GR), the best prospects for success lie in formulating quantum theory in a fundamentally relativistic way using the path integral (see e.g. [17][18][19]). To understand the reasoning behind this, recall that the Hilbert space formalism for quantum theory arises from canonical quantization of the Hamiltonian form of classical mechanics: in order to perform a canonical quantization of GR it must be formulated as a Hamiltonian theory based on a foliation of spacetime with spacelike hypersurfaces of constant time coordinate. General covariance implies that such a time coordinate and such hypersurfaces are not physical and the canonical quantization of this Hamiltonian form of GR leads to the infamous 'problem of time' [20,21] or, more accurately, multiple problems of time [22]. These difficulties suggest that quantum gravity requires a formulation of quantum theory based fundamentally on physical, covariant, spacetime quantities. Early in the history of quantum mechanics (QM), Dirac [23] argued that the canonical theory, based as it is on a choice of time variable, is essentially non-relativistic and that the alternative is to base QM on the Lagrangian form of classical mechanics which 'can easily be expressed relativistically on account of the action functional being a relativistic invariant'. Dirac showed that this leads to the path integral or sum-over-histories which was brought to prominence by Feynman in his paper, 'A spacetime approach to non-relativistic quantum mechanics' (our emphasis) [24]. An approach to quantum theory based on the path integral and free from any fundamental reference to states or measurements was pioneered by Caves, Hartle, Sorkin and others [25][26][27][28][29]. The many advantages for quantum gravity of a histories approach over a canonical approach have been set out by Sorkin [19].
In quantum gravity we must decide which principles from current theories to preserve and which to jettison in the search for the new theory and a principle of relativistic causality would seem to be an excellent candidate for keeping. However, the explicitly operational, no signalling condition is of no use in quantum theories of closed systems such as the whole universe as often considered in quantum gravity. A principle analogous to Bellʼs local causality, suitably generalized to apply in quantum theories of closed systems, which bans superluminal signalling as a special case and yet allows violations of the Bell inequalities, is what is needed [30]. Most concretely, in causal set quantum gravity Bellʼs local causality has been applied as a constraint on classical stochastic dynamical models of a growing discrete spacetime (causal set), with promising and suggestive results [31,32]. If we can find a principle of quantum Bell quantum measure. One proposal for such a condition made in [35] has not, thus far, been successful in this regard.
In this paper we continue the investigation of the quantum Fine trio. We have not achieved the goal of discovering quantum Bell causality but report on progress in understanding the quantitative strength of the condition of the existence of a strongly positive joint quantum measure. We use the the machinery of the NPA hierarchy and show that the strongly positive joint quantum measure condition closely resembles conditions from the hierarchy. In particular this is facilitated by new results on the + Q AB 1 condition in [38]. This allows the set of behaviours satisfying the quantum measure condition to be positioned rather precisely within the hierarchy.

Joint measurement scenarios
We describe here the framework of joint measurement scenarios. It makes contact with the path integral/sum-over-histories approach to QM, is suitable for discussing and resolving questions of common interest between quantum information, foundations of QM and quantum gravity and is closely related to the formalism proposed by Liang, Spekkens and Wiseman (LSW) [39]. The framework can be conceived of purely operationally but the measurement outcomes which which it deals can also be thought of as physical events which are to be (eventually) included in the physics-quantum or otherwise.
In this conceptual framework, there is a set of devices which can perform measurements. To each measurement is associated an exhaustive set of possible, exclusive outcomes. Certain sets of measurements are compatible, that is they can be jointly performed. It may be useful to have in mind, for instance, spin measurements on a collection of atoms or polarization measurements on some photons. Let be the set of possible basic measurements. Let Ξ i be the set of labels for the possible outcomes for the measurement M i . We will assume Ξ i has finite cardinality for all = i p 1 ,... . We introduce the non-contextuality space (NC space), Ξ, an element of which specifies an outcome for each of the measurements M i even though they cannot, in general, be jointly performed: . The NC space Ξ together with the set of all (doable) measurements specifies the joint measurement scenario . This framework is neither the most restrictive nor the most general that one might want to consider 6 . For example, one can imagine putting further constraints on the set of measurements,, such as, if all pairs of measurements in a set of measurements ⊂  K are jointly performable, then K is jointly performable [14,15,41]. One could consider more general NC spaces and/or more general partitions of the NC space as measurements. For example, one could consider scenarios in which some measurement choices can depend on the outcomes of other measurements, something we will investigate in section 5. probability function · P ( ) whose domain is the set of outcomes. P is not a probability measure because the set of outcomes is not a sigma algebra, however it is a measure when restricted to any algebra of outcomes A M for a measurement M. · P( ) can be referred to as the (set of) experimental probabilities.
The consistency of the probability function is a further non-contextuality assumption beyond the one mentioned above. There it was stressed that the formalism treats the outcome, X , of a measurement, M , as the same event no matter what measurement, ′ M , M may be included in. Now we are assuming that the probability of X is the same no matter what joint measurement M is included in. This would be most strongly, physically justified if the basic measurements have specific spacetime locations and those locations are such that any jointly performable basic measurements are spacelike separated from each other. Then the consistency of the probability function would be justified by relativistic causality: it is a no-signalling condition. Again, if the situation being modelled does not have this underlying spacetime structure, then the assumption of the consistency of the probability function would have to be argued for in some other way.
A key question in the analysis of contextuality and non-locality takes the form, in this framework, of whether a given behaviour Ξ can be consistently extended in some way to a larger subset of Ξ 2 than. set of outcomes along with a set of subsets of the outcomes representing measurements. While the NC space could be defined as the set of 'deterministic probabilistic models' (as defined by FLS) for an FLS scenario, measurements and their outcomes cannot always be expressed in terms of this set as they are above (it can be empty for valid FLS scenarios). There are more other differences in approach, most importantly that the FLS framework is not defined in terms of basic and joint measurements. Instead, the no-signalling and the commutation requirements for quantum models are enforced in non-locality scenarios by adding extra measurements. While this gives a simple and powerful formalism, without adding more structure it is difficult to define e.g. the original NPA set Q 1 , or to discuss different versions of our quantum measure-based conditions.
Example. Bell scenarios. To make contact with the familiar special case of non-locality setups we will need some additional concepts.
Composition of a set of scenarios can be defined if we are modelling a situation in which the choice of measurements on one system does not affect what can be done with the others: a particularly natural assumption if the systems are spacelike separated for the period of measurement. The NC space in this case is The set of measurements  contains all of these measurements for all members of  . Because the operations on one system do not affect the possibilities of measurement elsewhere, any set of these measurements all of which come from different scenarios in  can be jointly performed, and there are no further measurements iñ   .
Bell scenarios are a special type of joint measurement scenario suitable for the study of non-locality. First, consider a simple measurement scenario with m mutually incompatible measurements, each with d outcomes. Composing n of these 'wings' as above gives a Bell scenario, specified by the triple ( ) n m d , , . Each 'local' outcome for each wing X a x i i is thus uniquely specified by two integer indices, x i for the measurement 'setting' choice and a i for the outcome, where i labels the wing. Standard notation refers to such outcomes simply as | . The CHSH scenario, specified by ( ) 2, 2, 2 , is the simplest interesting Bell scenario. It is the most well-studied example, and also the one to which some previous results exclusively apply.
For Bell scenarios, consistency of the probabilities of outcomes over different measurements is equivalent to no signalling: the probabilities for outcomes of a measurement will be independent of any measurement choices made at spacelike separation.

The JPM condition, non-contextuality & Bell ʼs local causality
A behaviour gives probabilities for the outcomes of doable experiments. Let us now consider a joint probability measure P J on the sigma algebra, A, generated by and define the set of behaviours consistent with the existence of such a measure. In our case, = Ξ A 2 since Ξ is finite.
is in the set JPM if there exists a joint probability measure P J on the sigma algebra Ξ 2 such that J Since Ξ 2 is finite, P J is fixed by its values for the atoms of the algebra Ξ 2 , i.e. the singleton sets corresponding to the elements of Ξ itself: γ ( ) . JPM is therefore a non-contextuality condition. Each measurement outcome could be said to be determined by some pre-existing physical property which is not affected by, but merely revealed by the particular measurement performed. The measure P J can be interpreted as a measure over these properties. In the language of stochastic processes, outcomes are just some of the events, subsets of the 'sample space' Ξ, all of which are now covered by one probability measure.
For Bell scenarios, the JPM condition is equivalent to the existence of a Bell locally causal model [2] of the probabilities. For the CHSH setup this was proved by Fine [34] (see [36] for a generalization to all Bell scenarios, and further references). The JPM condition for Bell scenarios is also equivalent to the satisfaction of a set of (scenario dependent) linear inequalities between experimental probabilities, generically known as 'Bell inequalities' [36]. The equivalence between non-contextuality, Bellʼs local causality and constraints on experimental behaviours inspires our proposals at the quantum level, where we seek analogues of these conditions.

Ordinary quantum behaviours
The set of behaviours Q that can be derived from ordinary quantum models is of obvious interest. In the following, it will be useful to recall that the expression ∈ X M identifies X as a fine-grained outcome of the measurement M (X is a subset of Ξ in the measurement partition M ) while ∈ A X M refers to any outcome in the Boolean sigma algebra generated by the finegrained outcomes.
is in the set Q if there exists an ordinary quantum model for the behaviour, that is, a Hilbert space  , a pure state ψ | 〉 ∈  , and a projection operator E X on  for each fine-grained outcome, X , of each basic measurement such that can be jointly performed, then It follows that, for all basic measurements ∈  M i , the projectors defining an ordinary quantum model satisfy: for all ≠ X Y, because all these terms are positive operators.
is in the set Q if and only if there exists a Hilbert space  , a pure state ψ | 〉 ∈  , and a map˜→  E L : , from the set of outcomes to the lattice, L, of projection operators on  (equivalently, the lattice of subspaces of  ) such that is a Boolean algebra homomorphism ; • the probabilities are given by Proof. If the behaviour is in Q then we have projection operators for each of the fine-grained We then define a projector for a fine-grained outcome of a joint measurement by forming the product of the projectors for the relevant basic measurement outcomes: for outcome X of a joint measurement of the set of basic measurements … where the E X i k all commute and so this is indeed a projector. Finally, the projector, E(Y) for a coarse-grained outcome, ∈ A Y M , is defined to be the sum of the projectors for the fine-grained outcomes of M of which Y is the union. This defines a map˜→  E L : which has the properties above, by construction.
If the map E exists with the properties above, the projectors E(X) for the fine-grained outcomes of the basic measurements will satisfy the conditions in definition 2. Note here that we are using the definition of Boolean algebra homomorphism which includes the condition that the unit element is mapped to the unit element. □ We will refer to such a Hilbert space and map E as an ordinary quantum model for the behaviour. Note that, throughout this paper, we assiduously refer to such quantum models as 'ordinary' in order to distinguish them from other models that, from a histories perspective, are also classified as 'quantum', in particular the quantum measure theory models introduced below. In Bell scenarios, the condition in definition 2, equation (9), translates to the requirement that the projectors for spacelike separated measurements commute. The definition of ordinary quantum behaviours here follows the definition given in [8] in which it is not required that the Hilbert space be a tensor product. See that reference for a discussion of the fact that it is not known in general whether requiring a tensor product Hilbert space is a strictly stronger condition ('Tsirelsonʼs problem').
A number of questions arise regarding Q, including: what assumptions are sufficient to determine the shape of Q; can we derive the condition based on simple, more-or-less physical criteria rather than by invoking the above definition, which is considerably less conceptually simple than the definition of JPM; does the set of allowed behaviours remain the same when an alternative framework for QM is considered, such as the path integral?

NPA conditions
Tsirelson pioneered the study of bounds on Q, which can be thought of as defining sets of behaviours containing Q [3][4][5]. More recently, NPA introduced a hierarchy of conditions resulting in a nested sequence of sets of behaviours for bipartite Bell scenarios which contain and converge to Q [7,8]. The conditions defining these sets can be cast as semidefinite programmes which facilitates solution by computer, and allows e.g. estimation of violations of Bell inequalities in QM, beyond the CHSH scenario. Rather than review the whole hierarchy in detail here, two conditions that will be of particular relevance will be defined. The definitions have been generalized to joint measurement scenarios but are equivalent to the original ones given by NPA when we restrict to bipartite Bell scenarios.
The set of behaviours Q 1 is the first in the original NPA hierarchy. The basic observation used to define this set is that, in ordinary quantum models, the inner products between all pairs of vectors from the set ψ ψ ∪ an outcome of a basic measurement X must obey certain conditions, including relations to the experimental probabilities. If a model is in Q, there must therefore exist a Hilbert space and vectors in it satisfying these conditions. The set of behaviours obeying this condition is called Q 1 . This set of behaviours has received substantial attention in the literature. In the CHSH scenario it is equivalent to a simple bound, and the maximum violation of several Bell inequalities for Q 1 behaviours are known [8]. It is also equivalent to the 'macroscopic locality' condition proposed by Navascues and Wunderlich [12].
is in Q 1 if there exists a Hilbert space  and vectors ∈ { } X X indexed by the coarse-grained outcomes of the basic measurements, that span  , and satisfy the following conditions: It is clear from the definition that ⊆ Q Q 1 and it is known that Q 1 is actually strictly larger than Q [8].
The definition of Q 1 is adapted directly from the NPA definition in terms of positive matrices, and is equivalent to it for bipartite Bell scenarios. The proof is in appendix A. For the discussion below, it will be easier to work directly with this definition expressed in terms of vectors. Note that the the condition that the vectors span  can be dropped without altering the set Q 1 since if such a set of vectors exists that doesnʼt span the Hilbert space, then  can be redefined to be the span of the vectors.
The basic measurements play a key role in the definition of Q 1 and an obvious strengthening of the condition is to require that vectors corresponding to all outcomes of all joint measurements exist satisfying the appropriate conditions. This leads to the set of behaviours called + Q AB 1 in the NPA approach 7 , and is the first in the modified version of the hierarchy introduced later by FLS [42].
if there exists a Hilbert space  and a set of vectors ∈ { } X X indexed by the outcomes of the joint measurements, that span  and satisfy the following conditions: (ii) for each measurement ∈ M and all outcomes ∈ A X Y , , and there exists a basic measurement M i and outcomes Crucially, condition (iii) says that two vectors corresponding to outcomes X and Y in are orthogonal if they imply disjoint outcomes for any basic measurement, even if the two outcomes X and Y are not compatible and are not both elements of any single measurement sigma algebra A M .
The set + Q AB 1 contains Q, and it follows easily from the definitions that Q 1 contains + Q AB 1 . There is strong computational evidence that these are strict containments [8]. The condition is well-studied and has a number of remarkable properties. For instance, it can be shown to be equivalent to a bound on the Lovász number of the 'orthogonality graph' associated to the behaviour [42,43], an important quantity for classical information theory that was defined well before the advent of quantum information. Also, all of the currently proposed informationtheoretic principles that restrict non-locality can be shown to be implied by the + Q AB 1 bound [38].
Although the higher levels of the NPA hierarchy will not enter into the following discussion, note that level n is equivalent to the existence of vectors with the same inner product properties as those obtained from ordinary quantum models by applying all products of n projectors to the state ψ | 〉 (the projectors corresponding to basic measurements for the NPA version, and joint measurements for the FLS version of the hierarchy).

Quantum measure theory
We briefly review the histories approach to quantum theory. For more details see [29,[44][45][46]. In this approach quantum theory is understood as a species of generalized measure theory, a generalization of classical stochastic processes such as Brownian motion. In histories-based theories-classical, quantum or transquantum-the fundamental kinematical concept is the set of spacetime histories, Ω. For a particular physical system, each history in Ω is as complete a description of the system as is conceivable in the theory, over all time of interest. The kind of elements in Ω varies from theory to theory: in n-particle classical or QM, a history is a set of n trajectories; in a classical or quantum scalar field theory, a history is a real or complex function on spacetime; in GR, a history is a Lorentzian geometry. Discovering the appropriate set of histories Ω for a particular system is part of the business of physics. In quantum theory, conceived of as quantum measure theory, Ω is the set over which the integration of the path integral takes place. Even for the non-relativistic particle, the path integral remains to be defined rigorously as a genuine integral over paths [47], something that will not concern us here as we restrict ourselves to finite systems.
Once the set of histories has been settled upon, any proposition about the system is represented by a subset of Ω. For example in the case of the non-relativistic particle, if R is a particular spacetime region, the proposition 'the particle traverses R' corresponds to the set of all spacetime trajectories which intersect R. We follow the standard terminology of stochastic processes and refer to such subsets of Ω as events.
In this framework, a physical theory based on Ω is expressed as a generalized measure theory, specified by a sigma algebra, ⊆ Ω A 2 , of events and a measure, μ → A : , where μ is a non-negative function which encodes (a combination of) the dynamics and initial condition. Sorkin has identified a hierarchy of measure theories defined by a sequence of strictly weakening conditions on the measure, μ, of which the first is the Kolmogorov sum rule for probability measures. Classical stochastic theories such as Brownian motion therefore lie in the first level of the Sorkin hierarchy and classical deterministic theories are special cases.
The second level in the Sorkin hierarchy is quantum measure theory. A quantum measure on an event algebra A is a map μ → A :  such that: 1. for all α ∈ A, μ α ⩾ ( ) 0 (Positivity); The quantum sum rule expresses the lack of interference between triples of histories, just as the classical Kolmogorov rule expresses the lack of interference between pairs of histories. The higher levels of the Sorkin hierarchy generalize this to lack of interference between k histories. Each level of the hierarchy contains the previous levels, for example, a classical stochastic level 1 theory satisfies the level 2 condition.
There are, broadly, two interpretational frameworks for quantum measure theory in the literature. In both frameworks, the formalism of histories, events and the decoherence functional is the complete physics of the system: histories-based quantum theory is quantum theory of a closed system, a quantum theory without external observers, agents or measuring devices. Observers, agents and measuring devices, should they exist, are described within the theory. In decoherent or consistent histories [26,48,49] probabilities are considered to be fundamental and attention is focussed on subalgebras of A on which the quantum measure restricts to a probability measure. The fact that there are (infinitely) many such subalgebras must then be grappled with [50,51]. In the co-event framework [52,53], attention is focussed on events of measure zero and an interpretation sought based on the maxim 'events of measure zero do not happen'.
In all known quantum theories, the quantum measure is not given directly but via the agency of a decoherence functional on the event algebra. A decoherence functional is a map × → A A D:  such that [18,27] 1. For all α β ∈ A , , we have .

For all α ∈
The existence of a quantum measure is equivalent to the existence of a decoherence functional. If × → A A D:  is a decoherence functional then the map μ → A :  defined by μ α α α = D ( ) : ( , ) satisfies the conditions of being a quantum measure. And conversely, if μ is a quantum measure, there exists a (non-unique) decoherence functional D such that μ α α α = D ( ) ( , ) [29]. Given this, we will refer to a decoherence functional also as a quantum measure. We call a triple Ω A ( ) D , , a quantum measure system. It is also the case that in all quantum theories describing known physics the decoherence functional satisfies the condition of strong positivity [54]: a decoherence functional is strongly positive (SP) if for any finite collection of events, α α ,... The product of two independent finite, strongly positive quantum measure systems is a strongly positive quantum measure system.
Proof. The product decoherence functional on the atoms of the tensor product algebra is given by the Hadamard product of the two decoherence functionals of the subsystems. The Hadamard product of two positive matrices is positive. □ Strong positivity thus implies composability of subsystems and is related to complete positivity of quantum operations in the context of evolving systems in quantum measure theory [54]. There is therefore a good reason to adopt strong positivity as a physical condition on quantum measures, and if one were convinced by this reason, one might want to redefine quantum measures to include this condition. We will not do so in this paper as we want to discuss the consequences of adopting or not adopting the condition.

A quantum analogue of non-contextuality
One response to the Bell theorem, is to retreat to talk of the experimental settings and outcomes only, and to investigate operational principles that constrain the experimental probabilities in order to try to understand the bounds on Q. Loosely speaking, this is to start from the most general set of all behaviours and ask why the quantum constraints are so strong [6]. We propose a complementary perspective, continuing to focus on the NC space Ξ for a behaviour, but from a histories perspective within the framework of quantum measure theory. The approach then becomes one of generalizing the non-contextuality condition JPM, asking why the quantum constraints are so weak.
The JPM condition for a behaviour is that there exists a joint probability measure on the full sigma algebra A generated by. Probability measures are Sorkin level one measures and since quantum measures are Sorkin level two measures this suggests the following obvious quantum analogue of JPM: We can re-express the JQM condition in terms of decoherence functionals: is in the set JQM if and only if there exists a joint decoherence functional, D J over Ξ 2 such that for all ∈ Ξ X 2 [29]. Any decoherence functional satisfies for any two disjoint events, X and Y . For any measurement, M and follows. Note that the two conditions The converse is immediate on defining μ = ( ) ( ) An obvious strengthening of the JQM condition is: is in the set SPJQM if there exists a joint decoherence functional, D J over Ξ 2 such that • D J is strongly positive (SP).
Definitions 7 and 9 are generalizations of sets defined for the CHSH scenario in [35,37].
Recall that positive semidefiniteness of a Hermitian matrix is equivalent to the matrix being the inner product matrix, the Gram matrix, of a set of vectors spanning a Hilbert space  . Using this, we can present an equivalent definition of the SPJQM set which makes manifest its connection to the NPA heirarchy of conditions: is in SPJQM if and only if there exists a Hilbert space  spanned by a set of vectors indexed by elements of the sigma algebra Ξ 2 , , such that their inner products satisfy the following conditions: • for any ∈ Ξ X 2 , M In particular, if two outcomes, X and Y , are disjoint then 〈 | 〉 = X Y 0: disjoint outcomes decohere.
Although it is less simple to state, SPJQM is physically more interesting than JQM. A theory would be problematic if composition of independent behaviours allowed by the theory New J. Phys. 16 (2014) 033033 F Dowker et al yielded a behaviour not allowed by the theory (see e.g. [14,15,42] for applications of this). The JQM condition is not composable: there exist behaviours for the CHSH scenario that are in JQM but such that the composition of two independent copies is not [55]. The SPJQM condition on the other hand respects composition due to lemma 6. It is an open question whether the SPJQM condition follows from the JQM condition plus some simple principle expressing closure under composition of independent systems.
For more general scenarios this is not the case, for example the composition of two independent PR boxes is not in JQM [55]. The fact that two independent PR boxes when composed takes us outside the space of JQM is reminiscent to the situation for LO. This is not a coincidence, since one can show that JQM with some modification implies LO.
Corollary 12. Q is a proper subset of JQM.
The proofs of these two lemmas are in [35,37]. On comparison of the SPJQM condition with the NPA conditions, similarities immediately become apparent. Both require the existence of vectors with an inner product matrix that has a certain relationship to the experimental probabilities and a number of orthogonality relations. Much of the work has been done by formulating the conditions in this way. Based on this observation, the following relations can be discovered.  (14) and (15) from definition 4 of Q 1 is an easy task: the SPJQM vectors range over all subsets of Ξ, while the Q 1 vectors range over a subset of these, so we can define each Q 1 vector as equal to the corresponding SPJQM vector. This immediately gives (14) from (23) and (15) from (24). □ This theorem subsumes the earlier result given in lemma 13: the Tsirelson-Landau-Masanes bound is implied by Q 1 [8]. It should be noted that, although the SPJQM vectors are indexed by outcomes of all measurements in and not just the basic ones, one cannot use the same reasoning as above to prove that SPJQM ⊆ + Q AB

1
. This is because the relations (18) in the definition of + Q AB 1 are not implied by the conditions in the SPJQM definition. The significance of these extra orthogonality relations will be expanded on later.
Thus, the set of operators , indexed by the outcomes of M i , forms a projective measurement. Note this does not prove that there is an ordinary quantum model, because these operators may not satisfy (9). Using these projectors, we define the vector for an element γ Ξ ∈ as other order would have worked also. Thus there is no unique set of vectors γ { } { } satisfying the conditions.

A complication: branching measurements
In the formalism above, we have not allowed measurement choices to depend on other measurement outcomes, though one can imagine setting up such a situation in a lab. In the CHSH scenario, an example would be using the outcome of whatever measurement is done in the A wing to determine the measurement setting in the B wing, if the former is done in the causal past of the latter. Such 'branching' measurements play an important role in the alternative FLS formalism for contextuality [42], where they are necessary for the definition of Bell scenarios, and so a comparison with results in this formalism requires us to consider them.
We will investigate the inclusion of these branching measurements and the altering of the non-contextuality condition to reflect the assumption that if measurements M and ′ M are jointly performable then they are jointly performable in any causal order-either may causally precede the other, thus allowing branching, or they may be performed at spacelike separation-without altering an underlying measure on the NC space. This is to allow the classical communication between wings necessary for the branching choices. Note that this assumption is incompatible with an assumption that the basic measurements have fixed locations in spacetime and jointly performable basic measurements are spacelike separated from each other. We argued above that such fixed spacetime locations for the basic measurements justifies our joint measurement scenario and behaviour framework. Including branching measurements means that the noncontextuality assumptions inherent in the joint measurement scenario and behaviour framework must be made without the argument from locality.
We will show below that the altered non-contextuality condition in the classical case is no stronger than JPM. However in the quantum case the altered non-contextuality condition makes a difference to our conclusions. To illustrate the reason for this, consider two outcomes in the CHSH scenario, : these outcomes need not decohere with respect to each other because they are not alternative outcomes of a possible measurement. If, however, we add to the collection of possible measurements a branching measurement with both X and Y as alternative outcomesone in which the outcome, a, of the measurement = x 0 in the A wing is used to determine the y setting in the B wing-then the corresponding quantum non-contextuality condition requires that the joint decoherence functional restricts to a probability measure on the sigma algebra of outcomes for this branching measurement. This implies = ( ) , an additional condition on the joint quantum measure. All the extra conditions of this sort from branching measurements will allow us to prove a stronger result.
In general, given a joint measurement scenario Ξ ( ) , , the possible branching measurements form an extended set of measurement partitions. We will need to consider only branching measurements in which one decision is made; we call these single branching measurements. A single branching measurement consists of an initial (not necessarily basic) we also extend the probability function P to cover the larger set of outcomes b . Note here that each fine-grained outcome for a branching measurement is a fine-grained outcome for some joint measurement in, so the only elements in b and not in are coarsegrained outcomes of the new measurements. This means that the probability function, P, on can be uniquely extended to a probability function on b , by applying the Kolmogorov sum rule. We call the behaviour thus extended, Ξ ( ) , the branching extension of Ξ ( ) P , , . Does including these types of measurements make any difference to the strength of the conditions discussed above? That is, if in the definitions of our sets JPM, Q, JQM, etc., we were to replace with b and with b , would the sets remain the same? For the set of classically non-contextual behaviours JPM given by definition 1, it is not hard to see that nothing changes: the alternative definition would be the same except with equation (7) 8 One could also consider multiple branching measurements, the conceptually obvious (but notationally cumbersome) generalization of single branching measurements. However, this would have no effect of the argument below. With such an altered definition of b lemma 17 would still be true, and so the altered definition of SPJQMb would still be equivalent to + Q AB

1
. That is, once single-branching measurements have been added to set of possible measurements, adding multiple branching measurements does not alter the quantum non-contextuality condition.
whereas before P J is a probability measure over the sigma algebra Ξ 2 . Obviously this implies equation (7) because˜⊂  b . It is also implied by equation (7), because as just discussed, for sets ∈ X b which are not in, the value of P(X) is determined by the probabilities of sets in by the Kolmogorov rule, as is the value of ( ) It is a straightforward exercise to show that the same is true for the sets Q and Q 1 . Here we spell out how this works for + Q AB can be proved following the proof of theorem 16. □

Discussion
We have found close relations between the NPA hierarchy of sets of experimental behaviours, and sets of behaviours of interest in quantum measure theory. In particular a quantum noncontextuality condition, SPJQM, was defined, subsuming the previous definition for the CHSH scenario [35]. This condition was found to be intermediate in strength between Q 1 and + Q AB is strictly larger than Q, as indicated by the computational evidence, then nonlocal correlations beyond those achievable in ordinary QM are achievable within strongly positive quantum measure theory. However, it is possible that additional constraints may arise within this framework as it is developed.
We also saw that an interesting modification of SPJQM, which we called SPJQM b , is equal to the NPA set + Q AB

1
. The necessary change is to require that for any joint measurement behaviour allowed by the physics, its branching extension is also physical. Including branching where the outcome of one measurement determines which future measurement is performed, imposes more conditions because the quantum measure must restrict to a probability measure, and therefore decohere, on a larger collection of measurement subalgebras. The set SPJQM b is therefore 'closed under wiring' meaning that any behaviour created by applying 'classical operations'-coarse-graining measurement choices and/or outputs, postselection, composition or 'branching' as described above-to a collection of behaviours in the set is also in the set [38]. It is not yet known whether SPJQM b is strictly contained in SPJQM or whether they are equal. We will assume the former is the case, for the sake of the following discussion.
Returning to our main motivation, the use of quantum analogues of the Fine trio of conditions on a behaviour to discover quantum Bell causality, we see now that we have a choice for the 'joint measure' condition in the trio: SPJQM or SPJQM b . In contrast, as we saw, in the original Fine trio there is no choice to be made because the non-contextuality conditions are equivalent when the joint measure is a probability measure: ≡ JPM JPM b . For the purpose of identifying a quantum Bell causality condition, the relevant scenarios are Bell scenarios where the jointly performable measurements are spacelike separated which therefore rules out the possibility of branching measurements and it would seem therefore that SPJQM is the appropriate condition 9 . It would be prudent to bear SPJQM b in mind, however. We hope that the quantum non-contextuality condition will be a guide to discovering quantum Bell causality because the latter should imply the former. It is possible that if we find a scientifically fruitful condition of quantum Bell causality-perhaps by another route-it will turn out to imply the stronger condition and then we will have discovered that SPJQM b was the appropriate quantum non-contextuality condition after all.
Many other questions remain. The exact strength of the SPJQM condition has not been determined: it is not known, for example, whether or not it is equivalent to Q 1 . An exploration of the relation of the weaker, but simpler JQM condition to other principles such as local orthogonality and a proof that if JQM is strengthened to JQM b by including branching measurements it implies local orthogonality will appear elsewhere [55]. The assumption of strong positivity of the quantum measure was motivated by arguing that it guarantees composability of quantum measures on two independent systems. If it can be proved that strong positivity is necessary for composability then the assumption of strong positivity would have very strong physical motivation. The connection to local orthogonality/consistent exclusivity might lead to new insights here. What of the higher levels of the NPA hierarchy? As mentioned above, these conditions require the existence of a matrix with the same properties as the inner product matrix of vectors corresponding to strings of projection operators acting on the initial state in an ordinary quantum model. Strings of projection operators are known as class operators in the literature on decoherence functionals and are basic entities from which decoherence functionals are constructed for sequences of events ordered in time. For Bell scenarios therefore, it is possible that there is a connection between the higher level NPA conditions and decoherence functional based conditions arising from consideration of sequences of possible measurements. (A.1) • if M i is a basic measurement and ∈ X Y M , i are fine-grained outcomes, The translation to the terminology of NPA is fairly direct. For example, for two finegrained outcomes X and Y taken from jointly performable basic measurements, ∩ ( ) P X Y corresponds to ( ) P a b , , because the only jointly performable pairs of measurements in bipartite Bell scenarios are those taken from different wings. The following reformulation brings us closer to that used in the main text: , fine-grained outcomes of basic measurements or Ξ. Then Γ 1 is positive semidefinite, and equations (A.6, A.7) are implied by (14,15) respectively.
Conversely, consider a behaviour in Q NPA 1 . There exists a set of vectors indexed by the set of fine-grained basic measurement outcomes and Ξ, spanning a Hilbert space  , such that . We can extend this set of vectors to one indexed by -that is, all coarse-grained outcomes of basic measurements-by summation according to (14). This is consistent with the property (A.6) of the original vectors. Condition (15) follows from (A.7), noting that if this equation holds for the fine-grained basic measurement outcomes then it will also hold for unions of them, the coarse-grained outcomes. □ Very similar reasoning can be applied to the case of + Q AB

1
. Similar reasoning can be applied to the case of + Q AB

1
. There, the equation analogous to (A.6) is exactly the normalization condition of FLS [58] specialized to the first level of the hierarchy, while (A.7) implies their orthogonality condition (and also gives the connection to the experimental probabilities).