aa r X i v : . [ qu a n t - ph ] F e b Neither contextuality nor non-locality admits catalysts
Martti Karvonen
University of Ottawa, Canada bstract We show that the resource theory of contextuality does not admit catalysts. As a corollary, weobserve that the same holds for non-locality. This adds a further example to the list of “anomaliesof entanglement”, showing that non-locality and entanglement behave differently as resources. Wealso show that catalysis remains impossible even if instead of classical randomness we allow somemore powerful behaviors to be used freely in the free transformations of the resource theory.
Introduction.
Contextuality [1] and non-locality [2, 3] play a prominent role in a widevariety of applications of quantum mechanics, with non-locality being used for examplein quantum key-distribution [4], certified randomness [5] and randomness expansion [6].Similarly, contextuality powers quantum computation in some computational models [7–12]and even increases expressive power in quantum machine learning [13]. Consequently, it isvital to understand how non-locality and contextuality behave as resources.In this Letter, we show that neither contextuality nor non-locality admit catalysts: thatis, there are no correlations that can be used to enable an otherwise impossible conversionbetween correlations and still recovered afterwards. To state this slightly more precisely,let us write d, e, f . . . for various correlations (whether classical or not), d ⊗ e for havingindependent instances of d and e , and d e (read as “ d simulates e ”) for the existence ofa conversion d → e . Then our results state that, in suitably formalized resource theories ofcontextuality and non-locality, whenever d ⊗ e d ⊗ f , then e f already. This gives astrong indication that contextuality (and non-locality) are resources that get spent whenyou use them: there is no way of using a correlation d to achieve a task you could not dootherwise while keeping d intact. As entanglement theory famously allows for catalysts [14],this can be seen as yet another “anomaly of non-locality” [15] and thus further testamentto the fact that non-locality and entanglement are different resources.[16]We do this by working in precisely defined resource theories of contextuality and non-locality. These are not strictly speaking quantum resource theories [17], but resource theoriesin a more general sense [18, 19], as we allow resources such as PR-boxes [20] that are notquantum realizable. The kinds of conversions between correlations we have in mind cap-ture the intuitive idea of using one system to simulate another one, and have been studiedin earlier literature [21–29]. These roughly correspond to the local operations and sharedrandomness-paradigm ((LOSR)) or to wirings and prior-to-input classical communication2WPICC), depending on the precise definitions of these terms. However, existing formaliza-tions of these in the literature are often limited to the bipartite or tripartite settings, andat times overlook some technical issues resulting in non-convex sets of transformations [30,Appendix].More importantly, existing formalizations of the resource theory of non-locality tend tofocus on the case where each party has a discrete set of measurements, of which they canperform at most one. However, this is false even in relatively simple situations: for instance,if Alice shares one PR-box with Bob and one with Charlie, then she has four measurementsavailable, but is not restricted to only one measurement as she can choose a measurementfor each of her boxes. In particular, she might first measure one of the boxes, and use theoutcome (and possible auxiliary randomness) to choose what to measure next.To overcome such issues, we work in the general approach to contextuality initiatedin [31], and later extended in [32] to capture building some correlations from others usingsuch probabilistic and adaptive means, resulting in a resource theory for contextuality.We then obtain the resource theory of non-locality from this via a general mathematicalconstruction [33] that builds a resource theory of n -partite resources from a given resourcetheory. Working with such generality clarifies the relationship between resource theories ofcontextuality and non-locality, and captures the kinds of interconversions studied in earlierliterature in the exact, single-shot regime [34] in a precise yet tractable manner.We believe that our result could be phrased and proved in terms other approaches tocontextuality [35–39], as long as one formalizes such adaptive measurement protocols andtransformations between correlations within them. However, if one works with axiomaticallyrather than operationally defined transformations as in [40] the proof no longer carries overautomatically, as it could happen that operational transformations form a proper subset ofaxiomatically defined ones, as happens for instance in resource theories of entanglement [41]and magic [42]. No catalysis for contextuality
We begin by briefly reviewing the resource theory of con-textuality as defined in [32]. To start, we formalize the idea of a measurement scenario S : we imagine a situation where there is a finite set X S of measurements available, eachmeasurement x ∈ X S giving rise to outcomes in some finite set O S,x . However, only somemeasurements might feasible to perform together — other combinations may be ruled outby practical limitations or excluded by physical theory. We collect all jointly compatible3easurements into a single set Σ S , which we expect to satisfy two natural properties:1. Any measurement x ∈ X S induces a compatible set { x } ∈ Σ S
2. Any subset of a compatible set of measurements is compatibleCollecting all this data together results in a measurement scenario S = h X S , Σ S , O S i .Given two scenarios S and T , we let S ⊗ T denote the scenario that represents hav-ing access to S and T in parallel, so that a joint measurement is possible precisely if itscomponents in S and T are.An empirical model over a scenario S is given by specifying for each compatible σ ∈ Σ S a joint probability distribution e σ for measurements in σ . We only consider empirical modelsfor which the behavior of a joint measurement does not depend on what is measured withit, if anything. Thus, whenever τ ⊂ σ ∈ Σ, the distribution e τ over τ can be obtained bymarginalizing e σ to τ . We express this generalization of the usual no-signalling conditionsas e σ | τ = σ , so that more generally for a joint distribution d over outcomes of Y ⊂ X S and Z ⊂ Y , the expression d | Z denotes the marginal distribution on outcomes of Z .If the impossibility of measuring everything together is only a practical limitation, onecan contemplate the distribution d that would arise when measuring X S . If d explained themodel e we have at hand, we would expect it to satisfy d | σ = e σ for every σ ∈ Σ. If sucha distribution exists, we call e non-contextual . If no such distribution exists, we call e contextual , as we have reason to believe that is infeasible in principle to measure everythingtogether, unless one accepts that the observed joint distribution for a subset Y ⊂ X dependson what it is measured with i.e. its context.A simple example of scenario and a contextual empirical model on it, discussed in [43], isgiven by three measurements, any two of which are compatible but not all three together.Each measurement takes outcomes in { , } , and whenever two measurements are performedone observes (0 ,
1) and (1 ,
0) with equal probability, with the probability distributions forsingletons fixed by this and resulting in observing the two outcomes with equal probability.Too see that this is contextual, note that there is no joint outcome for all three measurementsthat is consistent with the observed marginals.More examples can be obtained from scenarios studied in non-locality, where one typicallyspecifies a scenario by giving the number of parties, the number of measurements availableto each of them and the size of the outcome sets, where it is then understood that maximal4 : S. . .. . . e : T. . .. . .
FIG. 1. Depiction of a deterministic simulation d → e compatible measurements are given by a choice of a single measurement by each party. Thisincludes for instance the famous CHSH model [44] and the PR-box [20] which goes beyondwhat is allowed in quantum mechanics, both models arising in the scenario with two partieshaving access to two dichotomic measurements.We now move on to transformations between scenarios and empirical models. We willbuild up to “wirings”, which in full generality capture the idea of simulating an empiricalmodel from another adaptively and with the help of non-contextual randomness. These willbe the free transformations of our resource theory, but we begin by considering the problemof building one scenario from another. A particularly simple way of building T from S , isby declaring that for each measurement x ∈ X T of T some measurement π ( x ) ∈ X S of S is to be performed instead. Moreover, each outcome o of π ( x ) is to be interpreted as theoutcome α x ( o ) instead. If for each compatible σ ∈ Σ S the corresponding measurement π ( σ )is jointly compatible in S , the pair h π, α = ( α x ) x ∈ X i describes a way of building T from S ,and we will denote this by writing h π, α i : S → T .Given such h π, α i : S → T , any empirical model e : S vs e : S induces a model h π, α i ∗ ( e ) : T , which describes the statistics one would see if one was to observe the statistics givenby e and then transform them according to h π, α i . The pair h π, α i is defined to be a deterministic simulation of e : T from d : S , denoted by h π, α i : d → e , precisely when h π, α i transforms d to e , i.e. if h π, α i ∗ ( d ) = e .There are two ways in which these deterministic simulations are weaker than one wouldwant in a general resource theory of contextuality: first of all, one might want to allow theusage of auxiliary (non-contextual) randomness, so that the dependence of measurementsand their outcomes of T on those of S is stochastic. The usage of auxiliary randomness iscaptured in [45] and discussed further in [46], and in our terms could be defined by allowing5robabilistic mixtures of transformations. However, this viewpoint leaves out another im-portant generalization—namely the possibility that a single measurement in T can dependon a joint measurement of S as in [47] or more generally on a measurement protocol on S that chooses which joint measurement to perform adaptively . The most general formula-tion of this idea would allow a measurement x in T to be simulated by a probabilistic and adaptive procedure, that first measures (depending on some classical randomness) some-thing in S , then based on the outcome (and possibly further classical randomness) chooseswhat to measure next (if anything) and so on. This idea is formalized carefully in [32] intwo stages, the first one adding adaptivity and the second adding randomness.To model adaptivity, one builds from a scenario S a new scenario MP ( S ) of (deterministic) measurement protocols over S . A measurement protocol is a procedure that, at anystage, either stops and reports all of the the measurement results obtained so far, or, basedon previously seen outcomes, performs a measurement in S that is compatible with all of theprevious measurements. The measurements of MP ( S ) are given by such protocols over S , anda set of measurement protocols is compatible if they can be performed jointly without havingto query measurements outside of Σ S . Then one can define adaptive (but still deterministic)transformations S → T between scenarios as deterministic transformations MP ( S ) → T as inFigure 2. In [32] we show that the assignment S MP ( S ) defines a comonad on the categoryof scenarios: this abstract language is not needed here, but can be thought of as guaranteeingthat one has a well-behaved way of composing MP ( S ) → T with MP ( T ) → U to obtain a map MP ( S ) → U . Intuitively, the composite is obtained by first interpreting each measurement in U as a measurement protocol over T , and each measurement in that measurement protocol asan measurement protocol over S , and then “flattening” the resulting measurement protocolof measurement protocols (i.e. a measurement of MP ( MP (( S ))) into a measurement protocolin S .To model (non-contextual) randomness, one then defines a simulation d → e to be adeterministic simulation MP ( d ⊗ c ) → e for some non-contextual model c as in Figure 3.Again, such simulations can be shown to compose in a well-behaved manner [48].We will denote the existence of such a simulation d → e by d e , intended to be read as“ d simulates e ”. Simulations as thus defined interact well with contextuality. For instance, • [32, Theorem 21] states that the non-contextual fraction, studied in [12] is a monotone:that is, if d e then NCF ( d ) ≤ NCF ( e ).6 : S. . .. . . e : T. . .. . . MP FIG. 2. Depiction of an adaptive simulation d → ed : S. . .. . . c : U. . .. . . e : T. . .. . . MP FIG. 3. General simulation d → e , where we require c to be non-contextual. • [47, Theorem 4.1] implies that an empirical model is non-contextual iff it can be sim-ulated from a trivial model. In fact, the notions of logical and strong contextualitycan be captured along similar lines [49], by relaxing the equality of probability distri-butions in the definition of simulation by equality (or inclusion) of supports of thesedistributions.We can now state our main result for this resource theory of contextuality. Theorem 1. If d ⊗ e d ⊗ f in the resource theory of contextuality, then e f . The key ideas of the proof are simple, even if the full details get technical: if d can catalyzea transformation e → f once, it can do so arbitrarily many times. Choosing a big enoughnumber of copies of e to transform to copies of f , using the pigeonhole principle then endsup guaranteeing that one only needs a compatible subset of S d (or rather, a compatible setof measurement protocols). Making this precise requires formalizing our framework morecarefully, and we do this in the Appendix. Our result subsumes [32, Theorem 22], as setting d = f and letting e be a trivial model implies that if d d ⊗ d the model d must benon-contextual. 7 o-catalysis for non-locality We know explain how to interpret non-locality within thisframework. At the level of scenarios and models on them, non-locality can be seen as aspecial case of contextuality: for non-locality, the measurement scenario typically arisesby considering n parties, with the i -th party choosing one measurement from a set X i ofmeasurements available to them, with a measurement x ∈ X i giving outcomes in someoutcome set O i,x . Often one restricts the situation even further and assumes that each partyhas the same number of measurements available and each measurement takes outcomes in aset of the same size, so that the scenario is specified (up to isomorphism) by three numbers:the number of parties, the number of measurements available to each of them and the size ofthe outcome sets. Whether or not one imposes this further restriction, such scenarios are ofthe form S = N ni =1 S i where each S i is just a discrete set of measurements (so only singletonmeasurements are possible in each S i ). In particular, a maximal measurement correspondsto a choice ( x , . . . x n ) of a measurement at each site, and a set of correlations can be givenas a family p ( o , . . . o n | x , . . . x n ) of conditional probabilities for each such measurement. Ifthe family p is (fully) no-signalling, it corresponds to a unique empirical model e : S (wherethe probabilities over non-maximal measurements are obtained by marginalization), and p is local iff e is non-contextual.However, if we allow parties to share different non-local resources, we move away fromthe situation where each party chooses one measurement from a set of mutually exclusivemeasurements. For instance if Bob shares one box with Alice and one with Charlie, Bob isnot limited to a single measurement: he can choose one for each box. As quantum theoryallows for arbitrary joint measurability graphs in the case of PVMs [50] and arbitrary simpli-cial complexes in the case of POVMs [51], we impose no restrictions on the measurementscenarios available at each site. However, if we are thinking about a n -partite scenario, it isreasonable to expect that measurement choices done at one site do not affect the measure-ments available at another one. Thus we model an n -partite scenario as a tuple ( S i ) ni =1 ofscenarios, thought of as representing the n parties sharing the scenario N ni =1 S i .At the level of transformations between models and scenarios, non-locality is no longera special case of contextuality, as observed for example in [30, Appendix A.1]. This isbecause our wirings are slightly too general as they allow the i -th party to wire some of theirmeasurements to measurements belonging to other parties, whereas operationally speakingit is reasonable to require each party to have access only to measurement available to them8and shared randomness). Nevertheless, the resource theories are very closely related, asone can obtain the resource theory for n -partite non-locality from that of contextuality viaa general construction, used in the context of cryptography in [33], that builds a resourcetheory of n -partite resources from a given resource theory.In our setting, this amounts to defining the resource theory of n -partite non-localityas follows: an n -partite scenario is an n -tuple ( S i ) ni =1 of scenarios (some of which may beempty), and an n -partite empirical model e : ( S i ) ni =1 is an empirical model on N ni =1 S i . Theparallel composite ⊠ of scenarios is defined pointwise, i.e. by setting ( S i ) ni =1 ⊠ ( T i ) ni =1 =( S i ⊗ T i ) ni =1 . To define ⊠ for two n -partite models e : ( S i ) ni =1 and d : ( T i ) ni =1 , note thatthe scenarios ( N ni =1 S i ) ⊗ ( N ni =1 T i ) and N ni =1 ( S i ⊗ T i ) are canonically isomorphic so thatthe model e ⊗ d : ( N ni =1 S i ) ⊗ ( N ni =1 T i ) induces a model e ⊠ d on N ni =1 ( S i ⊗ T i ) via thisisomorphism. Finally, an n -partite simulation d → e is defined as an n -tuple of simulations( MP ( T i ⊗ P i → S i )) that, when taken together transform d ⊠ c to e where c is some non-contextual shared correlation. In this manner the resource theory of n -partite non-localityis derived from that of contextuality by keeping track of the n -partite nature of scenariosand transformations between them.Our proof of Theorem 1 readily implies that non-locality admits no catalysts. Theorem 2. If d ⊠ e d ⊠ f in the resource theory of n -partite non-locality, then e f .Quantum-assisted transformations and beyond We briefly discuss a further generaliza-tion of our main result. One could consider resource theories with even more expressivesimulations than the one’s defined above. In the above, we define simulations as determin-istic simulations assisted by non-contextual randomness. As suggested in [32], one couldallow more general correlations to be used in transformations, and study e.g. quantum-assisted simulations. More specifically, one could define quantum-assisted simulations e → f as deterministic simulations MP ( e ⊗ q ) → f where q is a quantum-realizable empiricalmodel. More generally, for any class X of empirical models that is closed under ⊗ one getsa well-defined notion of X -assisted simulations [33], and we will write d X e to denote theexistence of an X -assisted simulation from d to e . If the class X contains all non-local corre-lations, the resulting resource theory has no catalysts. A similar result holds for X -assisted n -partite transformations between n -partite correlations, but we contend ourselves to statethe theorem for contextuality. 9 heorem 3. Let X be a class of empirical models that is closed under ⊗ and contains allclassical correlations. Then d ⊗ e X d ⊗ f implies e X f . For instance, if we take X to be the class of quantum-realizable empirical models, thisresult implies that, if by some miracle we got access to a single PR-box, we couldn’t use itto catalyze a quantum-assisted transformation that was hitherto impossible. Put anotherway, no matter what class of correlations we can use freely, the only way to get mileage outof a box that goes beyond our powers is to spend it—so we must choose wisely. ACKNOWLEDGMENTS
We wish to thank Marcelo Terra Cunha for suggesting the problem, and for EhtibarDzhafarov for organizing the Quantum Contextuality in Quantum Mechanics and Beyondworkshop in 2019, where we met Marcelo and got interested in this question. We arealso thankful for Samson Abramsky, Rui Soares Barbosa and Shane Mansfield for helpfuldiscussions.
APPENDIX
In this Appendix we provide proofs of our main theorems and some technical detailsconcerning the background that is needed in order to formalize these. The background isformalized in section A, and mostly explains the framework of [32]. The proofs take placein Appendix B, where we first prove Theorem 1 in B 1 and then extend the result to proveTheorem 2 in B 2 and Theorem 3 in B 3.
Appendix A: Background
We mostly follow the development of [32] with some notational changes. For a scenario S and subset Y ⊂ X S of measurements, we let E S ( Y ) (or E ( Y ) when S is clear fromcontext) denote the set of possible joint outcomes for the (not necessarily comeasurable) set Y . Formally, E S ( Y ) is defined as the cartesian product Q x ∈ Y O S,x .We now define empirical models more carefully. Given a probability distribution d on E S ( Z ), which we think of as a joint probability distribution for measurements in Z , and a10ubset Y ⊂ Z , we denote by d | Y the marginalization of d to measurements in Y . Now, anempirical model e : S consists of a family ( e σ ) σ ∈ Σ , where each e σ is a probability distributionover the set E ( σ ), and whenever τ ⊂ σ ∈ Σ we have e σ | τ = σ The parallel composite S ⊗ T of S and T is defined by X S ⊗ T = X S ⊔ X T , Σ S ⊗ T = { σ ⊔ τ | σ ∈ Σ S , τ ∈ Σ T } and setting the outcome set at x ∈ X S ⊔ X T to be O S,x if x ∈ X S and O T,x if x ∈ X T . Here X ⊔ Y denotes the disjoint union of the sets X and Y , and canbe defined e.g. as X ⊔ Y := X × { } ∪ Y × { } .A deterministic procedure S → T consists of a • A simplicial function π : X T → X S , i.e. a function π satisfying π ( σ ) ∈ Σ S for every σ ∈ Σ S • A family α = ( α x : O S,π ( x ) → O T,x ) x ∈ X of functions between outcome sets.A procedure h π, α i : S → T induces a mapping between empirical models so that each e : S is pushed forward to h π, α i ∗ e : T . Mathematically, the empirical model h π, α i ∗ e is definedat measurement σ ∈ Σ T by h π, α i ∗ e ) σ = α ∗ ( e πσ )where the right hand side denotes the pushforward of the probability distribution e πσ alongthe function E S ( π ( σ )) → E T ( σ ) whose x -th coordinate projection is given by E S ( π ( σ )) →E S ( π ( x )) α x −→ E T ( x ).If one imagines S as corresponding to some particular experimental setup, and a model e : S as corresponding to empirically observed propensities of outcomes for a fixed statepreparation, then one can interpret a procedure h π, α i : S → T as a way of (deterministically)building the experimental setup T out of S : the map π tells what measurement in S eachmeasurement in T corresponds to, and α tells how to interpret outcomes in S as outcomes in T . Then the model h π, α i ∗ e describes the statistics one would see if one was to observe thestatistics given by e but transform them according to h π, α i . In other words, the probability h π, α i ∗ e gives to some fixed outcome over the joint measurement σ ∈ Σ T is the sum theprobabilities e gives to all outcomes of π ( σ ) that are mapped to s by α .A deterministic simulation d → e where d : S and e : T is a deterministic procedure h π, α i : S → T that transforms d to e , i.e. that satisfies h π, α i ∗ e = d .11e now formalize measurement protocols carefully. These were used in [38, Appendix D]to relate the sheaf-theoretic approach [31] to contextuality to the hypergraph approach [38],and then later used in [32] to extend the former. Intuitively, a (deterministic) measurementprotocol is a set of rules that tells at each stage, what to measure next given the previousmeasurements and their outcomes. At any stage of the measurement protocol, we thushave a sequence ( x i , o i ) ni =1 of measurement-outcome pairs. A technical insight of [32] is thatmeasurement protocols can be defined in terms of sets of such sequences: if you know all suchsequences that could happen during a deterministic protocol, you also know the protocolitself. Definition 4. A run on a measurement scenario S is a sequence ¯ x := ( x i , o i ) ni =1 such that x i ∈ X S are distinct, { x , . . . , x n } ∈ Σ S , and each o i ∈ O S,x i . We denote the empty run byΛ. A run ¯ x determines a context σ ¯ x := { x , . . . , x n } ∈ Σ S and a joint assignment on thatcontext s ¯ x : : x i o i ∈ E ( σ ¯ x ). Two runs ¯ x and ¯ y are said to be consistent if they agreeon common measurements, i.e. for every z ∈ σ ¯ x ∩ σ ¯ y we have s ¯ x ( z ) = s ¯ y ( z ).Given runs ¯ x and ¯ y , we denote their concatenation by ¯ x · ¯ y . Note that ¯ x · ¯ y might not bea run. Definition 5. A measurement protocol on S is a non-empty set Q of runs satisfying thefollowing conditions:(i) if ¯ x · ¯ y ∈ Q then ¯ x ∈ Q ;(ii) if ¯ x · ( x, o ) ∈ Q , then ¯ x · ( x, o ′ ) ∈ Q for every o ′ ∈ O S,x ;(iii) if ¯ x · ( x, o ) ∈ Q and ¯ x · ( x ′ , o ′ ) ∈ Q , then x = x ′ .Identifying a measurement protocol with its set of possible runs, the first condition guar-antees that a prefix of a possible run is a possible run. The second condition ensures that,if x might be measured at some stage, then any outcome of x is an (in-principle) possibleoutcome at that stage, and the third condition guarantees that at any given stage, the nextmeasurement (if any), is uniquely determined. Definition 6.
Given a scenario S , we build a scenario MP ( S ) as follows:12 its set of measurements is the set MP ( S ) of measurement protocols on S ; • the outcome set O Q of a measurement protocol Q ∈ MP ( S ) is its set of maximal runs,i.e. those ¯ x ∈ Q that are not a proper prefix of any other ¯ y ∈ Q ; • a set { Q , . . . Q n } of measurement protocols is compatible whenever for any choice ofpairwise consistent runs ¯ x i ∈ Q i with i ∈ { , . . . , n } , we have S i σ ¯ x i ∈ Σ.Given an empirical model e : S , we define the empirical model MP ( e ) : MP ( S ) as fol-lows. For a compatible set σ := { Q , . . . , Q n } of measurement protocols and an assignment s : : Q i ¯ x i ∈ E ( σ ), we set MP ( e ) σ ( s ) := e S i σ ¯ xi ( ∪ i s ¯ x i ) if { ¯ x i } pairwise consistent0 otherwise.The intuition behind a set of measurement protocols being compatible is that they canbe performed together, without ever having to perform a measurement not allowed by Σ S .We next make this intuition precise. Definition 7.
We say that a measurement protocol P contains implicitly a protocol Q ,if any outcome of P determines the outcome of Q , and we denote this by P (cid:23) Q . Formally, P (cid:23) Q if for any maximal run ¯ x ∈ P there is a (necessarily unique) maximal run ¯ y ∈ Q suchthat the assignment s ¯ y determined by ¯ y is a restriction of s ¯ x . We say that a measurementprotocol P contains implicitly a set of protocols { Q , . . . Q n } if P (cid:23) Q i for each i . Lemma 8. If { Q , . . . Q n } is a compatible set of measurement protocols, there is a measure-ment protocol P that implicitly contains them all.Proof. We define a protocol P that first performs Q , then (whatever is left of) Q and soon. Formally, we first define inductively a merge operation ∗ for compatible runs:¯ x ∗ Λ := x ¯ x ∗ (( y, o ) · ¯ y ) := ¯ x ∗ ¯ y if y ∈ σ ¯ x (¯ x · ( y, o )) ∗ ¯ y otherwise.We extend ∗ to all pairs of runs by setting ¯ x ∗ ¯ y = Λ whenever ¯ x and ¯ y are not compatible.13e then define P by taking the closure of the set (cid:8) ¯ x ∗ · · · ∗ ¯ x n | ¯ x i ∈ O MP ( S ) ,Q i (cid:9) under prefixes. Now property (i) is true by construction, and properties (ii) and (iii) followfrom each Q i being a measurement protocol Finally, each outcome of P is of the form¯ x ∗ · · · ∗ ¯ x n , so that P (cid:23) Q i for each Q i . Remark. If P is a measurement protocol over S and we fix the outcomes of some subset Y ⊂ X S of measurements to equal t ∈ E ( Y ), this determines a measurement protocol P ( t ) that does not measure anything in Y as follows: it proceeds exactly as P except thatwhenever P is supposed to measure some x ∈ Y it behaves as if t ( x ) was observed andproceeds accordingly. In particular, if P is a measurement protocol over S ⊗ T and t is anelement of E T ( X T ), then P ( t ) is a measurement protocol over S .We can describe P ( t ) more formally as follows. Let us say that a run ¯ x ∈ P is compatiblewith t ∈ E ( Y ) if s ¯ x agrees with t on common measurements, i.e. if s ¯ x ( z ) = t ( z ) for every z ∈ σ ¯ x ∩ Y . Given a run ¯ x and t ∈ E ( Y ), we define ¯ x \ t to be the run that omits measurementsin Y . Formally, we can define this inductively on ¯ x by settingΛ \ t = Λ(¯ x · ( y, o )) \ t := ¯ x \ t if y ∈ Y (¯ x \ t ) · ( y, o )) otherwise.Then we can define P ( t ) by P ( t ) := { ¯ x \ t | ¯ x ∈ P is compatible with t } Appendix B: Proofs1. Proof of Theorem 1
Consider models d : S , e : T and f : U and an adaptive simulation d ⊗ e → d ⊗ f . Wewish to produce a simulation e → f . Heuristically speaking, our overall strategy is to showthat the composite d ⊗ e → d ⊗ f → f needs to use d only in a non-contextual manner.More precisely, we show that we only need a compatible set of measurement protocols over14 to carry out this simulation. There are several compatible sets one could use for this: wechoose one that works both for the resource theory of contextuality and for non-locality.We first reduce to the deterministic case: by definition, there is a deterministic map d ⊗ e ⊗ c → d ⊗ f with c non-contextual. Let us set g := e ⊗ c and denote the scenarioof e ⊗ c by R . We then have a deterministic map d ⊗ g → d ⊗ f . Now, if d can catalyzethe transformation g → f once without getting spent in the process, it can do so arbitrarilymany times by first catalyzing the first copy of g to f , then the second one and so on. Thisresults in a deterministic simulation d ⊗ g ⊗ n → d ⊗ f ⊗ n for any n with the property thatsimulating the i -th copy of f uses only d and the i -th copy of g .Thus the composite d ⊗ g ⊗ n → d ⊗ f ⊗ n → f ⊗ n can be equivalently described by n deterministic simulations d ⊗ g → f with the property that they can be run in parallel witha single copy of d to obtain a valid simulation d ⊗ g ⊗ n → f ⊗ n . As there are only finitelymany deterministic simulations d ⊗ g → f , we can force as many as we wish to coincideby choosing a large enough n . This implies that there is a fixed deterministic simulation d ⊗ g → f that yields a simulation d ⊗ g ⊗ n → f ⊗ n for any n .Consider now a fixed measurement x and the measurement protocol π ( x ) over S ⊗ R used to simulate it. Enumerate E R ( X R ) as t , . . . t k . Since all the copies of of x in U ⊗ n are compatible with each other, so are their images under the simulation d ⊗ g ⊗ n → f ⊗ n .When simulating different copies of x one queries a single copy of d but different copies of g , this means that whatever results one gets in g the measurements done in d are alwayscompatible. This implies that { π ( x )( t i ) | i = 1 , . . . k } is a compatible set of measurementprotocols, where π ( x )( t i ) is defined as in Remark A. Thus they can be combined to a singlemeasurement protocol P x by Lemma 8. In particular, when simulating the i -th copy of x we can first measure P x obtaining an outcome s , and then perform π ( x )( s ) (which is ameasurement protocol over R ) in the i -th copy of R . We can then replace d by the restrictionˆ d of MP ( d ) to MP ( S ) | { P x | x ∈ X U } . Moreover, considering distinct measurements x belong todifferent copies of f : U , we see that this set has to be a compatible set of measurementprotocols. As the set { P x | x ∈ X U } is comeasurable, the model ˆ d is non-contextual, so thatthe deterministic simulation ˆ d ⊗ g = ˆ d ⊗ e ⊗ c → f gives a probabilistic simulation e → f asdesired. 15 . Proof of Theorem 2 We now show how the above proof implies that there are no catalysts in the resourcetheory of non-locality. Indeed, assume a simulation ( π i , α i ) ni =1 : d ⊠ e → d ⊠ f in the resourcetheory of non-locality. Then N ( π i , α i ) defines a simulation d ⊠ e → d ⊠ f in the resourcetheory of contextuality. Pre- and postcomposing by the canonical isomorphisms( n O i =1 S i ) ⊗ ( n O i =1 T i ) ∼ = n O i =1 ( S i ⊗ T i )then implies that we have a simulation d ⊗ e → d ⊗ f , so that the proof gives us a simulation( π ′ , α ′ ) : e → f . However, we must check that this map is of the form N ( π ′ i , α ′ i ) in order forit to give a simulation e → f in the resource theory of non-locality. That this is true followsby inspecting the preceding proof. In fact, when simulating a measurement x in the T oneperforms a single measurement protocol in S that captures everything one might need whenmeasuring x , and then proceeds to R . If x is a measurement at the i -th measurement site,so is this “everything” and the protocol following it in R , so that the constructed simulation e → f is indeed of the required form.
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