A histories perspective on characterizing quantum non-locality

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 Bell's local causality condition [2], they do not allow all sets of experimental probabilities or 'behaviours' [3–5] 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 non-local computation' [10], 'information causality' [11], 'macroscopic locality' [12], and 'local orthogonality'/'consistent exclusivity' [13–15].

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 Bell's 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. [17–19]). 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 [25–29]. 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 Bell's 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 Bell's 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 causality, quantum dynamical models of discrete causal set spacetimes could then be constructed using a similar approach.

At this point, this important question in this approach to quantum gravity intersects with the programme of characterizing quantum non-locality in quantum information research in the following way. There is a trio of equivalent conditions on a behaviour in the Clauser–Horne–Shimony–Holt (CHSH) scenario: the existence of a Bell locally causal model (also known as a 'classical screening off model') of the probabilities, the satisfaction of all the CHSH (Bell) inequalities [33] and the existence of a joint probability measure (JPM) on all the possible (counterfactual) outcomes [34, 35]. This is also the case for more general experimental scenarios, with appropriate analogues, usually referred to simply as Bell inequalities, of the CHSH inequalities [36]. Thus, Bell's local causality, obedience of certain bounds on experimental probabilities and a non-contextuality condition are equivalent ⁴ . The path integral approach to QM reveals quantum theories to be generalized measure theories in which probability measures are generalized to quantum measures by weakening the Kolmogorov sum rule to an analogous rule that expresses the absence of genuine three-way interference between alternative histories for a system [29]. This immediately raises the question of what becomes of the 'Fine trio' of equivalent conditions—local causality (classical screening off), bounds on experimental probabilities and non-contextuality—when we enlarge the category of physical models we consider to ones in which dynamics is represented by quantum measures. A natural analogue of non-contextuality is the existence of a joint quantum measure (JQM) over all the outcomes [35]. The existence of such a measure in the CHSH scenario implies no restriction on the experimental probabilities, indeed a Popescu–Rohrlich (PR) box is possible [35, 37]. However, requiring the joint quantum measure to be strongly positive (SPJQM) implies the Tsirelson inequalities on the correlations [35, 37] and thus we have the beginnings of a quantum analogue of the Fine trio, with an implication of the form 'joint measure implies bounds on probabilities'. This raises the hope that this will be a guide to discovering the quantum Bell causality condition, the most physically important condition in the trio. In particular, quantum Bell causality should be a condition that implies the existence of a strongly positive joint 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 ${{Q}^{1+AB}}$ condition in [38]. This allows the set of behaviours satisfying the quantum measure condition to be positioned rather precisely within the hierarchy.

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 $\mathcal{M}:=\left\{ {{M}_{1}},{{M}_{2}},...{{M}_{p}} \right\}$ be the set of possible basic measurements. Let $\Xi_{i}$ be the set of labels for the possible outcomes for the measurement ${{M}_{i}}$. We will assume $\Xi_{i}$ has finite cardinality for all $i=1,...p$. We introduce the non-contextuality space (NC space), $\Xi$, an element of which specifies an outcome for each of the measurements ${{M}_{i}}$ even though they cannot, in general, be jointly performed:

$$\begin{eqnarray}&&\Xi:=\Xi_{1}\times \Xi_{2}\times \ldots \times \Xi_{p}.\end{eqnarray} \tag{ 1 }$$

Each basic measurement, ${{M}_{i}}$, corresponds to a partition of $\Xi$. Each possible outcome of the measurement ${{M}_{i}}$ will be represented by one of the subsets in this partition, $X\in {{M}_{i}}$. Note that we refer to the measurement and the partition with the same symbol. For a basic measurement ${{M}_{i}}$, an outcome ${{X}_{i}}\in {{M}_{i}}$ is a set of the form

$$\begin{eqnarray}&&{{X}_{i}}=\Xi_{1}\times ...\Xi_{i-1}\times \left\{ {{a}_{i}} \right\}\times \Xi_{i+1}\cdots \times \Xi_{p}\;\subset \;\Xi,\end{eqnarray} \tag{ 2 }$$

where ${{a}_{i}}\in \Xi_{i}$. The set of all such subsets, ranging over all elements of $\Xi_{i}$, is the partition of $\Xi$ identified with basic measurement ${{M}_{i}}$.

A joint measurement is specified by a set of basic measurements $\left\{ {{M}_{{{i}_{1}}}},{{M}_{{{i}_{2}}}},...{{M}_{{{i}_{n}}}} \right\}$ and the partition, $M$, corresponding to the joint measurement, is the set of all sets of the form

$$\begin{eqnarray}&&{{X}_{{{i}_{1}}}}{{X}_{{{i}_{2}}}}...{{X}_{{{i}_{n}}}},\end{eqnarray} \tag{ 3 }$$

where ${{X}_{{{i}_{k}}}}\in {{M}_{{{i}_{k}}}}$. Note that each outcome of one of the basic measurements ${{M}_{{{i}_{k}}}}$ is a union of outcomes for the joint measurement $M$. Also, for each joint outcome $X\in M$, and each outcome ${{X}_{{{i}_{k}}}}$ of basic measurement ${{M}_{{{i}_{k}}}}$, either $X{{X}_{{{i}_{k}}}}=X$ or $X{{X}_{{{i}_{k}}}}=\varnothing$.

A joint measurement scenario, based on the set of basic measurements $\mathcal{M}$ and NC space $\Xi$, is fixed by specifying a collection of joint measurements, to be thought of as those joint measurements which are physically possible to do. We will denote the set of all joint measurements in the scenario by $\hat{M}\,$. Each measurement in $\hat{M}\,$ corresponds to a subset of $\mathcal{M}$. All the basic measurements are elements of $\hat{M}\,$ and if there is a joint measurement in $\hat{M}\,$ of a set $K$ of basic measurements then there is a joint measurement in $\hat{M}\,$ of each subset of $K$. Endowing $\hat{M}\,$ with the corresponding order by inclusion and adding the empty measurement to $\hat{M}\,$ makes $\hat{M}\,$ a meet semilattice. A maximal joint measurement is a measurement that is maximal in this order and the basic measurements are minimal in the order if the empty measurement is removed. We say that measurement $M$ is included in measurement $M\prime$ if $M$ precedes $M\prime$ in the order. We say that a collection of measurements in $\hat{M}\,$ can be jointly performed if there is a joint measurement in $\hat{M}\,$ which includes them all.

Note that this formalism already includes some kind of non-contextuality assumption: an outcome, $X$, of a basic measurement, ${{M}_{i}}$ is dealt with by the formalism as the same event, no matter which joint measurement ${{M}_{i}}$ is included in. This would be physically most justifiable were the basic measurements all to have specified spacetime locations such that jointly performable basic measurements are in different spacetime locations. Then the non-contextuality is justified by a form of locality, namely the assumption that the basic measurement outcomes are local events whose description is in terms of local variables associated to the spacetime region in which the outcome occurs ⁵ . If the situation being modelled is supposed not to be of this form then the non-contextuality assumption inherent in the formalism has to be justified in some other way. We will return to these considerations later.

For each measurement $M\in \hat{M}\,$, there is a set of outcomes which are the elements of the partition $M$. We will refer to these as the fine-grained outcomes of $M$. We consider also all coarse-grained outcomes i.e. all unions of the fine-grained outcomes of $M$. For each $M\in \hat{M}\,$ the set of all coarse-grained outcomes together with the empty outcome form the (Boolean) sigma algebra, ${{\mathfrak{A}}_{M}}$, generated by the fine-grained outcomes. We will use the term sigma algebra even though the case in hand is finite because one might want to extend the formalism to infinite scenarios. The union of the algebras of outcomes over all measurements in a scenario will be denoted $\hat{C}\,:={{\mathop{}^{}}_{M\in \hat{M}\,}}{{\mathfrak{A}}_{M}}$ and the union of the algebras over all basic measurements will be denoted $\mathcal{C}:=\mathop{}^{}_{i=1}^{p}\;{{\mathfrak{A}}_{{{M}_{i}}}}$.

Both $\hat{C}\,$ and $\mathcal{C}$ are subsets of the power set, ${{2}^{\Xi}}$ of $\Xi$ but neither are sigma algebras: in general they are not closed under union or intersection. While $\hat{C}\,$ is a poset, ordered by inclusion, it is not a lattice. However, $\hat{C}\,$ is an orthocomplemented poset, meaning for each outcome $X\in \hat{C}\,$ there exists $\bar{X}\in \hat{C}\,$ such that: (i) $\overline{{\bar{X}}}=X$; (ii) $X\subset Y$ implies $\bar{Y}\subset \bar{X}$; (iii) $X\subset Y$ and $\bar{X}\subset Y$ implies $Y=\Xi$.

The NC space $\Xi$ together with the set $\hat{M}\,$ of all (doable) measurements specifies the joint measurement scenario $\mathcal{S}=\{\Xi,\hat{M}\,\}$. 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, $\hat{M}\,$, such as, if all pairs of measurements in a set of measurements $K\subset \mathcal{M}$ 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 5.1.

An experimental behaviour is a joint measurement scenario $(\Xi,\hat{M}\,)$ together with a probability measure $P\left( \cdot |M \right)$ on the sigma algebra ${{\mathfrak{A}}_{M}}$, for each $M\in \hat{M}\,$. The probabilities in an experimental behaviour are required to be consistent i.e, if an outcome $X\in \hat{C}\,$ is an element of ${{\mathfrak{A}}_{M}}$ and of ${{\mathfrak{A}}_{M\prime }}$ then $P\left( X|M \right)=P\left( X|M\prime \right)$. A behaviour is therefore equivalent to a consistent probability function $P\left( \cdot \right)$ whose domain is the set of outcomes $\hat{C}\,$. $P$ is not a probability measure because the set of outcomes $\hat{C}\,$ is not a sigma algebra, however it is a measure when restricted to any algebra of outcomes ${{\mathfrak{A}}_{M}}$ for a measurement $M$. $P\left( \cdot \right)$ 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, $M\prime$, $M$ 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 $(\Xi,\hat{M}\,,P)$ can be consistently extended in some way to a larger subset of ${{2}^{\Xi}}$ than $\hat{C}\,$.

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 $\mathcal{T}=\left\{ {{\mathcal{S}}_{1}},{{\mathcal{S}}_{1}},\ldots ,{{\mathcal{S}}_{n}} \right\}$ into one $\mathcal{S}\left( \mathcal{T} \right)=\left\{ \Xi_{\mathcal{T}},{{\hat{M}\,}_{\mathcal{T}}} \right\}$ 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

$$\begin{eqnarray}&&\Xi_{\mathcal{T}}=\Xi_{{\mathcal{S}}_{1}}\times \Xi_{{\mathcal{S}}_{2}}\times \ldots \times \Xi_{{\mathcal{S}}_{n}}\end{eqnarray} \tag{ 4 }$$

where $\Xi_{{\mathcal{S}}_{i}}$ is the NC space for scenario ${{\mathcal{S}}_{i}}$. For each outcome $X$ of a measurement $M$ in a scenario ${{\mathcal{S}}_{i}}\in \mathcal{T}$ there is a set ${{X}_{\mathcal{T}}}\in \Xi_{\mathcal{T}}$ defined as

$$\begin{eqnarray}&&{{X}_{\mathcal{T}}}=\Xi_{{\mathcal{S}}_{1}}\times \ldots \times \Xi_{{\mathcal{S}}_{i-1}}\times X\times \Xi_{{\mathcal{S}}_{i+1}}\times \ldots \times \Xi_{{\mathcal{S}}_{n}},\end{eqnarray} \tag{ 5 }$$

and the set

$$\begin{eqnarray}&&{{M}_{\mathcal{T}}}={{\left\{ {{X}_{\mathcal{T}}} \right\}}_{X\in M}}\end{eqnarray} \tag{ 6 }$$

is a new measurement partition ${{M}_{\mathcal{T}}}$ on $\Xi_{\mathcal{T}}$. The set of measurements ${{\hat{M}\,}_{\mathcal{T}}}$ contains all of these measurements for all members of $\mathcal{T}$. 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 $\mathcal{T}$ can be jointly performed, and there are no further measurements in ${{\hat{M}\,}_{\mathcal{T}}}$.

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 $\left( n,m,d \right)$. Each 'local' outcome for each wing ${{X}_{{{a}_{i}}{{x}_{i}}}}$ 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 $\left( {{a}_{i}}|{{x}_{i}} \right)$. Similarly, 'global' outcomes of a joint measurement on all systems are refereed to as $\left( {{a}_{1}},{{a}_{2}},\ldots ,{{a}_{n}}\;|\;{{x}_{1}},{{x}_{2}},\ldots ,{{x}_{n}} \right)$. The CHSH scenario, specified by $\left( 2,2,2 \right)$, 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.

3.1. The JPM condition, non-contextuality & Bell's 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, $\mathfrak{A}$, generated by $\hat{C}\,$ and define the set of behaviours consistent with the existence of such a measure. In our case, $\mathfrak{A}={{2}^{\Xi}}$ since $\Xi$ is finite.

Definition 1. (JPM) A behaviour $(\Xi,\hat{M}\,,P)$ is in the set JPM if there exists a joint probability measure ${{P}_{J}}$ on the sigma algebra ${{2}^{\Xi}}$ such that

$$\begin{eqnarray}&&P\left( X \right)={{P}_{J}}\left( X \right),\quad \forall X\in \hat{C}\,\;.\end{eqnarray} \tag{ 7 }$$

Since ${{2}^{\Xi}}$ is finite, ${{P}_{J}}$ is fixed by its values for the atoms of the algebra ${{2}^{\Xi}}$, i.e. the singleton sets corresponding to the elements of $\Xi$ itself: ${{P}_{J}}\left( \left\{ \gamma \right\} \right)$, $\gamma \in \Xi$. So, if a behaviour is in JPM then $P\left( X \right)={{\mathop{\sum }^{}}_{\gamma \in X}}{{P}_{J}}\left( \left\{ \gamma \right\} \right)$. 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' $\Xi$, 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, Bell's local causality and constraints on experimental behaviours inspires our proposals at the quantum level, where we seek analogues of these conditions.

3.2. 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 $X\in M$ identifies $X$ as a fine-grained outcome of the measurement $M$ ($X$ is a subset of $\Xi$ in the measurement partition $M$) while $X\in {{\mathfrak{A}}_{M}}$ refers to any outcome in the Boolean sigma algebra generated by the fine-grained outcomes.

Definition 2. (Q) A behaviour $(\Xi,\hat{M}\,,P)$ is in the set Q if there exists an ordinary quantum model for the behaviour, that is, a Hilbert space $\mathcal{H}$, a pure state $|\psi \rangle \in \mathcal{H}$, and a projection operator ${{E}^{X}}$ on $\mathcal{H}$ for each fine-grained outcome, $X$, of each basic measurement such that

• for each basic measurement ${{M}_{i}}\in \mathcal{M}$,
  $$\begin{eqnarray}&&\underset{X\in {{M}_{i}}}{\mathop{\mathop{\sum }^{}}}\,{{E}^{X}}=\mathbf{1}\end{eqnarray} \tag{ 8 }$$
  where $\mathbf{1}$ is the identity on $\mathcal{H}$;
• if basic measurements ${{M}_{i}},{{M}_{j}}\in \mathcal{M}$ can be jointly performed, then
  $$\begin{eqnarray}&&\left[ {{E}^{X}},{{E}^{Y}} \right]=0\quad \forall \;X\in {{M}_{i}},Y\in {{M}_{j}};\end{eqnarray} \tag{ 9 }$$
• If outcome $X$ of a joint measurement of the set of basic measurements $\left\{ {{M}_{{{i}_{1}}}},{{M}_{{{i}_{2}}}},\ldots {{M}_{{{i}_{n}}}} \right\}$ is $X=_{k=1}^{n}{{X}_{{{i}_{k}}}}$ where ${{X}_{{{i}_{k}}}}\in {{M}_{{{i}_{k}}}}$ then
  $$\begin{eqnarray}&&\langle \psi |{{E}^{{{X}_{{{i}_{1}}}}}}{{E}^{{{X}_{{{i}_{2}}}}}}\ldots {{E}^{{{X}_{{{i}_{n}}}}}}|\psi \rangle =P\left( X \right).\end{eqnarray} \tag{ 10 }$$

It follows that, for all basic measurements ${{M}_{i}}\in \mathcal{M}$, the projectors defining an ordinary quantum model satisfy:

$$\begin{eqnarray}&&{{E}^{X}}{{E}^{Y}}={{\delta }_{XY}}{{E}^{X}}\quad \forall \;X,Y\in {{M}_{i}}.\end{eqnarray} \tag{ 11 }$$

Since, from (8) we have that ${{\mathop{\sum }^{}}_{Y\in M}}{{E}^{X}}{{E}^{Y}}{{E}^{X}}={{E}^{X}}$ for all $X\in M$, from which it follows that ${{E}^{X}}{{E}^{Y}}{{E}^{X}}=0$ for all $X\ne Y$, because all these terms are positive operators.

Lemma 3. A behaviour $(\Xi,\hat{M}\,,P)$ is in the set Q if and only if there exists a Hilbert space $\mathcal{H}$, a pure state $|\psi \rangle \in \mathcal{H}$, and a map $E:\hat{C}\,\to L$, from the set of outcomes to the lattice, $L$, of projection operators on $\mathcal{H}$ (equivalently, the lattice of subspaces of $\mathcal{H}$) such that

• for each measurement $M\in \hat{M}\,$, the image of $E$ restricted to ${{\mathfrak{A}}_{M}}$, $E\left( {{\mathfrak{A}}_{M}} \right)$, is a Boolean algebra and the restriction of $E$,
  $$\begin{eqnarray*}&&E:\;{{\mathfrak{A}}_{M}}\to E({{\mathfrak{A}}_{M}}),\end{eqnarray*}$$
  is a Boolean algebra homomorphism ;
• the probabilities are given by
  $$\begin{eqnarray}&&\langle \psi |E\left( X \right)|\psi \rangle =P\left( X \right),\ \ \forall X\in \hat{C}\,\;.\end{eqnarray} \tag{ 12 }$$

Proof. If the behaviour is in Q then we have projection operators for each of the fine-grained outcomes, ${{X}_{{{i}_{k}}}}$, of each basic measurement so define $E\left( {{X}_{{{i}_{k}}}} \right):={{E}^{{{X}_{{{i}_{k}}}}}}$. 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 $\left\{ {{M}_{{{i}_{1}}}},{{M}_{{{i}_{2}}}},\ldots {{M}_{{{i}_{n}}}} \right\}$ is $X=_{k=1}^{n}{{X}_{{{i}_{k}}}}$ where ${{X}_{{{i}_{k}}}}\in {{M}_{{{i}_{k}}}}$,

$$\begin{eqnarray}&&E\left( X \right):={{E}^{{{X}_{{{i}_{1}}}}}}{{E}^{{{X}_{{{i}_{2}}}}}}\ldots {{E}^{{{X}_{{{i}_{n}}}}}}\;,\end{eqnarray} \tag{ 13 }$$

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, $Y\in {{\mathfrak{A}}_{M}}$, 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 $E:\hat{C}\,\to L$ 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 ('Tsirelson's 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?

3.3. NPA conditions

4.1. Quantum measure theory

4.2. 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 $\Xi$ 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 $\mathfrak{A}$ generated by $\hat{C}\,$. 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:

Definition 7. (JQM) A behaviour $(\Xi,\hat{M}\,,P)$ is in the set JQM if there exists a joint quantum measure, ${{\mu }_{J}}$ on sigma algebra ${{2}^{\Xi}}$ such that

$$\begin{eqnarray}&&{{\mu }_{J}}\left( X \right)=P\left( X \right),\quad \forall \ X\in \hat{C}\,\quad \text{(Experimental probabilities)}.\end{eqnarray} \tag{ 20 }$$

This condition says that the marginal quantum measure—${{\mu }_{J}}$ restricted to the subalgebra ${{\mathfrak{A}}_{M}}$—for each measurement $M$, coincides with the behaviour's probability measure $P\left( \cdot |M \right)$ on ${{\mathfrak{A}}_{M}}$. JQM is a quantum non-contextuality condition.

We can re-express the JQM condition in terms of decoherence functionals:

Lemma 8. A behaviour $(\Xi,\hat{M}\,,P)$ is in the set JQM if and only if there exists a joint decoherence functional, ${{D}_{J}}$ over ${{2}^{\Xi}}$ such that

• ${{D}_{J}}\left( X,X \right)=P\left( X \right),\quad \forall \ X\in \hat{C}\,$ (Experimental probabilities);
• ${{D}_{J}}\left( X,Y \right)=0,\quad \forall \ X,Y\in {{\mathfrak{A}}_{M}},\ XY=\varnothing ,\quad \forall \ M\in \hat{M}\,$ (Decoherence of alternative experimental outcomes).

Proof. Suppose a behaviour is in JQM so there exists a quantum measure ${{\mu }_{J}}:{{2}^{\Xi}}\to \mathbb{R}$ such that ${{\mu }_{J}}\left( X \right)=P\left( X \right)$ for all $X\in \hat{C}\,$. Then, there exists a real decoherence functional ${{D}_{J}}$ such that ${{D}_{J}}\left( X,X \right)={{\mu }_{J}}\left( X \right)$ for all $X\in {{2}^{\Xi}}$ [29]. Any decoherence functional satisfies

$$\begin{eqnarray}&&2\operatorname{Re}\left[ D\left( X,Y \right) \right]=D\left( XY,XY \right)-D\left( X,X \right)-D\left( Y,Y \right),\end{eqnarray} \tag{ 21 }$$

for any two disjoint events, $X$ and $Y$. For any measurement, $M$ and $X,Y\in {{\mathfrak{A}}_{M}},\ X\ne Y$ we also have

$$\begin{eqnarray}&&{{D}_{J}}\left( XY,XY \right)=P\left( XY \right),\quad {{D}_{J}}\left( X,X \right)=P\left( X \right)\quad \text{and}\quad {{D}_{J}}\left( Y,Y \right)=P\left( Y \right).\end{eqnarray} \tag{ 22 }$$

Since $D$ is real, the decoherence condition ${{D}_{J}}\left( X,Y \right)=0$ follows. Note that the two conditions together are equivalent to ${{D}_{J}}\left( X,Y \right)=P\left( XY \right),\quad \forall \ X,Y\in {{\mathfrak{A}}_{M}},\quad \forall \ M\in \hat{M}\,$.

The converse is immediate on defining ${{\mu }_{J}}\left( X \right)={{D}_{J}}\left( X,X \right)$ for all $X\in {{2}^{\Xi}}$.□

An obvious strengthening of the JQM condition is:

Definition 9. (SPJQM) A behaviour $\left( \Xi,\hat{M}\,,P \right)$ is in the set SPJQM if there exists a joint decoherence functional, ${{D}_{J}}$ over ${{2}^{\Xi}}$ such that

• ${{D}_{J}}\left( X,Y \right)=P\left( XY \right),\quad \forall \ X,Y\in {{\mathfrak{A}}_{M}},\quad \forall \ M\in \hat{M}\,$;
• ${{D}_{J}}$ 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 $\mathcal{H}$. Using this, we can present an equivalent definition of the SPJQM set which makes manifest its connection to the NPA heirarchy of conditions:

Lemma 10. A behaviour $\left( \Xi,\hat{M}\,,P \right)$ is in SPJQM if and only if there exists a Hilbert space $\mathcal{H}$ spanned by a set of vectors indexed by elements of the sigma algebra ${{2}^{\Xi}}$, ${{\left\{ \left| X \right\rangle \right\}}_{X\in {{2}^{\Xi}}}}$, such that their inner products satisfy the following conditions:

• for any $X\in {{2}^{\Xi}}$,
  $$\begin{eqnarray}&&|X\rangle =\underset{\gamma \in X}{\mathop{\mathop{\sum }^{}}}\,\left| \left\{ \gamma \right\} \right\rangle;\end{eqnarray} \tag{ 23 }$$
• for each measurement $M\in \hat{M}\,$,
  $$\begin{eqnarray}&&\langle X|Y\rangle =P\left( XY \right)\quad \forall \;X,Y\in {{\mathfrak{A}}_{M}}.\end{eqnarray} \tag{ 24 }$$
  In particular, if two outcomes, $X$ and $Y$, are disjoint then $\langle X|Y\rangle =0$: 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 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.

Lemma 11. Every behaviour for the CHSH (Bell (2,2,2)) scenario is in JQM.

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.

Lemma 13. If a behaviour for the CHSH scenario is in SPJQM, then the experimental probabilities satisfy the Tsirelson–Landau–Masanes bound [4, 56, 57].

The proofs of these two lemmas are in [35, 37].

Lemma 14.  $Q\subseteq SPJQM$

Proof. Suppose a behaviour is in Q. Then projection operators $\left\{ {{E}^{X}} \right\}$ indexed by the fine grained outcomes of basic measurements and satisfying the conditions of definition 2 exist. We define a vector $\left| \left\{ \gamma \right\} \right\rangle$ for each atom $\left\{ \gamma \right\}$ of ${{2}^{\Xi}}$:

$$\begin{eqnarray}&&\left| \left\{ \gamma \right\} \right\rangle:={{E}^{{{X}_{1}}}}...{{E}^{{{X}_{p}}}}\left| \psi \right\rangle\end{eqnarray} \tag{ 25 }$$

where $\left\{ \gamma \right\}={{X}_{1}}...{{X}_{p}}$ and ${{X}_{i}}\in {{M}_{i}}$. We define vectors for each $X\in \hat{C}\,$ by addition:

$$\begin{eqnarray}&&\left| X \right\rangle:=\underset{\gamma \in X}{\mathop{\mathop{\sum }^{}}}\,\left| \left\{ \gamma \right\} \right\rangle.\end{eqnarray} \tag{ 26 }$$

These vectors satisfy the conditions of lemma 10.□

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.

Theorem 15. SPJQM $\subseteq \;{{Q}^{1}}$.

Proof. Given a behaviour in SPJQM, there exists a set of vectors obeying the conditions (23)–(24). To show that there exists a set of vectors with the properties (14) and (15) from definition 4 of ${{Q}^{1}}$ is an easy task: the SPJQM vectors range over all subsets of $\Xi$, 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 $\hat{M}\,$ and not just the basic ones, one cannot use the same reasoning as above to prove that SPJQM $\subseteq {{Q}^{1+AB}}$. This is because the relations (18) in the definition of ${{Q}^{1+AB}}$ are not implied by the conditions in the SPJQM definition. The significance of these extra orthogonality relations will be expanded on later.

Theorem 16.  ${{Q}^{1+AB}}\;\subseteq$ SPJQM.

Miguel Navascues found this result for bipartite non-locality scenarios (private communication) using a new reformulation of ${{Q}^{1+AB}}$ [38]. The proof presented here for all joint measurement scenarios is based on his idea.

Proof. Consider a behaviour in ${{Q}^{1+AB}}$. By definition 5 there exists of a set of vectors ${{\left\{ |X\rangle \right\}}_{X\in \hat{C}\,}}$ associated to the outcomes of coarse-grained measurements, obeying the properties (16)–(18). From these vectors we will construct a set of vectors ${{\left\{ |X\rangle \right\}}_{X\in {{2}^{\Xi}}}}$ indexed by all subsets of the NC space, that obeys (23) and contains the original set ${{\left\{ |X\rangle \right\}}_{X\in \hat{C}\,}}$ as its restriction to $\hat{C}\,$. With this established, lemma 10 shows that the behaviour is in SPJQM.

We first construct vectors $|\left\{ \gamma \right\}\rangle$ for all singleton sets of histories, $\gamma \in \Xi$. For each basic measurement ${{M}_{i}}\in \mathcal{M}$, we select, arbitrarily, one outcome ${{X}^{*}}\in {{M}_{i}}$. For every other outcome $X\in {{M}_{i}}$, $X\ne {{X}^{*}}$ we define the projection operator,

$$\begin{eqnarray}&&{{E}^{X}}=\text{proj}\left[ {{\mathcal{H}}^{X}} \right],\end{eqnarray} \tag{ 27 }$$

where proj $\left[ \cdot \right]$ projects onto the given subspace, and

$$\begin{eqnarray}&&{{\mathcal{H}}^{X}}=\text{span}\left\{ \left| X\prime \right\rangle:X\prime \in \hat{C}\,\;\text{s}\text{.t}\text{.}\;X\prime \subset X \right\}.\end{eqnarray} \tag{ 28 }$$

That is, ${{\mathcal{H}}^{X}}$ is the subspace spanned by the vectors for all measurement outcomes that imply the outcome $X$ for basic measurement ${{M}_{i}}$. The orthogonality relation (18) implies that ${{\mathcal{H}}^{X}}$ and ${{\mathcal{H}}^{Y}}$ are orthogonal subspaces for $X\ne Y$ and so we have ${{E}^{X}}{{E}^{Y}}={{\delta }_{X,Y}}{{E}^{X}}$ for all outcomes $X,Y\in {{M}_{i}}$, such that $X,Y\ne {{X}^{*}}$. We then define ${{E}^{{{X}^{*}}}}$, for the remaining outcome ${{X}^{*}}$ of ${{M}_{i}}$, by

$$\begin{eqnarray}&&\underset{X\in {{M}_{i}}}{\mathop{\mathop{\sum }^{}}}\,{{E}^{X}}=\mathbf{1},\end{eqnarray} \tag{ 29 }$$

where $\mathbf{1}$ is the identity operator on the Hilbert space $\mathcal{H}$. Since the other ${{E}^{X}}$'s are orthogonal projectors, so is ${{E}^{{{X}^{*}}}}$. ${{E}^{{{X}^{*}}}}$ projects onto the orthogonal complement, ${{\mathcal{H}}^{{{X}^{*}}}}$, of the space spanned by all the subspaces ${{H}^{X}}$ with $X\in {{M}_{i}}$ such that $X\ne {{X}^{*}}$. We have, therefore,

$$\begin{eqnarray}&&{{E}^{X}}{{E}^{Y}}={{\delta }_{XY}}{{E}^{X}},\quad \forall X\in {{M}_{i}}.\end{eqnarray} \tag{ 30 }$$

Thus, the set of operators ${{\left\{ {{E}^{X}} \right\}}_{X\in {{M}_{i}}}}$, 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 $\gamma \in \Xi$ as

$$\begin{eqnarray}&&\left| \left\{ \gamma \right\} \right\rangle:={{E}^{{{X}_{1}}}}{{E}^{{{X}_{2}}}}\ldots {{E}^{{{X}_{p}}}}\left| \Xi \right\rangle,\end{eqnarray} \tag{ 31 }$$

where ${{X}_{i}}$ is the outcome for ${{M}_{i}}$ such that $\left\{ \gamma \right\}={{X}_{1}}{{X}_{2}}...{{X}_{p}}$.

We now show that the original ${{Q}^{1+AB}}$ vectors are sums of the vectors $|\left\{ \gamma \right\}\rangle$. Let $M\in \hat{M}\,$ be a joint measurement of the set of $m$ basic measurements $\left\{ {{M}_{{{i}_{1}}}},{{M}_{{{i}_{2}}}},...{{M}_{{{i}_{m}}}} \right\}$. Let $X\in M$ be a fine-grained outcome of $M$, $X={{X}_{{{i}_{1}}}}{{X}_{{{i}_{2}}}}...{{X}_{{{i}_{m}}}}$ where ${{X}_{{{i}_{k}}}}\in {{M}_{{{i}_{k}}}}$. By using (29) for all basic measurements not in $M$, we obtain,

$$\begin{eqnarray}&&\underset{\gamma \in X}{\mathop{\mathop{\sum }^{}}}\,\left| \{\gamma \} \right\rangle={{E}^{{{X}_{{{i}_{1}}}}}}{{E}^{{{X}_{{{i}_{2}}}}}}\ldots {{E}^{{{X}_{{{i}_{m}}}}}}|\Xi\rangle \end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&=\;{{E}^{{{X}_{{{i}_{1}}}}}}{{E}^{{{X}_{{{i}_{2}}}}}}\ldots {{E}^{{{X}_{{{i}_{m}}}}}}\underset{X\prime \in M}{\mathop{\mathop{\sum }^{}}}\,|X\prime \rangle ,\end{eqnarray} \tag{ 33 }$$

where the second line is an application of (16). Note that $X$ is one of the fine-grained outcomes of $M$ being summed over here. Each ${{E}^{{{X}_{{{i}_{k}}}}}}$ either leaves $|X\prime \rangle$ invariant or annihilates it depending on whether or not $X\prime \subset {{X}_{{{i}_{k}}}}$. Of all the outcomes of $M$ only $X$ itself is left invariant by all the projectors and so

$$\begin{eqnarray}&&\underset{\gamma \in X}{\mathop{\mathop{\sum }^{}}}\,\left| \gamma \right\rangle=\left| X \right\rangle,\quad \forall X\in M.\end{eqnarray} \tag{ 34 }$$

From (16) we can see that this holds for all coarse-grained outcomes,

$$\begin{eqnarray}&&\underset{\gamma \in X}{\mathop{\mathop{\sum }^{}}}\,\left| \gamma \right\rangle=\left| X \right\rangle,\quad \forall X\in {{\mathfrak{A}}_{M}},\end{eqnarray} \tag{ 35 }$$

and, since we chose an arbitrary measurement $M$ to consider, this holds for all outcomes $X\in \hat{C}\,$. Therefore we have

$$\begin{eqnarray}&&\left| X \right\rangle=\underset{\gamma \in X}{\mathop{\mathop{\sum }^{}}}\,\left| \left\{ \gamma \right\} \right\rangle,\quad \forall X\in {{2}^{\Xi}},\end{eqnarray} \tag{ 36 }$$

where this serves as the definition of $|X\rangle$ for each $X\in {{2}^{\Xi}}\hat{C}\,$.

This set of vectors ${{\left\{ |X\rangle \right\}}_{X\in {{2}^{\Xi}}}}$ satisfies (23) by construction. Because it restricts to the original ${{Q}^{1+AB}}$ vectors ${{\left\{ |X\rangle \right\}}_{X\in \hat{C}\,}}$, condition (17) in the definition of ${{Q}^{1+AB}}$ ensures that it also obeys (24). Lemma 10 states that if there exists a set of vectors ${{\left\{ |X\rangle \right\}}_{X\in {{2}^{\Xi}}}}$ satisfying these two conditions, the behaviour is in SPJQM.□

Note that the order in which the projectors are applied to the state $|\Xi\rangle$ in (31) was chosen in the proof to be the reverse of the order of the labelling of the basic measurements, $1...p$, but any other order would have worked also. Thus there is no unique set of vectors $\left\{ \left| \left\{ \gamma \right\} \right\rangle \right\}$ satisfying the conditions.

5.1. 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 $M\prime$ 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 non-contextuality 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, $X=\left( a=+1,b=+1;x=0,y=0 \right)$ and $Y=\left( a=-1,b=+1;x=0,y=1 \right)$. These are disjoint in the sense that $XY=\varnothing$ but there is no measurement partition in $\hat{M}\,$ that contains them both. Thus, a joint decoherence functional, ${{D}_{J}}$, for a behaviour for this scenario, as defined in definition 8 or 9, need not satisfy ${{D}_{J}}\left( X,Y \right)=0$: 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 outcomes—one 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 ${{D}_{J}}\left( X,Y \right)=0$, 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 $(\Xi,\hat{M}\,)$, 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) measurement ${{M}^{1}}\in \hat{M}\,$ and, depending on its outcome ${{X}^{1}}$, a second measurement ${{M}^{2}}({{X}^{1}})$ from the set of all those measurements in $\hat{M}\,$ that include ${{M}^{1}}$. This defines a new measurement partition of $\Xi$, ${{M}_{b}}({{M}^{1}},{{M}^{2}}({{X}^{1}}))$. Explicitly, ${{M}_{b}}$ contains all ${{X}^{2}}\in {{M}^{2}}\left( {{X}^{1}} \right)$ such that ${{X}^{2}}\subset {{X}^{1}}$, for all ${{X}^{1}}\in {{M}^{1}}$. We denote the set of all such single branching measurements by ${{\hat{M}\,}_{b}}$. Note that ${{\hat{M}\,}_{b}}$ is uniquely determined given $\hat{M}\,$. Also $\hat{M}\,\subseteq {{\hat{M}\,}_{b}}$ since each measurement, $M\in \hat{M}\,$, is (trivially) a single branching measurement where ${{M}^{1}}=M$ and ${{M}^{2}}\left( {{X}^{1}} \right)=M$ for all ${{X}^{1}}$. ⁸

For each ${{M}_{b}}\in {{\hat{M}\,}_{b}}$, the sigma algebra ${{\mathfrak{A}}_{{{M}_{b}}}}$ is that generated by the fine-grained outcomes in the partition ${{M}_{b}}$. The set of all outcomes for all measurements in ${{\hat{M}\,}_{b}}$ will be denoted ${{\hat{C}\,}_{b}}\;:=\;{{\mathop{}^{}}_{M\in {{\hat{M}\,}_{b}}}}\;{{\mathfrak{A}}_{M}}$.

Only one consequence of these new definitions, expressed in the following lemma, will be salient to the discussion below.

Lemma 17. If $X,Y\in \hat{C}\,$ and there exists a basic measurement ${{M}_{i}}$ and outcomes $X\prime ,Y\prime \in {{M}_{i}}$ such that $X\prime Y\prime =\varnothing$, $X\subset X\prime$ and $Y\subset Y\prime$ then there exists a single branching measurement ${{M}^{*}}\in {{\hat{M}\,}_{b}}$ such that $X,Y\in {{\mathfrak{A}}_{{{M}^{*}}}}$.

Proof. If the premise holds, there exist two measurements ${{M}_{X}},{{M}_{Y}}\in \hat{M}\,$ of which $X$ and $Y$ are, respectively, outcomes. Both ${{M}_{X}}$ and ${{M}_{Y}}$ must include basic measurement ${{M}_{i}}$. Consider the branching measurement ${{M}^{*}}$ in which ${{M}^{1}}={{M}_{i}}$ and ${{M}^{2}}\left( X\prime \right)={{M}_{X}}$, ${{M}^{2}}\left( Y\prime \right)={{M}_{Y}}$ and ${{M}^{2}}\left( Z \right)={{M}_{i}}$ for $Z\in {{M}_{i}}$, $Z\ne X\prime$ and $Z\ne Y\prime$. Then $X,Y\in {{\mathfrak{A}}_{{{M}^{*}}}}$.□

We have extended the joint measurement scenario $\left( \Xi,\hat{M}\, \right)$ to $\left( \Xi,{{\hat{M}\,}_{b}} \right)$. Given a behaviour $\left( \Xi,\hat{M}\,,P \right)$ we also extend the probability function $P$ to cover the larger set of outcomes ${{\hat{C}\,}_{b}}$. Note here that each fine-grained outcome for a branching measurement is a fine-grained outcome for some joint measurement in $\hat{M}\,$, so the only elements in ${{\hat{C}\,}_{b}}$ and not in $\hat{C}\,$ are coarse-grained outcomes of the new measurements. This means that the probability function, $P$, on $\hat{C}\,$ can be uniquely extended to a probability function on ${{\hat{C}\,}_{b}}$, by applying the Kolmogorov sum rule. We call the behaviour thus extended, $\left( \Xi,{{\hat{M}\,}_{b}},P \right)$, the branching extension of $\left( \Xi,\hat{M}\,,P \right)$.

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 $\hat{M}\,$ with ${{\hat{M}\,}_{b}}$ and $\hat{C}\,$ with ${{\hat{C}\,}_{b}}$, 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) replaced with

$$\begin{eqnarray}&&P\left( X \right)={{P}_{J}}\left( X \right),\quad \forall X\in {{\hat{C}\,}_{b}}\;,\end{eqnarray} \tag{ 37 }$$

whereas before ${{P}_{J}}$ is a probability measure over the sigma algebra ${{2}^{\Xi}}$. Obviously this implies equation (7) because $\hat{C}\,\subset {{\hat{C}\,}_{b}}$. It is also implied by equation (7), because as just discussed, for sets $X\in {{\hat{C}\,}_{b}}$ which are not in $\hat{C}\,$, the value of P(X) is determined by the probabilities of sets in $\hat{C}\,$ by the Kolmogorov rule, as is the value of ${{P}_{J}}\left( X \right)$.

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 ${{Q}^{1+AB}}$.

Lemma 18. A behaviour $\left( \Xi,\hat{M}\,,P \right)$, whose branching extension is $\left( \Xi,{{\hat{M}\,}_{b}},P \right)$, is in ${{Q}^{1+AB}}$ if and only if there exists a Hilbert space $\mathcal{H}$ and a set of vectors ${{\left\{ |X\rangle \right\}}_{X\in {{\hat{C}\,}_{b}}}}$ indexed by all the coarse-grained outcomes of all branching measurements, that span $\mathcal{H}$ and satisfy the following conditions:

• (i)  
  if ${{M}_{b}}\in {{\hat{M}\,}_{b}}$ and $X,Y\in {{\mathfrak{A}}_{{{M}_{b}}}}$ are disjoint outcomes, $XY=\varnothing$, then

  $$\begin{eqnarray}&&\left| XY \right\rangle=\left| X \right\rangle+\left| Y \right\rangle;\end{eqnarray} \tag{ 38 }$$
• (ii)  
  for each branching measurement ${{M}_{b}}\in {{\hat{M}\,}_{b}}$ and all outcomes $X,Y\in {{\mathfrak{A}}_{{{M}_{b}}}}$

  $$\begin{eqnarray}&&\langle X|Y\rangle =P\left( XY \right).\end{eqnarray} \tag{ 39 }$$

Proof. Suppose a behaviour is in ${{Q}^{1+AB}}$. Construct vectors ${{\left\{ |X\rangle \right\}}_{X\in {{2}^{\Xi}}}}$ as in the proof of theorem 16. These vectors satisfy (38) when $XY=\varnothing$. If $X$ and $Y$ are fine-grained outcomes of ${{M}_{b}}$ then either they disagree on the outcome of the first measurement ${{M}^{1}}$ in the branching measurement in which case their corresponding vectors are orthogonal, or they agree on that first outcome in which case they are outcomes of the same measurement ${{M}^{2}}\in \hat{M}\,$. Therefore (39) holds for fine-grained outcomes of ${{M}_{b}}$. The vector for a coarse grained outcome is the sum of the vectors for the fine-grained outcomes of which it is the union and and so (39) also holds for coarse-grained outcomes of ${{M}_{b}}$.

Conversely, if vectors ${{\left\{ |X\rangle \right\}}_{X\in {{\hat{C}\,}_{b}}}}$ exist satisfying (38) and (39) then those for $X\in \hat{C}\,$ satisfy conditions (i) and (ii) in definition 5. It remains to show that these vectors satisfy (iii) in definition 5. Suppose $X,Y\in \hat{C}\,$ and there exists a basic measurement ${{M}_{i}}$ and outcomes $X\prime ,Y\prime \in {{M}_{i}}$ such that $X\prime Y\prime =\varnothing$, $X\subset X\prime$ and $Y\subset Y\prime$. Then lemma 17 implies there is a branching measurement ${{M}^{*}}\in {{\hat{M}\,}_{b}}$ such that $X,Y\in {{\mathfrak{A}}_{{{M}^{*}}}}$. $X$ and $Y$ are disjoint and so by (39) we have

$$\begin{eqnarray}&&\langle X|Y\rangle =0.\end{eqnarray} \tag{ 40 }$$

□

Thus, the conditions JPM, Q, ${{Q}^{1}}$ and ${{Q}^{1+AB}}$, although originally expressed in terms of the joint measurements $\hat{M}\,$, also contain the information necessary to incorporate branching measurements in a non-contextual manner: once you have a JPM—or Q, ${{Q}^{1}}$ or ${{Q}^{1+AB}}$—model for the non-branching behaviour there is a unique way to extend it to a model of the behaviour extended by adding all branching measurements.

In contrast, however, the corresponding alteration of the set SPJQM is not obviously the same.

Definition 19. (${\text{SPJQM}_{b}}$) A behaviour $\left( \Xi,\hat{M}\,,P \right)$, whose branching extension is $\left( \Xi,{{\hat{M}\,}_{b}},P \right)$, is in the set $SPJQ{{M}_{b}}$ if there exists a joint decoherence functional, ${{D}_{J}}$ over ${{2}^{\Xi}}$ such that

• (i)  
  ${{D}_{J}}\left( X,Y \right)=P\left( XY \right),\quad \forall \ X,Y\in {{\mathfrak{A}}_{{{M}_{b}}}},\quad \forall \ {{M}_{b}}\in {{\hat{M}\,}_{b}}$;
• (ii)  
  ${{D}_{J}}$ is strongly positive (SP).

$SPJQ{{M}_{b}}\subseteq SPJQM$, but as mentioned above there is reason to think that the two are not equivalent. There is a modified version of lemma 10.

Lemma 20. A behaviour $\left( \Xi,\hat{M}\,,P \right)$, whose branching extension is $\left( \Xi,{{\hat{M}\,}_{b}},P \right)$, is in $SPJQ{{M}_{b}}$ if and only if there exists a Hilbert space $\mathcal{H}$ spanned by a set of vectors indexed by elements of the sigma algebra ${{2}^{\Xi}}$, ${{\left\{ |X\rangle \right\}}_{X\in {{2}^{\Xi}}}}$, such that their inner products satisfy the following conditions:

• (i)  
  For any $X\in {{2}^{\Xi}}$,

  $$\begin{eqnarray}&&|X\rangle =\underset{\gamma \in X}{\mathop{\mathop{\sum }^{}}}\,|\left\{ \gamma \right\}\rangle ;\end{eqnarray} \tag{ 41 }$$
• (ii)  
  for each measurement ${{M}_{b}}\in {{\hat{M}\,}_{b}}$,

  $$\begin{eqnarray}&&\langle X|Y\rangle =P\left( XY \right)\quad \forall \;X,Y\in {{\mathfrak{A}}_{{{M}_{b}}}}.\end{eqnarray} \tag{ 42 }$$

  In particular, if two outcomes $X$ and $Y$ of a measurement are disjoint then $\langle X|Y\rangle =0$: disjoint outcomes decohere.

Theorem 21.  SPJQM $_{b}$ $={{Q}^{1+AB}}$.

Proof. Lemmas 20 and 18 imply that $SPJQ{{M}_{b}}\subseteq {{Q}^{1+AB}}$ and ${{Q}^{1+AB}}\subseteq SPJQ{{M}_{b}}$ can be proved following the proof of theorem 16.□

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 non-contextuality 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 ${{Q}^{1+AB}}$. Thus, if ${{Q}^{1+AB}}$ is strictly larger than Q, as indicated by the computational evidence, then non-local 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 $SPJQ{{M}_{b}}$, is equal to the NPA set ${{Q}^{1+AB}}$. The necessary change is to require that for any joint measurement behaviour allowed by the physics, its branching extension is also physical. Including branching measurements, 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 $SPJQ{{M}_{b}}$ is therefore 'closed under wiring' meaning that any behaviour created by applying 'classical operations'—coarse-graining measurement choices and/or outputs, post-selection, 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 $SPJQ{{M}_{b}}$ 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 $SPJQ{{M}_{b}}$. 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: $JPMJP{{M}_{b}}$. 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 $SPJQ{{M}_{b}}$ 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 $SPJQ{{M}_{b}}$ 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 $JQ{{M}_{b}}$ 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.

The authors thank Miguel Navascues for his comments on these ideas, especially those which led to theorem 16. Thanks are also due to Rafael Sorkin for detailed discussions of the work, and to Toni Acin, Adan Cabello, Belen Sainz and Tobias Fritz for helpful discussion on contextuality scenarios. Thanks also to Alex Wilce and Rob Spekkens for comments on earlier versions of the work. The authors are grateful to Perimeter Institute for Theoretical Physics, Waterloo Canada and to the Institute for Quantum Computing, University of Waterloo, Ontario, Canada for hospitality at several stages of this research. Research at Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. FD and PW are supported in part by COST Action MP1006. PW is supported by EPSRC grant No. EP/K022717/1. JH is supported by a grant from the John Templeton Foundation.

Here we demonstrate the equivalence between the original definition of the sets of behaviours ${{Q}^{1}}$ and ${{Q}^{1+AB}}$ given by NPA and the definitions in the main text.

Firstly we restate the original NPA form of the ${{Q}^{1}}$ condition adapting it to our joint measurement scenarios. For Bell scenarios, the first three conditions below are equivalent to equation (19) in [8] and the remaining two conditions to (20) in [8] for $n=1$. NPA's 'null string' $1$ is more naturally referred to as $\Xi$ in our terminology.

Definition 22. A behaviour $\left( \Xi,\hat{M}\,,P \right)$ is in $Q_{\text{NPA}}^{1}$ if there exists a positive semi-definite matrix ${{\Gamma }^{1}}$ indexed by the set of all fine-grained outcomes $X\in {{M}_{i}}$ of all basic measurements ${{M}_{i}}\in \mathcal{M}$, and by $\Xi$, such that:

• $$\begin{eqnarray}&&\Gamma _{\Xi,\Xi}^{1}=1;\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ \ \end{eqnarray} \tag{ A.1 }$$
• if ${{M}_{i}}$ is a basic measurement and $X\in {{M}_{i}}$ is a fine-grained outcome,
  $$\begin{eqnarray}&&\Gamma _{\Xi,X}^{1}=P\left( X \right);\end{eqnarray} \tag{ A.2 }$$
• if basic measurements ${{M}_{i}},{{M}_{j}}\in \mathcal{M}$ are jointly performable and ${{M}_{i}}\ne {{M}_{j}}$, and $X\in {{M}_{i}}$ and $Y\in {{M}_{j}}$ are fine-grained outcomes, then
  $$\begin{eqnarray}&&\Gamma _{X,Y}^{1}=P\left( XY \right);\end{eqnarray} \tag{ A.3 }$$
• if ${{M}_{i}}$ is a basic measurement and $X\in {{M}_{i}}$ is a fine-grained outcome,
  $$\begin{eqnarray}&&\Gamma _{\Xi,X}^{1}=\Gamma _{X,X}^{1};\end{eqnarray} \tag{ A.4 }$$
• if ${{M}_{i}}$ is a basic measurement and $X,Y\in {{M}_{i}}$ are fine-grained outcomes,
  $$\begin{eqnarray}&&\Gamma _{X,Y}^{1}=0\;\text{if}\;X\ne Y.\end{eqnarray} \tag{ A.5 }$$

The translation to the terminology of NPA is fairly direct. For example, for two fine-grained outcomes $X$ and $Y$ taken from jointly performable basic measurements, $P\left( XY \right)$ corresponds to $P\left( a,b \right)$, 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:

Lemma 23. A behaviour $\left( \Xi,\hat{M}\,,P \right)$ is in $Q_{\text{NPA}}^{1}$ if and only if there exists a positive semi-definite matrix ${{\Gamma }^{1}}$ indexed by the set of all fine-grained outcomes $X\in {{M}_{i}}$ of all basic measurements ${{M}_{i}}\in \mathcal{M}$, and by $\Xi$, such that:

• For all basic measurements ${{M}_{i}}\in \mathcal{M}$ and for all $X\in {{M}_{i}}\left\{ \Xi \right\}$,
  $$\begin{eqnarray}&&\Gamma _{X,\Xi}^{1}=\underset{Y\in {{M}_{i}}}{\mathop{\mathop{\sum }^{}}}\,\Gamma _{X,Y}^{1};\end{eqnarray} \tag{ A.6 }$$
• if basic measurements ${{M}_{i}},{{M}_{j}}\in \mathcal{M}$ are jointly performable and $X\in {{M}_{i}}$ and $Y\in {{M}_{j}}$ are fine-grained outcomes, then
  $$\begin{eqnarray}&&\quad \Gamma _{X,Y}^{1}=P\left( XY \right).\end{eqnarray} \tag{ A.7 }$$

Proof. First we show that (A.1)–(A.5) imply (A.6, A.7). Equation (A.7) is the same as (A.3) in the case that ${{M}_{i}}\ne {{M}_{j}}$. When ${{M}_{i}}={{M}_{j}}$ and $X\ne Y$ (A.7) follows from (A.5) (in this case $XY=\varnothing$), and for $X=Y$ (A.7) follows from (A.4) and (A.2). To obtain (A.7) when $X=\Xi$, we apply (A.1) and then (A.2):

$$\begin{eqnarray}&&\Gamma _{\Xi,\Xi}^{1}=1=\underset{Y\in M}{\mathop{\mathop{\sum }^{}}}\,P\left( Y \right)=\underset{Y\in M}{\mathop{\mathop{\sum }^{}}}\,\Gamma _{\Xi,Y}^{1}\end{eqnarray} \tag{ A.8 }$$

for all basic measurements $M\in \mathcal{M}$. Otherwise $X$ is some outcome of the basic measurement ${{M}_{i}}$, and we apply (A.2) and then (A.7):

$$\begin{eqnarray}&&\Gamma _{X,\Xi}^{1}=P\left( X \right)=\underset{Y\in {{M}_{i}}}{\mathop{\mathop{\sum }^{}}}\,P\left( XY \right)=\underset{Y\in {{M}_{i}}}{\mathop{\mathop{\sum }^{}}}\,\Gamma _{X,Y}^{1}.\end{eqnarray} \tag{ A.9 }$$

Now we show the converse, that (A.6, A.7) imply (A.1)–(A.5). Equation (A.7) implies (A.3) and (A.5). Then (A.5) and (A.6) together imply (A.4). Then (A.4) and (A.7) together imply (A.2). Finally (A.2) and (A.6) together imply (A.1).□

It is an elementary result that a matrix is positive semidefinite (has no negative eigenvalues) if and only if it is the inner product matrix (or 'Gram matrix') of some set of vectors in a Hilbert space, and we rely on this to prove the following result.

Lemma 24.  ${{Q}^{1}}=Q_{\text{NPA}}^{1}$.

Proof. Consider a behaviour satisfying the conditions of definition 4 and define $\Gamma _{X,Y}^{1}:=\langle X|Y\rangle$ for $X,Y$ fine-grained outcomes of basic measurements or $\Xi$. Then ${{\Gamma }^{1}}$ is positive semidefinite, and equations (A.6, A.7) are implied by (14, 15) respectively.

Conversely, consider a behaviour in $Q_{\text{NPA}}^{1}$. There exists a set of vectors indexed by the set of fine-grained basic measurement outcomes and $\Xi$, spanning a Hilbert space $\mathcal{H}$, such that $\Gamma _{X,Y}^{1}=\langle X|Y\rangle$. We can extend this set of vectors to one indexed by $\mathcal{C}$—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 ${{Q}^{1+AB}}$. Similar reasoning can be applied to the case of ${{Q}^{1+AB}}$. 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).