Causation does not explain contextuality

Causation does not explain contextuality Sally Shrapnel and Fabio Costa Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia

arXiv:1708.00137v1 [quant-ph] 1 Aug 2017

2nd August 2017

Realist interpretations of quantum mechanics presuppose the existence of elements of reality that are independent of the actions used to reveal them. Such a view is challenged by several no-go theorems that show quantum correlations cannot be explained by noncontextual ontological models, where physical properties are assumed to exist prior to and independently of the act of measurement. However, all such contextuality proofs assume a traditional notion of causal structure, where causal influence flows from past to future according to ordinary dynamical laws. This leaves open the question of whether the apparent contextuality of quantum mechanics is simply the signature of some exotic causal structure, where the future might affect the past or distant systems might get correlated due to non-local constraints. Here we show that quantum predictions require a deeper form of contextuality: even allowing for arbitrary causal structure, no model can explain quantum correlations from non-contextual ontological properties of the world, be they initial states, dynamical laws, or global constraints.

Introduction The appeal of an operational physical theory is that it makes as few unwarranted assumptions about nature as possible. One simply assigns probabilities to experimental outcomes, conditioned on the list of experimental procedures required to realise these outcomes. Ideally, such operational theories are minimal: procedures that can not be statistically discriminated are given the same representation in the theory. Quantum mechanics is an example of such a minimal operational theory: all the statistically significant information about the preparation procedure is contained in the quantum state, and the probability of an event (labelled by a Positive Operator Valued Measure (POVM) element) does not depend on any other information regarding the manner in which the measurement was achieved (such as the full POVM). However, one of the most debated questions in the foundations of the theory is whether one can go beyond this statistical level and also provide an ontological description of some actual state of affairs that occurs during each run of an experiment. That is, a statement about the world that tells us what is responsible for the observed experimental outcomes. The task of providing such an ontological model for quantum theory has proven to be exceedingly difficult. A plethora of no-go theorems exists that describe the various natural assumptions one must forgo in order to produce an ontological model that accords with experiment. One such caveat is noncontextuality. Ultimately an apriori assumption, a non-contextual theory only posits the existence of elements of reality that can be distinguished experimentally, thus requiring that the physical properties relevant for a given phenomenon should not depend on the particular procedure used to observe or realise the phenomenon. It can be seen as an analogue of the no fine-tuning argument from causal modelling [1], an analogue of Leibniz’s principal of the Identity of Indiscernibles [2, 3], and a methodological assumption akin to Occam’s razor. Early notions of non-contextuality, for example those due Sally Shrapnel: [email protected] Fabio Costa: [email protected]

1

to Kochen-Specker [4] and Bell [5], prove that quantum measurements cannot be regarded as deterministically uncovering pre-existing, or ontic, properties of the system. More recent approaches have generalised the concept of non-contextuality in order to accomodate fundamentally indeterministic theories and to encompass generalised preparations, transformations and measurements [2, 3, 6–8]. Non-contextuality no-go theorems can serve as security proofs for a range of simple cryptographic scenarios [9, 10] and herald a quantum advantage for both computation [11], and state discrimination [8]. These results, however, all require the assumption of a fixed background causal structure; at the very minimum, a single causal arrow from preparation to measurement. In such approaches, non-contextual properties are assumed to exist before and independently of their measurement. This leaves open the question of whether one can produce a non-contextual ontological model by allowing for a suitably exotic causal structure. In particular, there has been growing interest in attempts to explain quantum correlations by positing backwards-in-time causal influences [12–17], or non-local constraints [18, 19]. The rationale is that non-contextuality could emerge naturally in such models: physical properties might well be “real” and “counterfactually definite”, but depend on future or distant measurements because of some physically motivated—although radically novel—causal influence. Such proposals do not fit neatly within the classical causal modelling framework, and so are not ruled out by recent work in this direction [1, 20], nor by any of the existing no-go theorems. In this paper, we further generalise the ontological models framework to show that even if one allows for arbitrary causal structure, ontological models of quantum experiments are necessarily contextual. Crucially, what is contextual is not just the traditional notion of “state”, but any supposedly objective feature of the theory, such as a dynamical law or boundary condition, which is responsible for the experimentally observed statistics. Our finding suggests that any model that posits unusual causal relations in the hope of saving “reality” will necessarily be contextual. Furthermore, this work also represents a possible approach to how we ought to think of the generalised quantum processes of recent work [21–35]. It is clear that any ontological reading of such processes will have to contend with the spectre of contextuality. The paper is organised as follows. In section 1 we introduce and justify the primitive elements required to define our operational model: local regions, local controllables, outcomes and an environment. In section 2 we introduce the operational equivalences, that is, the operationally indistinguishable elements of the model: events, instruments and processes. In Section 3 we characterise instrument and process non-contextuality according to these equivalences, and provide a generalised framework for a non-contextual ontological model. Using standard quantum theory and results from previous work [27, 35], in section 4 we characterise an operational model that accords with the experimental predictions of quantum theory. Section 5 puts these elements together to prove that one can not produce an ontological model that is both process and instrument non-contextual and accords with the predictions of quantum theory. In Section 6 we consider the constraints imposed on ontological models when one only assumes instrument non-contextuality. We finish with a discussion.

1 Operational primitives Typically in operational representations of quantum mechanics, the primitive elements are sets of preparations, transformations and measurements [36–39]. However, such primitives do not permit one to consider causal scenarios that move beyond the most simple causally ordered situations; in these models the notion of reality is defined in terms of properties that exist before a measurement takes place. The underlying ontology is therefore assumed to follow some ordinary causal structure, akin to the directed acyclic graphs of causal models [40]. In our model we wish to be able to consider more general situations, for example where we include any possible global dynamics, causal structure, space-time geometry or global constraints. In order to provide this alternative perspective we consider the primitive operational elements to be sets of labelled local regions, locally controllable properties and an environment. The key constraint here is that we wish to be able to talk about a notion of reality that is independent of the manner in which we access it; hence we must make a distinction between

2

local operations and an external environment. This means that the ontology we recover represents the part of the world that is invariant to whether we perform experiments or not. This is the type of requirement that typically underlies the realist perspective. We follow previous work on non-contextuality by first building an operational model (although with different primitive elements) and then using an apriori non-contextuality assumption to derive a plausible ontological model. We consider an experiment to consist of local labelled regions (A, B, C, . . . ) where one can perform controlled operations that can be associated with outcomes. The regions align with concepts such as local laboratories, communicating parties (e.g. Alice and Bob) and local spacetime regions (similar, e.g., to the operational framework of [41]). There is no apriori assumption that these regions be "fixed" or preassigned in some manner; they are simply labels for the locus of a set of controlled operations. Controlled operations generalise the notion of preparations, measurements, transformations, and can include the addition or subtraction of ancillary systems. Examples include the orientation of a wave-plate, the instigation of a microwave pulse, and the use of a photo-detector. ˜X , We call such local operations the local controllables. Each local controllable is represented as I where the superscript X = A, B, . . . labels the associated region. We consider outcomes as labels associated to the result of choosing a particular local controllable; the outcomes for region A are labelled a = 0, 1, 2, . . . . Examples include the number of detected photons, the result of a spin measurement or the time of arrival of a photon (we allow the outcomes to have infinite possible values as this enables us to use the same variable for local controllables that have different numbers of possible outcomes, although in general we expect that only a finite number of such outcomes is associated with non-zero probability). Finally, we consider all the possible properties that could account for correlations between outcomes in the local regions. These include any global properties, initial states, ˜ . If we connecting mechanisms, causal influence, or global dynamics. We call this the environment, W see a property change in relation to a choice of local controllable we label this as an outcome and do not classify it as part of the environment. A straightforward consequence is that in our operational model environments and local controllables are by construction always uncorrelated.

Figure 1: Operational primitives. It is clear that there is some ambiguity about what should pertain to a local controllable and what should be described as part of the environment. Indeed, the border between the two categories remains mobile and there is an ineliminable perspectival element that enters the particular modelling of any given experiment. What matters, however, is that there is consistency between each possible description of the same experiment (we return to this point in the discussion). It is also worth noting 3

that we do not make any apriori assumptions regarding the relevance (causal or otherwise) of each of these notions. For example, the colour of Alice’s shoes can be included as a local controllable, and the weather conditions during the experiment as an environment. It is the operational equivalences that we define in the next section that enforce a notion of relevance. We can thus describe a single run of an experiment by a set of regions, outcomes, local controllables and an environment. If we consider a particular run of an experiment there will in general be a collection of outcomes that occur, one for each local region. One can associate a joint probability to this set of outcomes and one can empirically verify probability assignments for each possible set of outcomes. An operational model for such an experiment allows one to calculate expected probabilities: ˜A , I ˜B , I ˜C , . . . , W ˜ ). P (a, b, c, . . . |I

(1)

˜A , I ˜B , . . . , The operational model thus specifies a distribution over outcomes for local controllables I ˜ . Note that it should be possible to have ignorance over part of the and a shared environment W environment and characterise this accordingly using the operational model. More explicitly, if ξ˜ represents the part of the environment about which we are ignorant, then the operational probabilities ˜ given the known part of the environment is obtained by marginalising over ξ: ˜A , I ˜B , I ˜C , . . . , W ˜)= P (a, b, c, . . . |I

Z Z

=

˜ (a, b, c, . . . , ξ| ˜I ˜A , I ˜B , I ˜C , . . . , W ˜) dξP

(2)

˜ (ξ| ˜W ˜ (a, b, c, . . . |I ˜A , I ˜B , I ˜C , . . . , W ˜ , ξ)P ˜ ), dξP

where the second equality comes from the assumption that the local controllables are uncorrelated ˜ can describe the axis along which a spin- 1 particle with the environment. As a concrete example, W 2 is prepared, while ξ˜ represents whether the spin is prepared aligned or anti-aligned with that axis.1 The marginal (2) then describes a scenario where there is some probabilistic uncertainty of the spin’s ˜W ˜ ) = 1, direction i.e. which value of ξ occurs in any given run. Note that, for the particular case P (ξ| 2 ˜ we obtain the maximally mixed state irrespective of the axis, making the variable W redundant. Such redundancies can be taken into account via operational equivalences.

2 Operational equivalences We next characterise the appropriate operational equivalences in order to refine our operational model of an experiment. Notationally, we omit the ‘tilde’ for each equivalence class.

2.1 Events ˜A ) is operationally We say that a pair composed of an outcome and the respective local controllable (a, I 0 0A 0 ˜ ) if the joint probabilities for a, b, c, . . . and a , b, c, . . . are the same for equivalent to the pair (a , I all possible outcomes and local controllables in the other regions B, C, . . . , and for all environments ˜. W ˜A , I ˜B , . . . W ˜0A , I ˜B , . . . W ˜ ) = P (a0 , b, c, . . . |I ˜ ), P (a, b, c, . . . |I (3) ˜B , I ˜C , . . . , W ˜ ). ∀(b, c, . . . , I We denote an equivalence class of such pairs of outcomes and local controllables as an event: ˜A )]. M A = [(a, I

(4)

1 Here (and again in Section 4), we take for simplicity a scenario with a single region where a measurement is performed, so the specification of a process is equivalent to the specification of a state. More generally, the variables W and ξ could describe quantum channels, quantum networks, or more general quantum processes.

4

2.2 Instruments ˜A , where an event We define an instrument as the list of possible events for a local controllable I A A ˜ A ˜ M = [(a, I )] is possible for I if ˜A , I ˜B , . . . W ˜ ) 6= 0, P (a, b, c, . . . |I

(5)

for some ˜B , I ˜C , . . . , W ˜ ). (b, c, . . . , I ˜A is equivalent to I ˜0A if they define the same list of possible events and we denote the We say that I ˜A ] ≡ {M A , . . . , M A }. Note that our definition allows distinct instruments equivalence class IA := [I n 1 to share one or more events. Note also, our definition implies that the probability for an event doesn’t depend on the particular instrument I, once we assume the event is possible given the instrument. This property we call operational instrument non-contextuality.

2.3 Process The process captures those physical features responsible for generating the joint statistics for a set of events, independently of the choice of local instruments. A process is defined as an equivalence class ˜ ], where W ˜ is equivalent to W ˜ 0 , if of environments, W := [W ˜ 0 ), ˜A , I ˜B , . . . , W ˜A , I ˜B , . . . W ˜ ) = P (a, b, c, . . . |I P (a, b, c, . . . |I

(6)

˜B , I ˜C , . . . , ). ∀(a, b, c, . . . , I A simple example is the spatio-temporal ordering of regions. It is clear that the operational statistics of events in regions A and B can be different for the following two causal orderings: (i) A is ˜ (i) and W ˜ (ii) , will not be equivalent. before B, (ii) B is before A; thus the respective environments, W On the other hand, for certain experiments we would not expect any difference in statistics for a simple rotation of the whole experiment by 45 degrees; these two environments will be represented by the same process W . The above equivalences allow us to define a joint probability distribution over the space of events (rather than outcomes) conditioned on instruments (rather than local controllables) and the process (rather than the environment). As discussed above, this distribution satisfies operational instrument non-contextuality, which means that the probability for some event is either zero or independent of the instrument. Therefore, it can be expressed in terms of a frame function fW that maps events to probabilities and is normalised for each instrument: P (M A , M B , . . . |IA , IB , . . . , W ) = fW (M A , M B , . . . )

Y

χIX (M X ),

(7)

X=A,B,...

where, for a set S, χS is the indicator function, χS (s) = 1 for s ∈ S and χS (s) = 0 for s 6∈ S. Note that the indicator functions are necessary to make the whole expression a valid probability distribution, normalised over the entire space of events2 . Furthermore, and in contrast to similar expressions involving POVMs, the dependency on the instruments is crucial to allow for causal influence across the regions: Integrating over the events of, say, region A, can result in a marginal distribution that still depends on A’s instrument and displays signalling from A to other regions. However, the fact that the dependency on the instruments is solely through the indicator functions tells us that the causal relations can be attributed to the particular events realised in each experimental run, rather than to the whole instruments (which include the specification of events that did not happen). 2

The indicator functions cannot be absorbed in the definition of the frame function, as the latter carries no dependency on the instruments.

5

3 Ontological model The purpose of an ontological model is to introduce possible elements of reality. The typical approaches assume that the ontology is encoded in a “state”, representing the physical properties of a system at a given time. Here we consider a weaker notion of ontology, which captures any physical properties of the environment that are responsible for mediating correlations between regions. We represent the collection of all such properties by a single variable ω, named the ontic process. We wish to clarify at this point that our ontic process captures the physical properties of the world that remain invariant under our local operations. That is, although we allow local properties to change under specific operations, we wish our ontic process to capture those aspects of reality that are independent of this probing. We elaborate more on the interpretation of ontic processes and on the relation with ontic states in the discussion section. The ontological model then specifies a joint probability for a set of outcomes, one at each local region, given the ontic process, the environment and the set of local controllables. This joint probability reduces to the operational joint probability when the value of the ontic process is unknown: ˜A , I ˜B , . . . W ˜)= P (a, b, c, . . . , |I

Z

˜A , I ˜B , . . . W ˜ ). dωP (a, b, c, . . . , ω|I

(8)

There are three natural assumptions one might require of an ontological model: Assumption 1. ω-mediation3 The ontic process mediates all the correlations between regions, thus ω screens off outcomes from the environment, and we have: ˜A , I ˜B , . . . , W ˜)= P (a, b, c, . . . |I

Z

˜A , I ˜B , . . . )P (ω|W ˜ ). dωP (a, b, c, . . . |ω, I

(9)

Assumption 2. Instrument non-contextuality. Operationally indistinguishable pairs of outcomes and local controllables should remain indistinguishable at the ontological level. That is, for operation˜A ), (a0 , I ˜0A ), ally equivalent pairs (a, I ˜A , I ˜B , . . . ) = P (a0 , b, c, . . . , |ω, I ˜0A , I ˜B , . . . ), P (a, b, c, . . . , |ω, I (10) which means that we can define a probability distribution on the space of events, conditioned on instruments and on the ontic process, in terms of a frame function fω , such that: P (M A , M B , . . . |IA , IB , . . . , ω) =

Y

χIX (M X )fω (M A , M B . . . ),

(11)

X

where χ is the indicator function, χX (x) = 1 for x ∈ X and χX (x) = 0 for x 6∈ X, and fω maps events to probabilities: fω (M A , M B , . . . ) ∈ [0, 1], (12) and is normalised for each set of events that corresponds to a particular instrument: X

fω (M A , M B , M C , . . . ) = 1.

(13)

M A ∈IA M B ∈IB M C ∈IC ...

Assumption 3. Process non-contextuality. ˜ ,W ˜ 0 ) the assumption of process non-contextuality implies: For operationally equivalent processes (W ˜ ) = P (ω|W ˜ 0 ), P (ω|W

(14)

and we can define a function gW (ω) that maps ontic processes to probabilities, given each process W : ˜ ), gW (ω) = P (ω|W

h

i

˜ , W = W

(15)

that is normalised for all ω: Z

dω gW (ω) = 1. 3

This assumption is similar to “λ-mediation” from Ref. [16].

6

(16)

For an ontological model that satisfies the above three assumptions, the operational probability can now be expressed in terms of events, instruments and processes as: A

B

A

Y

B

P (M , M , . . . |I , I , . . . , W ) =

X

χIX (M )

Z

dω fω (M A , M B , . . . )gW (ω).

(17)

X=A,B,...

4 Quantum models If one assumes that the results of experiments in local regions accord with quantum mechanics, then events can be associated to completely positive trace-non-increasing (CP) maps MA : AI → AO , where input and output spaces are the spaces of linear operators over input and output Hilbert spaces of the local region, AI ≡ L(HAI ), AO ≡ L(HAO ) respectively [42]. Each set of CP maps that sums to a completely positive trace preserving map is a quantum instrument [43]. An instrument thus represents the collection of all possible events that can be observed given a specific choice of local controllable. Given a local region A, an instrument is formally defined as a set IA of CP maps that sum up to a completely positive trace-preserving (CPTP) map: 

Tr 

 X

MA (ρ) = Tr(ρ).

(18)

MA ∈IA

Given these definitions of events and instruments, one can predict the joint probability over possible events using a generalised form of the Born rule: P (M A , M B , . . . |IA , IB , . . . , W ) =fW (M A , M B , . . . )

Y

χIX (M X ),

(19)

X

h

i

fW (M A , M B . . . ) = Tr (M A ⊗ M B ⊗ . . . )W ,

(20)

where M A , M B . . . are the Choi-Jamiolkowski representations of the local CP maps associated to particular events, and W is a positive, semi-definite operator associated to the relevant process [21, 24, 27]. We call W the process matrix, using the terminology of Ref. [27]. It is possible to derive this trace rule for probabilities by assuming linearity [27], or alternatively one can derive linearity (and the trace rule) from the assumption of operational instrument noncontextuality alone [35]. The significance of this latter derivation is that it is then clear that the trace rule is the unique probability assignment consistent with the assumption of instrument noncontextuality. By assuming instrument non-contextuality at the ontological level, this implies that, for each ontic process ω, the corresponding frame function can be expressed as: fω (M A , M B , . . . ) = Tr [σ(ω)M] ,

(21)

where we introduced the short-hand notation M ≡ M A ⊗ M B ⊗ . . . and σ(ω) is a process matrix [35]. We now wish to show that the function gW (ω) that features in our ontological model, under the assumption of process non-contextuality, can be represented as gW (ω) = Tr [η(ω)W ] ,

(22)

where {η(ω)}ω∈Ω , Ω being the set of ontic processes, is a quantum instrument. It is common in non-contextuality no-go theorems (as well as in the process matrix formalism) to assume preservation of probabilistic mixtures as an assumption that is independent of the assumption of non-contextuality. Here we rather derive it from our assumption of process non-contextuality. Consider two classical variables ξ, W used to describe the process, where we already take operational equivalences into account. Following the earlier example, we can think of W as describing a cartesian axis, while ξ—the aspect of the process about which we are ignorant—describes whether a spin- 21

7

particle is prepared aligned or anti-aligned to this axis. The operational probabilities given W , and the corresponding decomposition for ontological probabilities, are obtained by marginalisation: P (M A , M B , . . . |IA , IB , . . . , W ) dξ dω P (M A , M B , . . . |ω, IA , IB , . . . , W, ξ)P (ω|IA , IB , . . . , W, ξ)P (ξ|IA , IB , . . . , W )

Z

dξ dω P (M A , M B , . . . |ω, IA , IB , . . . W, ξ)P (ω|W, ξ)P (ξ|W ),

= =

(23)

Z

where, in the last identity, we use the fact that P (ω|W, ξ) does not depend on the local controllables (and thus on the instruments) due to the assumption of ω-mediation; and P (ξ|W ) is due to our assumption that the environment and local controllables (and thus process and instruments) are uncorrelated. Now let us write Wξ for the process corresponding to the pair W, ξ. We have Z

gR

dξ Wξ P (ξ|W ) (ω)

= gW (ω) = P (ω|W ) =

dξ gWξ (ω)P (ξ|W ),

(24)

thus gW (ω) is convex-linear in W . The first identity in Eq. (24) comes from the fact that probabilistic R mixtures of quantum processes are represented as convex combinations, thus W = dξ Wξ P (ξ|W ). This in turn is a consequence of the trace formula for operational quantum probabilities (which is itself a consequence of operational instrument non-contextuality): h

P (M A , M B , . . . |IA , IB , . . . , W ) = Tr (M A ⊗ M B ⊗ . . . )W

iY

χIX (M X )

(25)

X

Z

=

dξ P (M A , M B , . . . |IA , IB , . . . , W, ξ)P (ξ|W )

Y

χIX (M X )

X



A

B

= Tr (M ⊗ M ⊗ . . . )

Z

dξ Wξ P (ξ|W )

Y

χIX (M X )

(26)

X

for all CP maps M A , M B , . . . Using standard linear-algebra arguments, gW (ω) can be extended to a linear function over W , leading to the representation (22), gW (ω) = Tr [η(ω)W ]. Positivity and normalisation of probabilities then imply

gW (ω) ≥ 0 ⇒ η(ω) ≥ 0 ∀ω, Z

Z

dωgW (ω) = 1 ⇒ Tr

(27) 

dωη(ω)W = 1 ∀W.

(28)

Operators η(ω) as defined above can be understood as the Choi representation of CP maps that sum up to a trace preserving map, namely {η(ω)}ω∈Ω defines an instrument. In general, the CP maps η(ω) do not have to factorise over the separate regions, therefore it might not be possible to interpret them as local operations. This is not an obstacle, as such an interpretation is not required for the rest of the argument.

5 A quantum contradiction To summarise the results so far, we have an operational rule for the predictions of the joint probabilities of outcomes according to quantum theory: P (M A , M B , . . . |IA , IB , . . . , W ) =

Y X

8

χIX (M X ) Tr [M W ] .

(29)

We also have an ontological model for predicting the joint probabilities under the assumptions of ω-mediation, instrument non-contextuality and process non-contextuality: A

B

A

B

P (M , M , . . . |I , I , . . . , W ) =

Y

X

Z

χIX (M )

dωfω (M A , M B , . . . )gW (ω),

(30)

dω [Tr σ(ω)M ] [Tr η(ω)W ] .

(31)

X

which given the results of the last section, becomes: A

B

A

B

P (M , M , . . . |I , I , . . . , W ) =

Y

X

Z

χIX (M )

X

If this accords with quantum predictions then we should have: Z

dω [Tr σ(ω)M ] [Tr η(ω)W ] ∀M, W.

Tr [M W ] =

(32)

It has been noted [44] that a decomposition of the form (30) is akin to the expression of expectation values in terms of quasi-probability distributions [45,46]. However, the non-contextuality assumptions force both fω and gW to be ordinary, positive probability distributions. It is well known that quantum expectation values cannot be expressed in such a way, it is however instructive to consider an explicit contradiction within the present process framework. From (32) 



Z

Tr [M W ] = Tr M

∀M

(33)

dω σ(ω) gW (ω),

(34)

dω σ(ω) gW (ω)

→W =

Z

which follows from the fact that M span a complete set of the joint linear space AI ⊗AO ⊗B I ⊗B O , . . . Eq. (34) tells us that W is a convex mixture of the operators σ(ω). If we choose W to be extremal, by which we mean that it can not be decomposed into a non-trivial convex combination of other processes, then W ∝ σ(ω) for gW (ω) 6= 0. Consider a process W that can be decomposed into two distinct mixtures of two sets of extremal processes Wj and Wk0 (we take discrete sets for simplicity): W =

X

qj W j =

X

j

pk Wk0 .

(35)

k

We define ΩW as the support of gW . From (22) and (34) we have Z

W =

dω σ(ω) gW (ω)

(36)

ΩW

Z

dω σ(ω)

= ΩW

qj gWj (ω)

(37)

pk gWk (ω),

(38)

j

Z

dω σ(ω)

=

X

ΩW

X k

and (by definition of ΩW ) for all ω ∈ ΩW there exists a j such that gWj (ω) 6= 0, likewise for all ω ∈ ΩW there exists a k such that gWk (ω) 6= 0. This implies W ∝ σ(ω) ∝ Wj ∝ Wk0 .

(39)

However, one can find many examples where no process in one decomposition is proportional to any process in the other. This implies a contradiction and shows that a decomposition such as (32) cannot exist for all CP maps and quantum processes. As a particular example to show the above contradiction,

9

consider a process W corresponding to a quantum channel from a region with a two-level output, AO to a region with a two-level input, BI : W =

X

qj W j =

j

X

pk Wk0 ,

(40)

k

formed from the following two combinations of extremal processes: W1 =[[1]] = 1 + X ⊗ X − Y ⊗ Y + Z ⊗ Z,

(41)

W2 =[[X]] = 1 + X ⊗ X + Y ⊗ Y − Z ⊗ Z,

(42)

W3 =[[Y ]] = 1 − X ⊗ X − Y ⊗ Y − Z ⊗ Z,

(43)

W4 =[[Z]] = 1 − X ⊗ X + Y ⊗ Y + Z ⊗ Z.

(44)

W10 =[[U 1]] = 1 + X ⊗ U XU † − Y ⊗ U Y U † + Z ⊗ U ZU † ,

(45)

W20 =[[U X]] = 1 + X ⊗ U XU † + Y ⊗ U Y U † − Z ⊗ U ZU † ,

(46)

W30 W40

=[[U Y ]] = 1 − X ⊗ U XU − Y ⊗ U Y U − Z ⊗ U ZU ,

(47)

=[[U Z]] = 1 − X ⊗ U XU † + Y ⊗ U Y U † + Z ⊗ U ZU † .

(48)

†

†

†

where X, Y and Z are the Pauli matrices, U is a unitary, and we used the notation [[V ]] := rs |ri hs|⊗ V |ri hs| V † for the Choi representation of a unitary V . It is clear that no Wj is proportional to any Wk for an appropriate choice of U , and we have a contradiction with (39). P

6 Process-contextual extensions of quantum theory The above no-go theorem requires two independent assumptions: instrument non-contextuality and process non-contextuality. At the formal level, these assumptions play the roles of measurement and preparation non-contextuality [2], respectively. Many contextuality no-go theorems focus only on the requirement of measurement non-contextuality; it is thus interesting to see what are the consequences of instrument non-contextuality in our framework, while dropping process non-contextuality. It is easy to see that instrument non-contextual, process-contextual models are possible. An example is a model where the ontic process is directly identified with the quantum process: gW (ω) = δ (W − ω) .

(49)

Operational probabilities are the recovered simply by using the “quantum process rule”, Eq. (20), for the ontic frame function: fω (M A , M B . . . ) = Tr [M ω] . (50) This “crude” ontological model is similar to the Beltrametti-Bugajski (BB) model [47], which identifies elements of reality with the quantum wave functions. A difference is that the BB model only identifies pure quantum states with elements of reality, while in Eq. (49) any process counts as ontic, including those corresponding to mixed states or noisy channels. One could refine the above model by only allowing an appropriately defined “pure process” to be ontic. (See however Ref. [48] for possible ambiguities regarding such a definition.) Even without preparation non-contextuality, measurement non-contextuality imposes strong constraints on the ontology: essentially, any non-contextual ontology must reduce to the BB model [6]. A similar result holds in our case. As already discussed above, the only instrument non-contextual frame function must be given by Eq. (21), namely to every ontic process ω is associated a process matrix σ (ω). The implication is that an instrument non-contextual hidden variable cannot provide 10

more information than that contained in a process matrix. We thus conclude that, even without any assumption about the causal structure of the ontological theory, quantum mechanics admits no nontrivial, non-contextual extension. This is analogous to the result of [6], where it was proved that no measurement non-contextual extension of quantum theory exists that can provide more accurate predictions of experimental outcomes. Therefore, even instrument non-contextuality alone poses strong restrictions on hidden variable models that attempt to leverage exotic causal structures to recover a non-contextual notion of reality.

7 Discussion We have shown that it is not possible to construct an ontological model that is both instrument and process non-contextual and also accords with the local predictions of quantum mechanics. We take both forms of non-contextuality to be very reasonable assumptions if one wishes "reality" to be describable in a manner that is independent of the act of experimentation. Thus our work shows that models that posit unusual causal, global or dynamical relations will not solve a key quantum mystery, that of contextuality. Our model also respects the mobility of the boundary between local instruments and processes that one sees in ordinary applications of quantum theory. As a simple example, consider a preparation P of a quantum system, followed by a measurement M . This can be modelled in three different ways: (i) ˜ , and M as an instrument associated to a single local region, (ii) with P as part of the environment W ˜ capturing both a channel between with P and M as instruments in two distinct local regions, and W preparation and measurement, plus any additional information about the environment, or (iii) with both P and M characterising the instruments in a single local region and all other information about ˜ . For classical processes characterised as causal models, such a shift the environment modelled as W in perspective is formalised by the notion of “latent variables” [40]. An analogue notion of "latent laboratories" exists for quantum processes characterised as quantum causal models, and this formal structure likewise characterises the mobility of the boundary to which we refer [32]. It is interesting to consider the relationship between the ontic states posited in previous work on ontological models, usually denoted λ, and our ontic process, ω. In particular, it is important to clarify the manner in which our framework allows for more general ontologies. We can consider, as an example, a classical process with a simple ontological interpretation. Here two regions, A and B, are identified with localised space-time regions, each delimited by a past and future boundary. B For a classical system, we can assign input states λA I and λI to the past boundaries of A and B, A B respectively, and output states λO and λO to the respective future boundaries. A deterministic local X operation is a function   f that maps the input state of each region to the corresponding output, X X X X λI 7→ λO = f λI , where X characterises the respective local region, A or B. A local event can 



X then be characterised by an input-output state pair, λX := λX O , λI , and the collection of possible

events given a choice of operation,

n



X λX λX I , f I

o

, represents an instrument (which here is

λX I X f ) (see

simply equivalent to the specification of the function figure 2a). As an example of a process, we can consider a situation where region A is in the past of region B, and the input of B is fully determined by the output of A. In this case, the consists of a  process  B λA = λB , where the specification of the input state for A and of the functional relation λA → 7 w O O I function wB represents the relevant classical dynamics. Ontological and operational elements  can be B . It is directly identified in this example; in particular, the ontic process is just ω ≡ W ≡ λA , w I clear that the usual ontological models are recovered in the case of a single region (or in case wB is a constant), as in that case the ontic process is simply the input ontic state. Our ontological models can capture scenarios well beyond the above example. We could consider a situation where we still have ontic events λ, as above, but the ontic process is not limited to evolve from the past into the future. Processes of this type have been proposed to describe classical closed 11

(a)

(b)

Figure 2: Classical process with ontological interpretation. (a) We assign input states λA I and A and λB to the respective λB to the past boundaries of A and B, respectively, and output states λ I O O future boundaries. A deterministic local operation is a function f X that maps the input state of each region to the corresponding output. (b) An example of ontic closed  process  is one describingclassical  A B B A B time-like curves [49], defined by a pair of functions ω = w , w , where λO 7→ w λO = λA I and similarly for wB . 







A λB = time-like curves [49], and are described by a pair of functions ω = wA , wB , where λB O 7→ w O B (as in the example above). Such processes (for three or more regions) λA I and similarly for w have been shown to allow violation of causal inequalities [49, 50] (device-independent constraints on probabilities imposed by a definite causal order [27,51]), but still have a clearly defined non-contextual, classical ontology. One could also consider more general ontological models, where instruments are not associated with local input-to-output functions but with more general sets of input-output pairs. In such models, a local operation may well change the state of the system in the past, as well as in the future, of the region. One could also remove the distinction between the past and future state, or even the notion of “ontic event” λ altogether. Still, such models cannot reproduce the predictions of quantum mechanics as long as the relations between events are mediated by some non-contextual process. A consequence of our results is that an explanation of quantum contextuality will not come from choosing particular causal, global or dynamical relations. Finally, we draw attention to the fact that our results rely on complete matching to the operational predictions of quantum theory. This leaves open the possibility that particular ontological models might allow for some experimentally testable, different predictions. So, for proponents of particular models, the door remains open to develop their ontology such that they can predict some possible deviation from quantum statistics. In the face of such statistical deviation, the possibility of a noncontextual ontological model remains open.

Acknowledgments We thank Časlav Brukner, Eric Cavalcanti, Matthew Leifer, Gerard Milburn, Alberto Montina, Ravi Kunjwal, David Schmid, and Robert Spekkens for helpful discussions. This work was supported by an Australian Research Council Centre of Excellence for Quantum Engineered Systems grant (CE 110001013), and by the Templeton World Charity Foundation (TWCF 0064/AB38). F.C. acknowledges support through an Australian Research Council Discovery Early Career Researcher Award (DE170100712). This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. We acknowledge the traditional owners of the land on which the University of Queensland is situated, the Turrbal and Jagera people.

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References [1] E. G. Cavalcanti, “Quantum nonlocality and contextuality as fine-tuning,” arXiv:1705.05961 [quant-ph]. [2] R. W. Spekkens, “Contextuality for preparations, transformations, and unsharp measurements,” Phys. Rev. A 71, 052108 (2005). [3] R. Kunjwal, “Contextuality beyond the Kochen-Specker theorem,” arXiv:1612.07250 [quant-ph]. [4] S. Kochen and E. Specker, “The problem of hidden variables in quantum mechanics,” J. Math. Mech. 17, 59–87 (1967). [5] J. S. Bell, “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys. 38, 447–452 (1966). [6] Z. Chen and A. Montina, “Measurement contextuality is implied by macroscopic realism,” Phys. Rev. A 83, 042110 (2011). [7] M. D. Mazurek, M. F. Pusey, R. Kunjwal, K. J. Resch, and R. W. Spekkens, “An experimental test of noncontextuality without unphysical idealizations,” Nat. commun. 7, (2016). [8] D. Schmid and R. W. Spekkens, “Contextual advantage for state discrimination,” arXiv:1706.04588 [quant-ph]. [9] A. Chailloux, I. Kerenidis, S. Kundu, and J. Sikora, “Optimal bounds for parity-oblivious random access codes,” New Journal of Physics 18, 045003 (2016). [10] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde, “Preparation contextuality powers parity-oblivious multiplexing,” Physical review letters 102, 010401 (2009). [11] M. Howard, J. Wallman, V. Veitch, and J. Emerson, “Contextuality supplies the ‘magic’ for quantum computation,” Nature 510, 351–355 (2014). [12] H. Price, “Does time-symmetry imply retrocausality? How the quantum world says “Maybe”?,” Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 43, 75–83 (2012). [13] P. W. Evans, H. Price, and K. B. Wharton, “New Slant on the EPR-Bell Experiment,” Brit. J. Philos. Sci. 64, 297–324 (2013). [14] K. Wharton, “Quantum States as Ordinary Information,” Information 5, 190–208 (2014). [15] Y. Aharonov, E. Cohen, and T. Shushi, “Accommodating Retrocausality with Free Will,” Quanta 5, 53–60 (2016). [16] M. S. Leifer and M. F. Pusey, “Is a time symmetric interpretation of quantum theory possible without retrocausality?,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 473, (2017). [17] R. I. Sutherland, “How retrocausality helps,” AIP Conference Proceedings 1841, 020001 (2017). [18] A. Carati and L. Galgani, “Nonlocality of classical electrodynamics of point particles, and violation of Bell’s inequalities,” Nuovo Cimento B 114, 489–500 (1999). [19] S. Weinstein, “Nonlocality Without Nonlocality,” Foundations of Physics 39, 921–936 (2009). [20] C. J. Wood and R. W. Spekkens, “The lesson of causal discovery algorithms for quantum correlations: Causal explanations of Bell-inequality violations require fine-tuning,” New J. Phys. 17, 033002 (2015). [21] G. Gutoski and J. Watrous, “Toward a general theory of quantum games,” in Proceedings of 39th ACM STOC, pp. 565–574. 2006. arXiv:quant-ph/0611234. [22] G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Quantum Circuit Architecture,” Phys. Rev. Lett. 101, 060401 (2008). [23] G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Memory Effects in Quantum Channel Discrimination,” Phys. Rev. Lett. 101, 180501 (2008). [24] G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Theoretical framework for quantum networks,” Phys. Rev. A 80, 022339 (2009). [25] A. Bisio, G. Chiribella, G. D’Ariano, and P. Perinotti, “Quantum networks: General theory and applications,” Acta Physica Slovaca. Reviews and Tutorials 61, 273–390 (2011). [26] A. Bisio, G. M. D’Ariano, P. Perinotti, and M. Sedlák, “Optimal processing of reversible quantum channels,” Physics Letters A 378, 1797 – 1808 (2014).

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[27] O. Oreshkov, F. Costa, and Č. Brukner, “Quantum correlations with no causal order,” Nat. Commun. 3, 1092 (2012). [28] K. Modi, “Operational approach to open dynamics and quantifying initial correlations,” Scientific Reports 2, (2012). [29] M. S. Leifer and R. W. Spekkens, “Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference,” Physical Review A 88, 052130 (2013). [30] M. Ringbauer, C. J. Wood, K. Modi, A. Gilchrist, A. G. White, and A. Fedrizzi, “Characterizing Quantum Dynamics with Initial System-Environment Correlations,” Phys. Rev. Lett. 114, 090402 (2015). [31] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, “Complete framework for efficient characterisation of non-Markovian processes,” arXiv:1512.00589 [quant-ph]. [32] F. Costa and S. Shrapnel, “Quantum causal modelling,” New Journal of Physics 18, 063032 (2016). [33] J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee, and R. W. Spekkens, “Quantum common causes and quantum causal models,” arXiv:1609.09487 [quant-ph]. [34] S. Milz, F. A. Pollock, and K. Modi, “Reconstructing open quantum system dynamics with limited control,” arXiv:1610.02152 [quant-ph]. [35] S. Shrapnel, F. Costa, and G. Milburn, “Updating the Born rule,” arXiv:1702.01845 [quant-ph]. [36] L. Hardy, “Quantum Theory From Five Reasonable Axioms,” quant-ph/0101012. [37] H. Barnum, C. A. Fuchs, J. M. Renes, and A. Wilce, “Influence-free states on compound quantum systems,” arXiv:quant-ph/0507108. [38] G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Probabilistic theories with purification,” Phys. Rev. A 81, 062348 (2010). [39] B. Dakić and Č. Brukner, “Quantum Theory and Beyond: Is Entanglement Special?,” arXiv:0911.0695 [quant-ph]. [40] J. Pearl, Causality. Cambridge University Press, 2009. [41] O. Oreshkov and C. Giarmatzi, “Causal and causally separable processes,” New Journal of Physics 18, 093020 (2016). [42] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, 2000. [43] E. Davies and J. Lewis, “An operational approach to quantum probability,” Comm. Math. Phys. 17, 239–260 (1970). [44] R. W. Spekkens, “Negativity and Contextuality are Equivalent Notions of Nonclassicality,” Phys. Rev. Lett. 101, 020401 (2008), arXiv:0710.5549 [quant-ph]. [45] E. Wigner, “On the Quantum Correction For Thermodynamic Equilibrium,” Phys. Rev. 40, 749–759 (1932). [46] M. Scully and M. Zubairy, Quantum Optics. Cambridge University Press, 1997. [47] E. G. Beltrametti and S. Bugajski, “A classical extension of quantum mechanics,” J. Phys. A: Math. Gen. 28, 3329 (1995). [48] M. Araújo, A. Feix, M. Navascués, and Č. Brukner, “A purification postulate for quantum mechanics with indefinite causal order,” Quantum 1, 10 (2017). [49] Ä. Baumeler and S. Wolf, “The space of logically consistent classical processes without causal order,” New Journal of Physics 18, 013036 (2016). [50] Ä. Baumeler, A. Feix, and S. Wolf, “Maximal incompatibility of locally classical behavior and global causal order in multi-party scenarios,” Phys. Rev. A 90, 042106 (2014). [51] C. Branciard, M. Araújo, A. Feix, F. Costa, and Č. Brukner, “The simplest causal inequalities and their violation,” New Journal of Physics 18, 013008 (2016).

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