For published version see PNAS 118 (17) e2020569118 (2021)

Violations of locality and free choice are equivalent resources in Bell experiments

Pawel Blasiaka,b,1, Emmanuel M. Pothosb, James M. Yearsleyb, Christoph Gallusc, and Ewa Borsuka

aInstitute of Nuclear Physics Polish Academy of Sciences, PL-31342 Krakow, Poland; bCity, University of London, London EC1V 0HB, United Kingdom; cTechnische Hochschule Mittelhessen, D-35390 Gießen, Germany

Bell inequalities rest on three fundamental assumptions: realism, lo- Surprisingly, nature violates Bell inequalities (8–15) which cality, and free choice, which lead to nontrivial constraints on cor- means that if the standard causal (or realist) picture is to be relations in very simple experiments. If we retain realism, then vio- maintained at least one of the remaining two assumptions, that lation of the inequalities implies that at least one of the remaining is locality or free choice, has to fail. It turns out that rejecting two assumptions must fail, which can have profound consequences just one of those two assumptions is always enough to explain for the causal explanation of the experiment. We investigate the ex- the observed correlations, while maintaining consistency with tent to which a given assumption needs to be relaxed for the other the causal structure imposed by the other. Either option poses to hold at all costs, based on the observation that a violation need a challenge to deep-rooted intuitions about reality, with a not occur on every experimental trial, even when describing corre- full range of viable positions open to serious philosophical lations violating Bell inequalities. How often this needs to be the dispute (16–18). Notably, quantum theory in its operational case determines the degree of, respectively, locality or free choice formulation does not provide any clue regarding the causal in the observed experimental behavior. Despite their disparate char- structure at work, leaving such questions to the domain of acter, we show that both assumptions are equally costly. Namely, the interpretation. It is therefore interesting to ask about the resources required to explain the experimental statistics (measured extent to which a given assumption needs to be relaxed, if we by the frequency of causal interventions of either sort) are exactly insist on upholding the other one (while always maintaining the same. Furthermore, we compute such defined measures of lo- realism). In this paper, we seek to compare the cost of locality cality and free choice for any nonsignaling statistics in a Bell exper- and free choice on an equal footing, without any preconceived iment with binary settings, showing that it is directly related to the conceptual biases. As a basis for comparison we choose to amount of violation of the so-called Clauser-Horne-Shimony-Holt in- measure the weight of a given assumption in terms of the equalities. This result is theory independent as it refers directly to following question: the experimental statistics. Additionally, we show how the local frac- How often a given assumption, i.e. locality or free choice, can tion results for quantum-mechanical frameworks with infinite num- be retained, while safeguarding the other assumption, in order ber of settings translate into analogous statements for the measure to fully reproduce some given experimental statistics within a of free choice we introduce. Thus, concerning statistics, causal ex- standard causal (or realist) approach? planations resorting to either locality or free choice violations are This question presumes that a Bell experiment is performed fully interchangeable. trial-by-trial and the observed statistics can be explained in the standard causal model (or hidden variable) framework (1– locality free choice causality Bell inequalities | | | | 7, 19–21), which subsumes realism. It means that the remaining measure of locality and free choice two assumptions of locality and free choice translate into condi- tional independence between certain variables in the model, "I would rather discover one true cause than gain the kingdom of Persia." – Democritus (c. 460-370 BC) Significance Statement he study of experimental correlations provides a win- Faced with a violation of Bell inequalities, a committed realist Tdow into the underlying causal mechanisms, even when might pursue an explanation of the observed correlations on their exact nature remains obscured. In his seminal works (1– the basis of violations of the locality or free choice (sometimes arXiv:2105.09037v1 [quant-ph] 19 May 2021 5), John Bell showed that seemingly innocuous assumptions called measurement independence) assumptions. The ques- about the structure of causal relationships leave a mark on tion of whether it is better to abandon (partially or completely) the observed statistics. The first assumption, called realism (or locality or free choice has been strongly debated since the counterfactual definiteness), presents the worldview in which inception of Bell inequalities, with ardent supporters on either physical objects and their properties exist, whether they are side. We offer a comprehensive treatment that allows a compar- observed or not. Note that realism allows a standard notion of ison of both assumptions on an equal footing, demonstrating a causality (6, 7), which in turn provides us with the language to deep interchangeability. This advances the foundational debate express the remaining two assumptions. The locality assump- and provides quantitative answers regarding the weight of each tion is a statement that physical (or causal) influences prop- assumption for causal (or realist) explanations of observed agate in accord with the spatio-temporal structure of events correlations. (i.e., neither backward in time nor instantaneous causation). P.B. designed and performed the research; P.B, E.M.P, J.M.Y. C.G. and E.B. discussed the results The free choice assumption asserts that the choice of measure- and wrote the paper.

ment settings can be made independently from anything in the The authors declare no competing interests. (causal) past. These three assumptions are enough to derive testable constraints on correlations called Bell inequalities. 1To whom correspondence should be addressed. E-mail: [email protected]

www.pnas.org/cgi/doi/10.1073/pnas.2020569118 PNAS 118 (17) e2020569118 (2021) | 1–10 whose causal structure is determined by their spatio-temporal relations (6, 22) (for some alternative approach endorsing in- definite causal structures see e.g. (23, 24), or (25) for discussion of retrocausality). Modelling of the experiment implies that in each run of the experiment all variables (including unobserved or hidden ones) always take definite values and the statistics accumulates over many trials. This leaves open the possibility that the violation of the assumptions do not have to occur on each run of the experiment to explain the given statistics. We can thus put flesh on the bones of the above question and seek the maximal proportion of trials in which a given assumption can be retained, while safeguarding the other assumption, so as to fully reproduce some given statistics. In the following, Fig. 1. Summary of the results. The main Theorem1 is the backbone of the paper, consolidating both measures of locality µ and free choice µ . Theorem2 is a theory- we shall denote so defined measure of locality (safeguarding L F independent result about both measures µL and µF. It offers a concrete interpretation freedom of choice) as µL and measure of free choice (safeguard- for the amount of violation of the CHSH inequalities. Theorems3 and 4 are specific to

ing locality) as µF. Also, without stating this in every instance, the quantum-mechanical statistics stated here for measure µF. They are translations ( ) we note that in all subsequent discussion realism is assumed.* of some remarkable local fraction results µL in the literature (marked with an ∗ , There has been some previous research on this theme. A cf. (29–32)).

measure of locality analogous to µL was first proposed by Elitzur, Popescu and Rohrlich (27) to quantify non-locality in infinite number of settings, utilising existing results for the a singlet state. Note, it seems that the original idea of a bound local fraction µ , which thus translate on the newly developed for such a locality measure was expressed earlier, in (28), but a L concept of the measure of free choice µ . Fig. 1 summarises bound was not worked out. In any case, Elitzur, Popescu, and F the results in the paper. Rohrlich’s measure was dubbed local fraction (or content) and shown (with improvements in (29–31)) to vanish in the limit of an infinite number of measurement settings. A substantial Results step was made in (32) where the local fraction is explicitly Bell experiment and Fine’s theorem. Let us consider the usual calculated for any pure two-qubit state for an arbitrary choice Bell-type scenario with two parties, called Alice and Bob, of settings. We note that those results concern measure µL only playing the roles of agents conducting experiments on two for the specific case of quantum-mechanical predictions. In separated systems (whose nature is irrelevant for the argu- this paper we go beyond this framework and consider the case ment). We assume that on each side there are two possi- of general experimental statistics (see (33) for extension to the ble outcomes labelled respectively a, b = 1 and M possible ± idea of contextuality). To avoid confusion, the term local fraction measurement settings labelled respectively x, y M where ∈ for measure µL will be only used in relation to the quantum M 1, 2 , . . . , M . A Bell experiment consists of a series of case. Furthermore, we propose a similar treatment of the free trials≡ in{ which Alice} and Bob each choose a setting and make choice assumption quantified by measure µF. Natural as it may a measurement registering the outcome. After many repeti- seem, this approach has not been pursued in the literature, tions, they compare their results described by the set of M M with some other measures proposed to this effect (34–42) (all × distributions Pab xy xy , where Pab xy denotes the probability of { | } | retaining locality as a principle, but departing from the original obtaining outcomes a, b, given measurements x, y were made notion of free choice introduced by Bell (6, 22)). on Alice and Bob’s side respectively. For conciseness, following We aim to comprehensively consider the extent to which a the terminology in (5), we will call Pab xy xy a "behaviour". Note | given assumption, i.e. locality or free choice, can be preserved that without assuming anything about{ the} causal structure un- through partial violation of the other assumption. To accom- derlying the experiment any behaviour is admissible (as long plish this, we provide similar definitions and discuss on an as the distributions are normalised, i.e. ∑ , Pab xy = 1 for each a b | equal footing both measures of locality µL and free choice µF. x, y M). In particular, quantum theory gives a prescription Then, we derive the following results. First, we prove a general ∈ for calculating the experimental statistics Pab xy for each choice | structural theorem about causal models explaining any given of settings x, y M based on the formalism of Hilbert spaces. ∈ experimental statistics in a Bell experiment (for any number of It is instructive to recall the special case of two measurement settings) showing that such defined measures are necessarily settings on each side x, y M = 0, 1 for which Bell derived µ = µ equal, L F. This result consolidates those two disparate his seminal result. Briefly,∈ this can{ be expressed} by saying that concepts demonstrating their deep interchangeability. Second, any local hidden variable model with free choice has to satisfy we explicitly compute both measures for any non-signalling the following four CHSH inequalities (43) statistics in a two-setting and two-outcome Bell scenario. This enables a direct interpretation to the amount of violation of the Si 6 2 for i = 1, ... , 4 ,[ 1] Clauser-Horne-Shimony-Holt (CHSH) inequalities (43). Third, | | we consider the special case of the quantum statistics with where

*As noted, realism is subsumed in the standard notion of causality, which is implicit in the defini- S = ab + ab + ab ab ,[ 2] tion of locality and free choice (1–6). So, henceforth, referring to the standard causal framework 1 h i00 h i01 h i10 − h i11 implies the realist approach. We also remark that, although, "realism" goes under different guises S = ab + ab ab + ab ,[ 3] in the literature (e.g. "counterfactual definiteness", ’"local causality", "hidden causes", etc.), for our 2 h i00 h i01 − h i10 h i11 purposes those distinctions are irrelevant and the underlying mathematics remains the same, i.e. S3 = ab 00 ab 01 + ab 10 + ab 11 ,[ 4] it boils down to the hidden variable framework (which beyond physics is frequently referred to as h i − h i h i h i the structural causal models (7)). See (3, 26) for some discussion. S = ab + ab + ab + ab ,[ 5] 4 −h i00 h i01 h i10 h i11

2 | www.pnas.org/cgi/doi/10.1073/pnas.2020569118 Blasiak et al. with ab xy = ∑ , ab Pab xy being correlation coefficients for a provided by hidden variable models (1–5). First, a hidden h i a b | given choice of settings x, y. Interestingly, by virtue of Fine’s variables model allows a formal statement of the realism as- theorem (44, 45), this is also a sufficient condition for a non- sumption, understood to mean that properties of a physical

signalling behaviour Pab xy xy to be explained by a local hidden system exist irrespective of an act of measurement (counterfac- { | } variable model with freedom of choice (for non-signalling see tual definiteness). Second, hidden variable models provide the Eqs. (16) and (17)). causal language in which the locality and free choice assump- It is crucial to observe that, although locality and freedom tions are expressed (6, 7). The locality assumption conveys the of choice are two disparate concepts with different ramifica- requirement that the propagation of physical (or causal) influ- tions for our understanding of the experiment, they are in ences have to follow the spatio-temporal structure of events a certain sense interchangeable. If locality is dropped with (i.e., preserve the arrow of time and respect that actions at a Alice and Bob freely choosing their settings, then the boxes, by distance require time). The free choice assumption concerns

influencing one another, can produce any behaviour Pab xy xy. the choice of measurement settings which are deemed cusally | Similarly, a violation of the free choice assumption{ can} be unaffected by anything in the past (and thus it is sometimes ‡ used to reproduce any behaviour Pab xy xy, without giving up called measurement independence). Both assumptions take the | locality. It is straightforward to see{ how} this might work if one form of conditional independencies between certain variables of the two assumptions fails on every experimental trial.† in a hidden variables model. However, such a complete renouncement of assumptions To make this idea more concrete, let us consider a given

so central to our view of nature may seem excessive, espe- set of probability distributions (behaviour) Pab xy xy which | cially when the CHSH inequalities are violated only by a little describes the statistics in a Bell experiment. Without{ } loss of amount (less than the maximal algebraic bound of Si 6 4), generality, by conditioning on λ in some a priori unknown leaving room for a possible explanation of the experimental| | hidden variable space Λ, one can always write (4, 5, 7) statistics by rejecting one of the assumptions sometimes only. Pab xy = Pab xyλ Pλ xy ,[ 6] Here we assess the cost of such a partial violation by asking | ∑ | | λ Λ · how often a given assumption can be retained in order to ∈

account for a behaviour Pab xy xy . We will investigate both { | } where Pλ xy and Pab xyλ are valid (i.e. normalised) conditional cases in parallel: ( ) full freedom of choice with occasional non- | | ♠ probability distributions. The role of the hidden variable (cause locality (communication), and ( ) the possibility of retaining ♣ in the past) λ Λ, distributed according to some Pλ, is to full locality at a price of compromising freedom of choice (by provide an explanation∈ of the observed experimental statistics. controlling or rigging measurement settings) on some of the This means that at each run of the experiment the outcomes trials. We shall use the least frequency of violation, required are described by the distribution Pab xyλ with λ Λ fixed in | to model some statistics with a hypothetical simulation, as a a given trial, so that the accumulated experimental∈ statistics natural figure of merit, guided by the principle that the less Pab xy obtains by sampling from some distribution Pλ xy over the | | the violation the better. Notably, such simulations should not whole hidden variable space Λ. It is customary to say that restrict possible distributions of measurement settings Pxy. In

other words, we define a measure of locality µL as the choice of space Λ and probability distribution Pλ

along with conditional distributions Pab xyλ and Pλ xy | | [?] the maximal fraction of trials in which Alice and Bob do satisfying Eq. (6) specify a hidden variable (HV) model not need to communicate trying to simulate a given be- of a given behaviour Pab xy xy . [ ] | haviour Pab xy xy , optimised over all conceivable strategies ♠ { } { | } with freely chosen settings. Note that such a model implicitly describes the distribution of settings chosen by Alice and Bob through the standard formula Similarly, we define a measure of free choice µF as

Pxy = Pxy λ Pλ .[ 7] the maximal fraction of trials in which Alice and Bob can ∑ | λ Λ · grant free choice of settings in trying to simulate a given ∈ [ ] behaviour Pab xy xy , optimised over all conceivable local ♣ So far the framework is general enough to accommodate any { | } strategies. causal explanation of the statistics observed in the experiment. The assumptions of locality and free choice take the form of In the quantum-mechanical context the measure µ is called L constraints on conditional distributions in (?). For a local hidden a local fraction (27–32). By analogy, when considering the variable (LHV) model, we require the following factorisation§ quantum-mechanical statistics the measure µF might be called a free fraction. This provides an equal basis for comparing Pab xyλ = Pa xλ Pb yλ ,[ 8] the two assumptions within the standard causal (or realist) | | · | approach, which we formalise in the following section. for each x, y M and all λ Λ. The freedom of choice as- sumption consists∈ of requiring∈ that λ does not contain any Causal models, locality and free choice. The appropriate information about variables x, y representing Alice and Bob’s framework for the discussion of locality and free choice is ‡As noted, the free choice assumption is sometimes called measurement independence. Instead †For the simulation of a given behaviour P in a Bell experiment one may proceed as of on the agent, measurement independence is focussed on the measurement devices and pos- { ab xy}xy follows. Upon rejection of locality, in each trial| the system on Alice’s side, one may not only use sible correlations between their settings, which can affect the observed statistics. Regardless of input x but also y to generate outcomes (and similarly for the box on Bob’s side) that comply interpretation, the mathematics remains the same, with the source of correlations traced to some with the appropriate distribution. On the other hand, when freedom of choice is abandoned, both common factor (in the causal past). § settings x, y may be specified in advance on each trial and the boxes can be instructed to provide Locality can be seen as a conjunction of two conditions: parameter independence Pa xyλ = Pa xλ | | the outcomes needed to simulate the appropriate distribution. It is however unclear how this might & Pb xyλ = Pb yλ, and outcome independence Pa bxyλ = Pa xyλ & Pb axyλ = Pb xyλ. One can | | | | | | work with occasional violation of the respective assumptions. show that such defined locality entails the factorisation condition Pab xyλ = Pa xλ Pb yλ (46). | | · |

Blasiak et al. PNAS 118 (17) e2020569118 (2021) | 3 Fig. 2. Causal model with some non-locality (communication). In a Bell scenario, Fig. 3. Causal model with some freedom of choice (rigging). In a Bell scenario, with free choice of settings, correlations between Alice and Bob’s outcomes have two with locality assumption, correlations between the outcomes on Alice and Bob’s side possible explanations: common cause in the past or causal influence between the can be explained by a common cause affecting choice or not (the latter implies freedom parties. In any causal model the space of hidden variables (representing common of choice). In any causal model the space of hidden variables (representing common causes) splits into two disjoint parts Λ = Λ Λ distinguished by whether, for a causes) splits into two disjoint parts Λ = Λ Λ distinguished by whether, for a 0 0L ∪ 0NL 00 00F ∪ 00NF given λ Λ , causal influence occurs or not, Eq. (10). Then, locality is measured by given λ Λ , the choice is free or not, Eqs. (11). Then, the parties enjoy freedom of ∈ 0 ∈ 00 the proportion of events when locality is maintained, which is equal to the probability choice only on the trials when λ Λ , which happens with a frequency equal to the ∈ 00F accumulated over subset Λ0L, i.e. Prob (λ Λ0L) ∑λ Λ Pλ. probability accumulated over subset Λ00F , i.e. Prob (λ Λ00F ) ∑λ Λ Pλ. ∈ ≡ ∈ 0L ∈ ≡ ∈ 00F

place (λ Λ ) and whether some influence from the past on choice of measurement settings. This boils down to the inde- ∈ NL pendence condition (6, 22) the measurement settings occurs (λ ΛNF). In other words, in a hypothetical simulation scenario∈ these possibilities corre-

Pλ xy = Pλ (or equivalently Pxy λ = Pxy),[ 9] spond to, respectively, communication or rigging measurement | | settings. How often this has to happen depends on the distri- holding for x, y M and all λ Λ. In the following, we will ∈ ∈ bution Pλ. This picture lends itself to quantifying the degree abbreviate a hidden variable model with freedom of choice as FHV of locality and freedom choice in a given HV model. model. ? The crucial point is the distinction between local vs non-local Remark 1. For a given HV model ( ) locality is measured by Prob (λ ΛL ) ∑λ Λ Pλ, and similarly freedom of choice is as well as free vs non-free situations in the individual runs of ∈ ≡ ∈ L measured by Prob (λ ΛF ) ∑λ Λ Pλ. the experiment modelled by Eq. (6). This means that each ∈ ≡ ∈ F condition Eq. (8) and Eq. (9) should be considered separately This remark captures the intuition of measuring locality and for each λ Λ, i.e. whenever the respective condition does freedom of choice by considering the proportion of trials when not hold for∈ a given λ the assumption fails on the correspond- the respective property is maintained across the whole ex- ing experimental trials. Such a distinction leads to a natural perimental ensemble. We note that this quantity is model- splitting of the underlying HV space into two unique partitions dependent, since it is a property of a particular HV model

Λ = ΛL ΛNL and Λ = ΛF ΛNF. The first one divides Λ by adopted to explain some given experimental statistics Pab xy xy ∪ ∪ { | } the locality property (including the distribution of measurement settings Pxy, cf. Eq. (7)). λ ΛL Eq. (8) holds for all x, y M , The concepts just introduced allow a precise expression for ∈ ⇔ ∈ [10] λ Λ Eq. (8) fails for some x, y M , the informal definitions ( ) and ( ) given above. ∈ NL ⇔ ∈ ♠ ♣ Definition 1. For a given behaviour Pab xy xy the measure of local- while the second one divides Λ by the free choice property { | } ity µL and freedom of choice µF are defined as

λ ΛF Eq. (9) holds for all x, y M , µL := min max Pλ ,[ 12] ∈ ⇔ ∈ [11] Pxy FHV ∑ λ Λ Eq. (9) fails for some x, y M . λ ΛL ∈ NF ⇔ ∈ ∈ µF := min max ∑ Pλ ,[ 13] Figs. 2 and 3 illustrate the causal structures for two extreme Pxy LHV λ ΛF cases: FHV and LHV models (in general built on different ∈ where the maxima are taken respectively over all hidden variable HV spaces Λ0 and Λ00). The first one grants full freedom of models with freedom of choice (FHV) or all local hidden variable choice (Λ0 = Λ0F) while allowing for partial violation of locality models (LHV) simulating given behaviour Pab xy xy, with a fixed (Λ Λ ). The second one retains full locality (Λ = Λ ) | 0 ⊃ 0L 00 00L { } while admitting some violation of free choice (Λ Λ ). distribution of settings Pxy , minimized over any choice of the latter. 00 ⊃ 00F Thus, for a given experimental trial (with λ Λ fixed) This definition follows the intuition of, respectively, locality the constraints in Eqs. (10) and (11) indicate, respectively,∈ or free choice as properties that can be relaxed only to the ex- whether some non-local influence between the parties takes tent that is required to maintain the other assumption in every

4 | www.pnas.org/cgi/doi/10.1073/pnas.2020569118 Blasiak et al. experimental situation (i.e., for any distribution of measure- This is a general structural theorem about causal modelling ment settings Pxy). Formally, the measures µL and µF count the of a given behaviour Pab xy xy. It means that the resources | maximal frequency of, respectively, local or free choice events measured by the frequency{ } of causal interventions of either optimised over all protocols simulating Pab xy xy without vio- sort, required to explain an experimental statistics, are equally { | } lating of the other assumption, cf. Remark 1. The minimum costly. Thus, as far as the statistics is concerned, causal expla- over all Pxy amounts to the worst case scenario, which takes nations resorting either to violation of locality or free choice into account the possibility that Pxy is a priori unspecified (i.e., (or measurement dependence) should be kept on an equal foot- this amount of freedom is enough to simulate an experiment ing. Preference should be guided by a better understanding with any arbitrary choice of distribution Pxy in compliance with of a particular situation (design of the experiment as well as Eq. (7)). ontological commitments in its description). At first glance, even if conceptually appropriate, such a Let us emphasise two features of Theorem 1. First, this is definition might seem too general to provide a manageable a theory-independent result in the sense that it applies directly notion, due to the range of experimental scenarios that need to experimental statistics irrespective of the design or theo- to be taken into account (i.e. arbitrariness of Pxy). However, retical framework behind the experiment (with the quantum the situation considerably simplifies because of the following predictions being just one example). Second, the connection lemma (see Methods section for further discussion and proof). between those two seemingly disparate quantities µL and µF This lemma also provides additional support for Definition 1. has a practical advantage: knowledge of one suffices to com- Lemma 1. In both Eqs. (12) and (13) in Definition 1 the first pute the other. Both features are illustrated by the following minimum can be omitted, i.e. we have results.

µL = max Pλ ,[ 14] FHV ∑ λ Λ Non-signalling behaviour with binary settings. Consider the ∈ L case of Bell’s experiment with only two measurement settings µF = max Pλ ,[ 15] LHV ∑ λ Λ on each side x, y M = 0, 1 . Let us recall that non-signalling ∈ F ∈ { } of some given behaviour Pab xy xy means that Alice cannot infer { | } where the respective maxima are taken for some fixed nontrivial Bob’s measurement setting (whether it is y = 0 or 1) from the distribution Pxy (i.e., the expression is insensitive to this choice statistics on her side alone, i.e. provided all settings are probed, P = 0 for all x, y). xy 6 It is in this way that the present measure of locality µL Pa x0 = Pab x0 = Pab x1 = Pa x1 for all a, x ,[ 16] | ∑ | ∑ | | extends the notion of local fraction (27–32) to arbitrary exper- b b imental behaviour Pab xy xy. Remarkably, the twin concept, { | } and similarly on Bob’s side (whether Alice chooses = 0 or which is the measure of free choice µF has not been considered x at all. Perhaps the reason for this omission is the issue of arbi- 1), i.e. trariness of the distribution Pxy, for which there are non-trivial constraints when freedom of choice is violated (note that for Pb 0y = Pab 0y = Pab 1y = Pb 1y for all b, y .[ 17] | ∑ | ∑ | | a a the measure µL this problem does not occur). Those concerns can be dismissed only after the proper treatment in Lemma 1. Now we can state another result which explicitly computes This allows a so defined measure of freedom µF on a par with both measures µL and µF in a surprisingly simple form (see the more familiar measure of locality µL. So far the concepts of violation of locality and freedom of Methods for the proof). choice, and the corresponding measures µL and µF, have been Theorem 2. For a given non-signalling behaviour Pab xy xy with kept separate. This is expected given their disparate character. { | } binary settings x, y M = 0, 1 both measures of locality µ and First, each concept plays a different role in the description ∈ { } L of an experiment and hence offers a different explanation for free choice µF from Definition 1 are equal to any observed correlations, this is, direct influence (communi- 1 cation during the experiment) vs measurement dependence 2 (4 Smax ) , if Smax > 2 , µL = µF = − [18] (employing common past for rigging measurement settings). ( 1 , otherwise , Second, on the level of causal modelling those assumptions are expressed differently, Eq. (8) vs Eq. (9). Third, violating where S = max S : i = 1, ... , 4 is the maximum absolute free choice gives rise to subtle issues regarding constraints on max {| i | } value of the four CHSH expressions in Eqs. (2)-(5). the distribution of settings Pxy (as noted, these concerns are addressed in Lemma 1). We thus obtain a systematic method for assessing the degree Having brought all those issues to the spotlight, it is sur- of locality and free choice directly from the observed statistics prising that the assumption of locality and free choice are Pab xy xy without reference to the specifics of the experiment | intrinsically connected. We now present the key result in (the{ only} requirement is non-signalling of the observed dis- this paper showing the exchangeability of both concepts, tributions). In this sense, this is a general theory-independent while maintaining the same degree of locality and freedom statement. of choice so defined. It holds for any number of settings Overall, Theorem 2 allows an interpretation of the amount x, y M = 0, 1, ... , M (see Methods for the proof). ∈ { } of violation of the CHSH inequalities in Bell-type experiments Theorem 1. For a given behaviour Pab xy xy the degree of locality as a fraction of trials violating locality (granted freedom of { | } and freedom of choice are the same, i.e. both measures in Definition 1 choice) or equivalently trials without freedom of choice (given coincide µL = µF. locality).

Blasiak et al. PNAS 118 (17) e2020569118 (2021) | 5 The quantum case: Binary settings and beyond. Let us re- quantum theory. The theorems should be contrasted with strict our attention to the special case of the quantum statistics. Theorem 2 which is a theory-independent statement, not limited Notably, various aspects of non-locality have been extensively to a particular theoretical framework. researched in relation to the quantum-mechanical predictions, see (4, 5) for a review. This includes the notion of local frac- Discussion tion (27–32), which is the same as measure µL here defined The ingenuity of Bell’s theorem lies in the fundamental nature for a general behaviour Pab xy xy. As noted, it may be thus { | } of the premises from which the result is derived. Within the surprising that the equally natural measure of freedom µF has not been explored. Theorem 1 bridges the gap between those standard causal (or realist) approach, it is hard to assume less two seemingly disparate notions: there is no actual need for about two agents than having free choice and their systems separate study. We next review some crucial results for the being localised in space. Yet in some experiments nature refutes local fraction in the quantum-mechanical framework, which al- the possibility that both assumptions are concurrently true (8– lows us to make similar statements for the measure of freedom 15). It is not easy to reject either one of them without carefully µ . rethinking the role of observers and how cause-and-effect man- F ¶ We first observe that Theorem 2 can be readily applied ifests in the world. Our objective in this paper is this: instead to the quantum-mechanical statistics (where non-signalling of pondering the question of how this could be possible, we ask about holds). In a Bell experiment, quantum probabilities obtain the extent to which a given assumption has to be relaxed in order to a b maintain the other. Expressed more colloquially, it is natural for through the standard formula Pab xy = Tr [ ρ P P ] where | x y ⊗ a= 1 a realist to ask what is the cost of trading one concept for the ρ is a (bipartite) mixed state with two PVMs Px ± and b= 1 { } P ± representing Alice and Bob’s choice of measurement other: Is it possible to save free choice by giving up on some locality? { y } settings x, y M = 0, 1 . Calculating the CHSH expressions Or, maybe is it better to forego a modicum of free choice in exchange Eqs. (2)-(5) in∈ each particular{ } case is straightforward, which for locality? These questions can be compared on equal footing by computing a proportion of trials across the whole experi- gives explicitly the expression for both measures µL and µF via Eq. (18). The result of special significance concerns the famous mental ensemble in which a given assumption must fail, when QM √ the other holds at all times. Surprisingly, the answer can be Tsirelson bound Smax = 2 2 for the maximal violation of the CHSH inequalities in quantum mechanics (47). By virtue of obtained by looking at the observed statistics alone (avoiding Theorem 2, this means that in order to locally recover the the specifics of the experimental setup). The first question quantum predictions in a Bell experiment with two settings, was formulated in the quantum-mechanical context by Elitzur, Alice and Bob can enjoy freedom of choice in the worst case, Popescu and Rohrlich (27) who introduced the notion of lo- cal fraction further elaborated in (29–32) (see (28) for an early at most, with a fraction µ = 2 √2 0.59 of all trials (cor- F indication of these ideas). Here, we generalise this notion to ar- responding to the choice of measurements− ≈ on a maximally bitrary experimental statistics (see also (33)). Furthermore, we entangled state that saturate the Tsirelson bound). Clearly, the answer the second question by adopting a similar approach to same applies to local fraction µ in a two-setting scenario. L measuring the amount of free choice (which by analogy may be Interestingly, relaxing the constraint on the number of called free fraction). The first main result, Theorem 1, compares settings for Alice and Bob’s measurements x, y M = ∈ such defined measures in the general case (arbitrary statistics 1, 2, 3, ... , M the quantum statistics forces us to further con- with any number of settings), showing that both assumptions strain,{ respectively,} locality or free choice. The case of local are equally costly. This demonstrates a deeper symmetry be- fraction µ with arbitrary number of settings M ∞ has been L tween locality and free choice, which may come as a surprise, thoroughly investigated for statistics generated→ by quantum given our intuition of a profound difference in the role these states. Let us refer to two interesting results in the literature concepts play in the description of an experiment. on local fraction µ which readily translate via Theorem 1 to L In this paper, the notions of locality and free choice are the measure of freedom µ . The first one concerns the statistics F understood in the usual sense required to derive Bell’s theo- of a maximally entangled state, cf. (27, 29) (see SI Appendix rem (6, 22). They are expressed in the standard causal model for a direct proof). framework (which subsumes realism) as unambiguous yes-no Theorem 3. For every local hidden variable (LHV) model that criteria for each experimental trial (i.e. when all past variables explains the statistics of a Bell experiment for a maximally entangled are fixed), determining whether there is a causal link between state the amount of free choice tends to zero with increasing number certain variables in a model (without pondering its exact na- of measurement settings M, i.e. µF / 0 . M ∞ ture). The measures µL and µF count the fraction of trials when → Apparently, for less entangled states the amount of freedom such a connection needs to be established, breaking locality or free choice respectively, in order to explain the observed statis- increases, reaching the maximal value µF = 1 for separable states. This is a consequence of the result in (32), which explic- tics. This problem is prior to a discussion of how this actually occurs, which is particularly relevant when the exact nature of itly computes the local fraction µL for all pure two-qubit states. the phenomenon under study is obscured. Theorem 1 shows Stated for measure µF this takes the following form. no intrinsic reason for a realist to favour one assumption vs Theorem 4. For a pure two-qubit state, which by appropriate choice the other. The minimal frequency of the required causal in- of the basis can always be written in the form ψ = cos θ 00 + 2 fluences of either sort, measured by µL and µF, is exactly the θ π | i | i sin 11 with θ [0, ], the amount of freedom is equal µF = 2 | i ∈ 2 ¶ cos θ, whatever the choice and number of settings on Alice and Bob’s We note that the conventional understanding of causality and the language of counterfactuals has recently gained a solid mathematical basis; see e.g. the work of J. Pearl (7). However, in view of side. the apparent difficulties with embedding quantum mechanics in that framework, the standard ap- proach to causality based on Reichenbach’s principle or claims regarding spatio-temporal structure Note that both Theorem 3 and Theorem 4 assume a spe- of events might need reassessment; see e.g. indefinite causal structures (23, 24) or retrocausal- cific form of behaviour Pab xy xy as obtained by the rules of ity (25). { | }

6 | www.pnas.org/cgi/doi/10.1073/pnas.2020569118 Blasiak et al. same. Notably, this is a general result which holds for any such a freedom can be retained in a model strictly consistent behaviour Pab xy xy. What remains is explicit calculation of with locality. It thus benefits from a direct interpretation within | those measures{ for} a given experimental statistics. the established causal framework of Bell inequalities and has The second main result, Theorem 2, evaluates both mea- a clear-cut operational meaning. sures µL and µF for any non-signalling behaviour in a Bell Regarding Theorem 3, which rules out any freedom of experiment with two outcomes and two settings. It provides a choice so defined, it is interesting to take an adversarial per- direct interpretation to the amount of violation of the CHSH spective on the problem of free choice in relation to quantum inequalities (43). The key motivation behind this result is that cryptography and device independent certification (59, 60). In the degree by which the inequalities are violated has not been this narrative an eavesdropper controls the devices trying to given tangible interpretation so far, beyond its use as a binary simulate the quantum statistics of a Bell test, which is impossi- test of whether the inequalities are obeyed or not in study of ble as long as the parties enjoy freedom of choice. However, Bell non-locality. Furthermore, Theorem 2 has the advantage any breach of the latter, i.e. control of measurement settings, of being theory-independent in the sense of being applicable to shifts the balance in favour of the eavesdropper in her mali- the experimental statistics regardless of its theoretical origin cious task. Taking the view that any causal influence comes (i.e., beyond the quantum-mechanical framework). This makes with a cost or danger of being uncovered there are two di- it suitable for quantitative assessment of the degree of locality verging strategies that reduce the cost/risk to be considered: and free choice across different experimental situations, with (a) resort to the use of control of choice as seldom as possible prospective applications beyond physics, e.g. in neuroscience, during the experiment, or (b) minimise the intensity of each act cognitive psychology, social sciences or finance (48–52). of control. Theorem 3 completely rules out the first possibil- ity when simulating quantum statistics, i.e., the eavesdropper We also state two results, Theorem 3 and Theorem 4, for needs to manipulate both settings on each trial in order to sim- the measure of free choice µF in the case of the quantum statistics generated by the pure two-qubit states. Both are ulate the quantum statistics. The question about the intensity of the control is left open in our discussion, but amenable to direct translation, via Theorem 1, of the corresponding results for the local fraction µ (27–32). information-theoretic methods (35–42). This gives additional L security criteria for quantum cryptography and device inde- It is worth noting a related idea of quantifying non-locality pendent certification by forcing the eavesdropper to a more through the amount of information transmitted between the challenging sort of attack (not only can she not miss a trial, parties that is required to reproduce quantum correlations (un- but the control has to be subtle enough). der free choice assumption). Together with the development We remark that the main Theorem 1 readily extends to the of the specific models (53–57), this has led to various results re- case of larger number of parties and outcomes Pabc... xyz... xyz.... garding communication complexity in the quantum realm (58). { | } This should be also possible for Theorem 2 when characteri- However, in this paper we take a different perspective on mea- sation of the local polytope is known, cf. (61–67). Yet another suring non-locality by changing the question from "how much" valuable avenue for research in that case consists of completing to "how often" communication needs to be established between the analysis to include signalling scenarios (68, 69). As for the the parties to simulate given correlations. Theorem 2 gives quantum case, we considered the simplest Bell-type scenario the exact bound in the case of non-signalling statistics in the with two parties involved in the experiment, but extensions two-setting and two-outcome Bell experiment. In the quantum may prove even more surprising (see (5) for a technical review case, such a simulation requires communication in at least 41 % of the vast field of Bell non-locality). In particular, in three- of trials (because of Tsirelson’s bound (47)) and for maximally party scenarios the methods discussed presently can be used to entangled states increases to 100 % of trials when the number eliminate freedom of choice already for two settings per party of settings is arbitrary (cf. Theorems 3 and 4). sharing the GHZ state (cf. Mermin inequalities which saturate Natural as it may seem, the idea of measuring freedom of in that case (70)). We should also mention an intriguing re- choice by measure µF has not been developed in the literature. sult (71) for a triangle quantum network in which non-locality The reason for this omission can be traced to the conceptual can be proved with all measurements fixed. Remarkably, there and technical issues with handling arbitrariness of the distribu- is nothing to choose in that setup, but there is another assump- tion of settings Pxy. Those concerns are properly addressed in tion of preparation independence which plays a crucial role in the present paper with Lemma 1, which considerably simpli- the argument. fies and supports Definition 1. We note that various measures In this paper we are trying to remain impartial as to which have been developed as a means of quantifying freedom of assumption — locality or free choice — is more important on choice (or measurement independence, as it is sometimes called). the fundamental level. This is certainly a strongly debated They include maximal distance between distributions (35, 37), subject in general, both among physicists and philosophers, mutual information (38, 42) or measurement dependent lo- with strong supporters on each side (16–18). As just one cality (39–41). Furthermore, some explicit models simulating example depreciating the role of freedom of choice let us quote correlations in a singlet state with various degrees of measure- Albert Einstein||: "Human beings, in their thinking, feeling and ment dependence have been proposed (34, 36) and analysed acting are not free agents but are as causally bound as the stars (e.g. see (42) for comparison of causal vs retrocausal models). in their motion." As a counterbalance, it is hard to resist the However, these attempts depart from the original understand- objection that was eloquently stated by Nicolas Gisin (72): "But ing of the free choice as introduced by Bell (6, 22) (strict in- for me, the situation is very clear: not only does free will exist, but dependence of choice from anything in the past) in favour of it is a prerequisite for science, philosophy, and our very ability to more sophisticated information-theoretic accounts. Notably, think rationally in a meaningful way." Without entering into this the proposed measure of free choice builds on the Bell’s origi- nal framework assessing the maximal frequency with which ||Statement to the Spinoza Society of America. September 22, 1932. AEA 33-291.

Blasiak et al. PNAS 118 (17) e2020569118 (2021) | 7 debate, we remark that both assumptions are interchangeable for λ Λ . Both are LHV models with marginals ∈ NF on a deeper level. Namely, for a given experimental statistics F F F Pab xy = Pab xyλ Pλ xy ,[ 23] | ∑ | | Pab xy xy in a Bell-type experiment the measure of locality µL λ Λ · { | } ∈ F and measure of free choice µF are exactly the same. This makes NF NF NF Pab xy = ∑ Pab xyλ Pλ xy ,[ 24] an even stronger case regarding the inherent impossibility of | λ Λ | · | ∈ NF inferring causal structure from experimental statistics alone. which provide a convex decomposition of the original behaviour Pab xy xy , i.e. { | } Materials and Methods F NF Pab xy = pF Pab xy + (1 pF ) Pab xy .[ 25] | · | − · | In order to facilitate the following discussion we begin with two The crucial point is a careful adjustment of these two models to technical lemmas. See SI Appendix for the proofs. recover some arbitrary distribution of settings P˜xy, while maintaining The first one holds for a Bell experiment with arbitrary number of the respective marginals Eqs. (23) and (24). For the first one (restriction settings x, y M = 1, 2, 3, ... , M . ∈ { } to ΛF) the situation is trivial as explained above: since it is a FHV ˜ F ˜ Lemma 2. For any behaviour P and distribution of settings P model, then it suffice to redefine Pxy λ := Pxy (in compliance with ab xy xy xy | there exists a local hidden variable{ model| } (LHV) which fully violates the Eq. (20)) and leave all rest intact. As for the second one (restriction ˜ freedom of choice assumption. [i.e. if Λ˜ is the relevant HV space, then we to ΛNF), we can use Lemma 2 for constructing another HV space ΛNF with a LHV model without any free choice, that simulates behaviour have Λ˜ = Λ˜ L = Λ˜ NF, cf. Eqs. (10) and (11))]. NF ˜ Pab xy xy with the required distribution of settings Pxy. Then, such { | } The second one concerns a Bell scenario with binary settings x, y modified models can be stitched back together on the compound M = ∈ 0, 1 . HV space Λ˜ := Λ Λ˜ with respective weights p and 1 p . { } F ] NF F − F Lemma 3. Each non-signalling behaviour Pab xy xy with binary settings This guarantees reconstruction of the original behaviour Pab xy xy { | } { | } x, y M = 0, 1 can be decomposed as a convex mixture of a local (see Eq. (25)) with the new distribution of settings P˜xy. The model ∈ { } behaviour P¯ab xy xy and a PR-box P˜ab xy xy in the form is local and has the same degree of freedom equal to pF (the first { | } { | } component has full freedom of choice, while in the second one it is ¯ ˜ Pab xy = p Pab xy + (1 p) Pab xy ,[ 19] entirely missing). | · | − · | 1 The above construction shows that for every LHV model of some with p = (4 Smax ) for all x, y 0, 1 . 2 − ∈ { } behaviour Pab xy xy there is always another one adjusted for any other { | } ˜ Recall that a PR-box (73) is a non-signalling behaviour for which distribution of settings Pxy with the same degree of freedom. This one of the CHSH expressions in Eqs. (2)-(5) reaches the maximal justifies the simpler expression for µF in Eq. (15) and hence concludes algebraic bound of S = 4. Here, local behaviour means existence of the proof of Lemma 1. | i | a LHV+FHV model of P¯ab xy xy and Smax = max Si : i = 1, ... , 4 . We are now ready to{ proceed| } with the proofs.{| | } Proof of Theorem1. Note that Lemma 1 Eqs. (14) and (15) can be taken as a definition of measures µL and µF. This is very convenient, since it Proof of Lemma1. Suppose we have a HV model (?) of some behaviour allows a discussion free from any concerns about the distribution of Pab xy xy for some nontrivial distribution of settings Pxy. The latter settings Pxy (this is particularly relevant in the case of µF as explained { | } obtains via Eq. (7) from the conditional probabilities Pxy λ which are above). | related to probabilities specified by the model, Pλ xy and Pλ , by the It is instructive to observe that the calculation of both measures µL | usual Bayes’ rule. The point at issue is whether a given HV model can and µF can be succinctly formulated as a convex optimisation problem. ˜ Suppose, we can decompose some given behaviour Pab xy xy as a simulate any other distribution of settings Pxy via Eq. (7) by changing { | } ˜ mixture Pxy λ Pxy λ , while keeping the remaining components of the HV | | L NL model (?) intact. This requires consistency with Bayes’ rule, i.e. Pab xy = pL Pab xy + (1 pL ) Pab xy ,[ 26] | · | − · | P P˜xy L λ xy where P xy is a local behaviour with full freedom of choice (i.e., P˜xy λ = | · ,[ 20] ab xy | Pλ { | } NL has a LHV+FHV model), and Pab xy xy is a free behaviour (i.e., has a { | } which should be a well-defined probability distribution for each λ. FHV model). And similarly, suppose that Since distributions Pλ xy and Pλ are fixed by the HV model (?), then | F NF the distribution of settings P˜ is arbitrary as long as the expression Pab xy = pF Pab xy + (1 pF ) Pab xy [27] xy | · | − · | in Eq. (20) is less then 1 for each λ Λ (normalisation is trivially where PF is a local behaviour with full freedom of choice (i.e., fulfilled). Now, whenever freedom of∈ choice from Eq. (9) holds, this ab xy xy { | } NF condition is always satisfied, and hence such a HV model can be has a LHV+FHV model), and Pab xy xy is a local behaviour (i.e., has a { | } LHV model). In both cases we assume that 0 p , p 1, and both trivially adjusted for any distribution P˜xy (by redefining P˜xy λ := P˜xy in 6 L F 6 | Eq. (26) and Eq. (27) have to hold for all a, b = 1 and x, y M. Then, compliance with Eq. (20), and keeping all the remaining components ± ∈ of the HV model (?) unchanged). Of course, for FHV models in the we have definition of µL in Eq. (12) this is the case, which thus entails the Remark 2. Measures µL and µF evaluate the maxima over all possible simpler expression for µL in Eq. (14). decompositions in Eqs. (26) and (27) of behaviour Pab xy xy, i.e. Clearly, such a simple argument falls apart for models without { | } freedom of choice, like those in the definition of µF in Eq. (13), when µL = max pL ,[ 28] decomp. (26) Pλ xy and Pλ do not cancel out and the probability in Eq. (20) may be | µF = max pF .[ 29] ill-defined. In that case, some deeper intervention into the model is decomp. (27) required as shown below. Let us take some LHV model (?) simulating a given behaviour Proof. We will justify only Eq. (28), since the argument for Eq. (29) is Pab xy xy with nontrivial distribution of settings Pxy. Then the related analogous. { | } HV space decomposes as Λ = ΛF ΛNF and the degree of freedom Let us observe that every HV model (?) of behaviour Pab xy xy as ] { | } is measured by pF := ∑λ Λ Pλ , cf. Remark 1. Now, consider a described by Eq. (6) splits into two components (cf. Eq. (10)) ∈ F restriction of the model to the respective subspaces ΛF and ΛNF which Pab xy = Pab xyλ Pλ + Pab xyλ Pλ xy ,[ 30] amounts to the following rescaling | ∑ | ∑ | | λ Λ · λ Λ · ∈ L ∈ NL F 1 F 1 F Pλ := p Pλ , Pλ xy := p Pλ xy , Pab xyλ := Pab xyλ ,[ 21] F | F | | | p PL (1 p ) PNL L ab xy L ab xy · | − · | for λ ΛF, and similarly | {z } | {z } ∈ which defines decomposition of the type in Eq. (26) with pL := NF 1 NF 1 NF Pλ := 1 p Pλ , Pλ xy := 1 p Pλ xy , Pab xyλ := Pab xyλ ,[ 22] ∑λ Λ Pλ. Therefore, by Eq. (14), we get µL 6 maxdecomp. (26) pL. − F | − F | | | ∈ L

8 | www.pnas.org/cgi/doi/10.1073/pnas.2020569118 Blasiak et al. To see the reverse, we note that every decomposition of the type in 10. LK Shalm, et al., Strong Loophole-Free Test of Local Realism. Phys. Rev. Lett. 115, 250402 L (2015). Eq. (26) implies existence of a LHV+FHV model of behaviour Pab xy xy { | } ˜ NL 11. B Hensen, et al., Loophole-free Bell inequality violation using electron spins separated by 1.3 on some HV space ΛL and a FHV model of behaviour Pab xy xy on kilometres. Nature , 682 (2015). { | } 526 some HV space Λ˜ NL. Those two models, when combined on a com- 12. A Aspect, Closing the Door on Einstein and Bohr’s Quantum Debate. Physics 8 (2015). 13. J Gallicchio, AS Friedman, DI Kaiser, Testing Bell’s Inequality with Cosmic Photons: Closing pound HV space Λ := Λ˜ L Λ˜ NL with the respective weights pL and ] the Setting-Independence Loophole. Phys. Rev. Lett. 112, 110405 (2014). 1 pL , provide a HV model of behaviour Pab xy xy. Since the local − ˜ { | } 14. D Rauch, et al., Cosmic Bell Test Using Random Measurement Settings from High-Redshift domain of such a model contains ΛL, then from Eq. (14) we have Quasars. Phys. Rev. Lett. 121, 080403 (2018). µL > pL , which entails µL > maxdecomp. (26) pL. This concludes the proof 15. C Abellán, et al., Challenging local realism with human choices. Nature 557, 212 (2018). of Eq. (28). 16. T Maudlin, Philosophy of Physics: Quantum Theory. (Princeton University Press), (2019). 17. F Laloë, Do we really understand quantum mechanics? (Cambridge University Press), 2nd Now, in order to prove Theorem 1 it is enough to show that for ev- edition, (2019). ery decomposition of the type in Eq. (26) there exists a decomposition 18. T Norsen, Foundations of Quantum Mechanics: An Exploration of the Physical Meaning of Quantum Theory, Undergraduate Lecture Notes in Physics. (Springer), (2017). of the type in Eq. (27) with the same weight pL = pF, and vice versa. A closer look at both expressions reveals that behaviours PL 19. CJ Wood, RW Spekkens, The lesson of causal discovery algorithms for quantum correlations: ab xy xy causal explanations of Bell-inequality violations require fine-tuning. New J. Phys. 17, 033002 F { | } and Pab xy xy are both local with full freedom of choice (i.e., share (2015). { | } the same LHV+FHV model). Thus, the problem can be reduced to 20. R Chaves, R Kueng, JB Brask, D Gross, Unifying Framework for Relaxations of the Causal NL Assumptions in Bell’s Theorem. Phys. Rev. Lett. 114, 140403 (2015). justifying that: (a) behaviour Pab xy xy also has a LHV model (possibly 21. EG Cavalcanti, Classical Causal Models for Bell and Kochen-Specker inequality Violations { | } NF a non-FHV model), and (b) behaviour Pab xy xy also has a FHV model Require Fine-Tuning. Phys. Rev. X 8, 021018 (2018). { | } (possibly a non-LHV model). 22. R Colbeck, R Renner, A short note on the concept of free choice. arXiv:1302.4446 (2013). Ad. (a) This readily follows from Lemma 2. 23. C Brukner, Quantum causality. Nat. Phys. 10, 259 (2014). 24. JMA Allen, J Barrett, D Horsman, CM Lee, RW Spekkens, Quantum Common Causes and Ad. (b) Here, a trivial model will suffice. Let us take Λ := λo (a { } Quantum Causal Models. Phys. Rev. X 7, 031021 (2017). single-element set) with Pλo Pλo xy := 1 and conditional distribution NF ≡ | 25. KB Wharton, N Argaman, Colloquium: Bell’s theorem and locally mediated reformulations of defined as Pab xyλo := P . Clearly, it is a FVH model of behaviour | ab xy quantum mechanics. Rev. Mod. Phys. 92, 021002 (2020). NF | Pab xy xy. 26. T Norsen, Against ‘Realism’. Found. Phys. 37, 311 (2007). { | } Thus, we have shown equivalence of both decompositions Eqs. (26) 27. AC Elitzur, S Popescu, D Rohrlich, Quantum nonlocality for each pair in an ensemble. Phys. and (27), which, by virtue of Remark 2, proves Theorem 1. Lett. A 162, 25 (1992). 28. L Hardy, A new way to obtain Bell inequalities. Phys. Lett. A 161, 21 (1991). 29. J Barrett, A Kent, S Pironio, Maximally Nonlocal and Monogamous Quantum Correlations. By virtue of it suffices to prove the Proof of Theorem2. Theorem 1 Phys. Rev. Lett. 97, 170409 (2006). result for one of the measures. Let it be measure µL evaluated by 30. R Colbeck, R Renner, Hidden Variable Models for Quantum Theory Cannot Have Any Local means of Eq. (28) in Remark 2. Part. Phys. Rev. Lett. 101, 050403 (2008). Consider some arbitrary decomposition Eq. (26) of behaviour 31. R Colbeck, R Renner, The Completeness of Quantum Theory for Predicting Measurement Pab xy xy. Then, by linearity, the four CHSH expressions Eqs. (2)-(5) Outcomes in Quantum Theory: Informational Foundations and Foils, eds. G Chiribella, RW decompose{ | } as well, i.e. we get Spekkens. (Springer), pp. 497–528 (2016). 32. S Portmann, C Branciard, N Gisin, Local content of all pure two-qubit states. Phys. Rev. A L NL Si = pL Si + (1 pL ) Si ,[ 31] 86, 012104 (2012). · − · 33. S Abramsky, RS Barbosa, S Mansfield, Contextual Fraction as a Measure of Contextuality. L NL L where Si and Si are calculated for the respective behaviours Pab xy xy Phys. Rev. Lett. 119, 050504 (2017). { | } NL 34. CH Brans, Bell’s theorem does not eliminate fully causal hidden variables. Int. J. Theor. Phys. and Pab xy xy. Since the first one is a local behaviour with full freedom { | } of choice (i.e. having a LHV+FHV model), then from the CHSH 27, 219 (1988). 35. MJW Hall, Local Deterministic Model of Singlet State Correlations Based on Relaxing Mea- inequalities Eq. (1) we have SL 2. For the second one there is | i | 6 surement Independence. Phys. Rev. Lett. 105, 250404 (2010). nothing interesting to be said other than noting the maximal algebraic 36. MJW Hall, The Significance of Measurement Independence for Bell Inequalities and Locality bound SNL 4. As a consequence, the following inequality obtains | i | 6 in At the Frontier of Spacetime, ed. T Asselmeyer-Maluga. (Springer), pp. 189–204 (2016). 37. MJW Hall, Relaxed Bell inequalities and Kochen-Specker theorems. Phys. Rev. A 84, 022102 Si 6 pL 2 + (1 pL ) 4 = 4 2 pL ,[ 32] | | · − · − (2011). 1 38. J Barrett, N Gisin, How Much Measurement Independence Is Needed to Demonstrate Nonlo- and we get pL 6 2 (4 Si ). Thus, by assumed arbitrariness of decom- position, Eq. (26) gives−| the| upper bound on expression in Eq. (28) cality? Phys. Rev. Lett. 106, 100406 (2011). 39. G Pütz, D Rosset, TJ Barnea, YC Liang, N Gisin, Arbitrarily Small Amount of Measurement 1 Independence Is Sufficient to Manifest Quantum Nonlocality. Phys. Rev. Lett. 113, 190402 µL 6 (4 Si ) ,[ 33] 2 − | | (2014). 40. D Aktas, et al., Demonstration of Quantum Nonlocality in the Presence of Measurement where Smax = max Si : i = 1, ... , 4 . By Lemma 3 we conclude that the bound is tight,{| which| ends the proof} of Theorem 2. Dependence. Phys. Rev. Lett. 114, 220404 (2015). 41. G Pütz, N Gisin, Measurement dependent locality. New J. Phys. 18, 055006 (2016). 42. MJW Hall, C Branciard, Measurement-dependence cost for Bell nonlocality: Causal versus ACKNOWLEDGMENTS. We thank R. Colbeck, J. Duda, N. Gisin, retrocausal models. Phys. Rev. A 102, 052228 (2020). M. Hall, L. Hardy, D. Kaiser, M. Markiewicz, S. Pironio, D. Rohrlich 43. JF Clauser, MA Horne, A Shimony, RA Holt, Proposed Experiment to Test Local Hidden- and V. Scarani for helpful comments. PB acknowledges support from Variable Theories. Phys. Rev. Lett. 23, 880–884 (1969). the Polish National Agency for Academic Exchange in the Bekker 44. A Fine, Hidden Variables, Joint Probability, and the Bell Inequalities. Phys. Rev. Lett. 48, 291–295 (1982). Scholarship Programme. PB and EMP were supported by Office of 45. JJ Halliwell, Two proofs of Fine’s theorem. Phys. Lett. A 378, 2945–2950 (2014). Naval Research Global grant N62909-19-1-2000. 46. JP Jarrett, On the Physical Significance of the Locality Conditions in the Bell Arguments. Nous , 569 (1984). 1. JS Bell, Speakable and unspeakable in quantum mechanics. (Cambridge University Press), 18 47. BS Tsirelson, Quantum generalizations of Bell’s inequality. Lett. Math. Phys. , 93–100 (1987). 4 (1980). 2. ND Mermin, Hidden variables and the two theorems of John Bell. Rev. Mod. Phys. 65, 803– 48. JR Busemeyer, PD Bruza, Quantum Models of Cognition and Decision. (Cambridge Univer- 815 (1993). sity Press), (2012). 3. HM Wiseman, The two Bell’s theorems of John Bell. J. Phys. A: Math. Theor. 47, 424001 49. E Haven, A Khrennikov, Quantum Social Science. (Cambridge University Press), (2013). (2014). 50. EM Pothos, JR Busemeyer, Can quantum probability provide a new direction for cognitive 4. N Brunner, D Cavalcanti, S Pironio, V Scarani, S Wehner, Bell nonlocality. Rev. Mod. Phys. modeling? Behav. Brain Sci. 36, 255–274 (2013). 86, 419 (2014). 51. L Hardy, Bell inequalities with retarded settings in Quantum [Un]speakables II: Half a Century 5. V Scarani, Bell Nonlocality. (Oxford University Press), (2019). of Bell’s Theorem, eds. R Bertlmann, A Zeilinger. pp. 261–272 (2017) arXiv:1508.06900. 6. JS Bell, Free variables and local causality in Speakable and unspeakable in quantum me- 52. L Hardy, Proposal to use humans to switch settings in a bell experiment. arXiv: 1705.04620 chanics. (Cambridge University Press), (1987). (2017). 7. J Pearl, Causality: Models, Reasoning, and Inference. (Cambridge University Press), 2nd 53. T Maudlin, Bell’s Inequality, Information Transmission, and Prism Models in PSA: Proceed- edition, (2009). ings of the Biennial Meeting of the Philosophy of Science Association 1992. (The University 8. A Aspect, J Dalibard, G Roger, Experimental Test of Bell’s Inequalities Using Time-Varying of Chicago Press) Vol. 1, pp. 404–417 (1992). Analyzers. Phys. Rev. Lett. 49, 1804 (1982). 54. G Brassard, R Cleve, A Tapp, Cost of Exactly Simulating Quantum Entanglement with Clas- 9. M Giustina, et al., Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons. sical Communication. Phys. Rev. Lett. 83, 1874 (1999). Phys. Rev. Lett. 115, 250401 (2015).

Blasiak et al. PNAS 118 (17) e2020569118 (2021) | 9 55. M Steiner, Towards quantifying non-local information transfer: finite-bit non-locality. Phys. Lett. A 270, 239 (2000). 56. BF Toner, D Bacon, Communication Cost of Simulating Bell Correlations. Phys. Rev. Lett. 91, 187904 (2003). 57. RD Gill, The Triangle Wave Versus the Cosine: How Classical Systems Can Optimally Ap- proximate EPR-B Correlations. Entropy 22, 287 (2020). 58. H Buhrman, R Cleve, S Massar, R de Wolf, Nonlocality and communication complexity. Rev. Mod. Phys. 82, 665–698 (2010). 59. J Kofler, T Paterek, C Brukner, Experimenter’s freedom in Bell’s theorem and quantum cryp- tography. Phys. Rev. A 73, 022104 (2006). 60. DE Koh, et al., Effects of Reduced Measurement Independence on Bell-Based Randomness Expansion. Phys. Rev. Lett. 109, 160404 (2012). 61. P Suppes, M Zanotti, When are probabilistic explanations possible? Synthese 48, 191–199 (1981). 62. A Garg, ND Mermin, Farkas’s Lemma and the nature of reality: Statistical implications of quantum correlations. Found. Phys. 14, 1 (1984). 63. M Zukowski, C Brukner, Bell’s Theorem for General N-Qubit States. Phys. Rev. Lett. 88, 210401 (2002). 64. AA Klyachko, MA Can, S Binicioglu,˘ AS Shumovsky, Simple Test for Hidden Variables in Spin-1 systems. Phys. Rev. Lett. 101, 020403 (2008). 65. J Barrett, et al., Nonlocal correlations as an information-theoretic resource. Phys. Rev. A 71, 022101 (2005). 66. J Barrett, S Pironio, Popescu-rohrlich correlations as a unit of nonlocality. Phys. Rev. Lett. 95, 140401 (2005). 67. NS Jones, L Masanes, Interconversion of nonlocal correlations. Phys. Rev. A 72, 052312 (2005). 68. JV Kujala, EN Dzhafarov, JA Larsson, Necessary and Sufficient Conditions for an Extended Noncontextuality in a Broad Class of Quantum Mechanical Systems. Phys. Rev. Lett. 115, 150401 (2015). 69. EN Dzhafarov, JV Kujala, JA Larsson, Contextuality in Three Types of Quantum-Mechanical Systems. Found. Phys. 45, 762–782 (2015). 70. ND Mermin, Quantum mysteries revisited. Am. J. Phys. 58, 731–734 (1990). 71. MO Renou, et al., Genuine Quantum Nonlocality in the Triangle Network. Phys. Rev. Lett. 123, 140401 (2019). 72. N Gisin, Quantum Chance: Nonlocality, Teleportation and Other Quantum Marvels. (Springer), (2014) [Ch. 9, p. 90]. 73. S Popescu, D Rohrlich, Quantum Nonlocality as an Axiom. Found. Phys. 24, 379 (1994).

10 | www.pnas.org/cgi/doi/10.1073/pnas.2020569118 Blasiak et al. Supplementary Information for

Violations of locality and free choice are equivalent resources in Bell experiments

Pawel Blasiak, Emmanuel M. Pothos, James M. Yearsley, Christoph Gallus and Ewa Borsuk

Corresponding Author: Pawel Blasiak. E-mail: [email protected]

This PDF file includes: Supplementary text Figs. S1 to S2 SI References arXiv:2105.09037v1 [quant-ph] 19 May 2021

Pawel Blasiak, Emmanuel M. Pothos, James M. Yearsley, Christoph Gallus and Ewa Borsuk 1 of5 Supporting Information Text xy), i.e., we have Λ˜ NF = Λ˜ . Finally, we define the distribution

of P˜λ on the whole HV space Λ˜ by the following condition In this Supplement we prove two technical results Lemma 2 and Lemma 3 from the Methods section in the main ˜ ˜(uv) ˜ (uv) Pλ := Pλ Puv for λ Λ ,[ 41] manuscript. We also give an alternative self-standing proof of · ∈

Theorem 3. where Puv is the desired distribution of settings (which for sake of generality is left unspecified). Observe that Eqs. (40) and [The numbering of equations follows the main text.] (41) allow us to reverse the conditioning, i.e., Bayes’ rule gives

˜(xy) ˜ (xy) ˜ Pλ if λ Λ , Pλ xy := ∈ [42] PROOF OF LEMMA2 | 0 otherwise .  For reference we quote Lemma 2 from the main manuscript Having defined in Eqs. (38)-(41) all components of the HV that we will shortly prove. Here, the choice of settings is model (local and fully violating freedom of choice) we need to arbitrary x, y M = 1, 2 , . . . , M . ∈ { } check whether it correctly reconstructs some given behaviour

Lemma 2. For any behaviour Pab xy xy and distribution of settings Pab xy xy and distribution of settings Pxy via Eqs. (6) and (7). { | } { | } Pxy there exists a local hidden variable model (LHV) which fully The following calculation violates the freedom of choice assumption (i.e., if Λ˜ is the relevant ˜ ˜ (42) ˜ ˜(xy) ˜ ˜ ˜ Pab xyλ Pλ xy = Pab xyλ Pλ [43] HV space, then we have Λ = ΛL = ΛNF ; cf. Eqs. (10) and (11)). ∑ | | ∑ | ˜ · · λ Λ λ Λ˜ (xy) ∈ ∈ Proof. Let us first introduce an auxiliary pair of indices uv (39) ˜(xy) ˜(xy) ∈ = ∑ Pab xyλ Pλ [44] M M where M = 1, 2 , . . . , M and for each pair uv define | · λ Λ˜ (xy) × {˜(uv) } ∈ a trivial behaviour Pab xy xy which consists of M M copies (37) (34) { | } × ˜(xy) = Pab xy = Pab xy ,[ 45] of the same distribution | | ˜(uv) confirms the validity of Eq. (6). Similarly, we check that Pab xy := Pab uv for all xy M M ,[ 34] | | ∈ × (40) i.e., this holds independently of the choice of settings xy. Then, P˜xy λ P˜λ = P˜λ [46] ∑ | ∑ ˜ · for each such behaviour there exists an LHV model. To see λ Λ λ Λ˜ (xy) ∈ ∈ this, for a given pair of indices uv it is enough to consider a (41) = P˜(xy) P = P ,[ 47] four-element HV space ∑ λ · xy xy λ Λ˜ (xy) ∈ Λ˜ (uv) := (α, β) : α, β = 1 ,[ 35] { ± } in compliance with Eq. (7). This concludes the proof. with probability distributions defined in the following way

˜(uv) ˜ ˜(uv) Pλ := Pαβ uv and Pab xyλ := δa=α δb=β ,[ 36] PROOF OF LEMMA3 | | · where λ = (α, β) Λ˜ (uv). By construction, the locality con- In the following we consider a Bell experiment with binary ∈ ˜ (uv) ˜ (uv) settings x, y M = 0 , 1 . dition Eq. (8) is satisfied for all λ, i.e., Λ = ΛL , and the ∈ { } marginals reproduce the behaviour in Eq. (34) Preliminaries. We begin with a useful observation about the ˜(uv) ˜(uv) ˜(uv) Pab xy = ∑ Pab xyλ Pλ .[ 37] CHSH expressions in Eqs. (2)-(5). | | · λ Λ˜(uv) ∈ Remark 3. We have Si + Sj 6 4 whenever i = j. ˜(uv) | | | | 6 Those LVH models of auxiliary behaviours Pab xy xy can be used to construct another local model on a compound{ | } HV Proof. This is a direct consequence of a simple algebraic in- space equality involving the four expressions S1, ... , S4 which fol- low the pattern of Eqs. (2)-(5), i.e., S1 = a + b + c d, (uv) − Λ˜ := Λ˜ ,[ 38] S2 = a + b c + d, S3 = a b + c + d and S4 = a + b + c + d. − − − u,v M It is straightforward to check that for 1 6 a, b, c, d 6 1, when- ]∈ − ever i = j we have Si Sj 6 4 and Si Sj 6 4 (since in each with conditional probabilities on each component left un- case four6 out of eight± ± terms cancel± out).∓ Clearly, in compact M changed, i.e., for each u, v we have form this is equivalent to the inequality S + S 4 which ∈ | i | | j | 6 ˜ ˜(uv) ˜ (uv) concludes the proof. Pab xyλ := Pab xyλ for λ Λ .[ 39] | | ∈ Note that in this way the locality condition is retained, i.e., the As an immediate consequence of Remark 3, note that for any ˜ behaviour Pab xy xy at most one out of four CHSH inequalities in factorisation of Eq. (8) holds for every λ Λ (cf. Eq. (36)), { | } ˜ ˜ (uv) ˜ ∈ Eq. (1) can be violated at the same time. and we have ΛL = u,v M ΛL = Λ. Then we impose the ∈ Without loss of generality, in the following proof we con- conditional distributions P˜xy λ in the form U | sider a given behaviour Pab xy xy with binary choice of settings | (uv) { } P˜xy λ := δx=u δy=v for λ Λ˜ .[ 40] x, y M = 0, 1 for which only the first CHSH inequality is | · ∈ violated,∈ i.e.,{ we} assume that Clearly, this violates the free choice assumption Eq. (9) for every λ Λ˜ (since the knowledge of λ fully determines the settings S = S > 2 ,[ 48] ∈ max 1

2 of5 Pawel Blasiak, Emmanuel M. Pothos, James M. Yearsley, Christoph Gallus and Ewa Borsuk of their role in the description of the non-signalling polytope in a two-setting and two-outcome Bell scenario [2]. For reference let us quote Lemma 3 from the main manuscript that we will shortly prove; see [3] for a related result. See Fig.S 1 for illustration.

Lemma 3. Each non-signalling behaviour Pab xy xy with binary { | } settings x, y M = 0, 1 can be decomposed as a convex mixture ∈ { } of a local behaviour P¯ab xy xy and a PR-box P˜ab xy xy in the form { | } { | }

Pab xy = p P¯ab xy + (1 p) P˜ab xy ,[ 52] | · | − · | with p = 1 (4 S ) for all x, y 0, 1 . 2 − max ∈ { } [In the case S1 > 2, the PR-box is that of Eq. (49)]. Here, local behaviour means existence of an LHV+FHV model

of P¯ab xy xy and Smax = max Si : i = 1, ... , 4 . { | } {| | } Proof. Let us consider a given non-signalling behaviour

Pab xy xy such that S1 > 2, and define the following four distri- | butions{ } ¯ 1 1 p ˜ (1) Pab xy := p Pab xy −p Pab xy ,[ 53] | | − | ¯ (1) for x, y 0, 1 , where Pab xy xy is the PR-box defined in ∈ { } { | } Fig. S1. Decomposition in Lemma3. A two-dimensional section of the non- Eq. (49). In order to prove Lemma 3 it is enough to show that signalling polytope for the two-setting and two-outcome Bell scenario. It is spanned by ¯ (j) ˜ (k) sets of local deterministic behaviours P and non-local PR-boxes P . The subset (a) P¯ab xy are well-defined probability distributions | of local behaviours (depicted as the inner green rectangle) is fully characterised by for each x, y 0, 1 , and the CHSH inequalities Eq. (1). For each non-signalling behaviour P which breaks ∈ { } the local bound, one finds a convex decomposition P = p P¯ + (1 p) P˜ (k) with (b) P¯ab xy xy defines a local behaviour, 1 ¯ ·˜ (k) − · | p = 2 (4 Smax ) for some local behaviour P and a PR-box P . On the picture, we { } − 1 have Smax = S1 > 2 and k = 1. for the choice p = (4 S ). 2 − 1 ¯ Ad. (a). The normalisation condition ∑ Pab xy = 1 is trivially ab | ˜ (1) and consequently Si < 2 for i = 2, 3, 4 . It is straightfor- satisfied, since Pab xy and Pab xy are normalised. It only remains | | | | ward to appreciate that all other cases reduce to this one by to check that P¯ab xy 0. | > relabelling the indices.* It is known that the non-signalling polytope for the two- setting and two-outcome scenario is characterised by 24 = 16 + 8 Decomposition of Lemma3 (proof). Let us start by recalling extremal points [2] (cf. Fig.S 1): the concept of a PR-box (due to Popescu-Rohrlich [1]) which P¯ (j) – sixteen local deterministic behaviours is a non-signalling behaviour P˜ab xy xy saturating the CHSH ab xy xy | { | } expressions. To be more precise,{ only} one of the expressions for j = 1, ... , 16 , ˜ Eqs. (2)-(5) can reach its maximal/minimal value Si = 4 and ˜ (k) ± Pab xy xy – eight PR-boxes for k = 1, ... , 8 . then the rest must be equal to zero (cf. Remark 3). There are { | }

eight possible PR-boxes out of which we select just two cases This means that we can decompose Pab xy xy as a convex com- { | } for illustration bination

1 16 8 ˜ (1) /2 for a b = x y , ¯ (j) ˜ (k) Pab xy = ⊕ · [49] Pab xy = ∑ pj Pab xy + ∑ qk Pab xy [54] | 0 otherwise , | · | · |  j=1 k=1 and 16 8 with proper normalisation ∑j=1 pj + ∑k=1 qk = 1 and pj, qk > 0 for all j, k. We can use Eq. (54) to calculate the first CHSH ˜ (2) 0 for a b = x y , Pab xy = ⊕ · [50] expression Eq. (2), which by linearity gives | 1/2 otherwise .  16 8 ˜ (k) ¯ (j) ˜ (k) All the remaining boxes Pab xy xy for k = 3, ... , 8 obtain by S1 = ∑ pj S1 + ∑ qk S1 ,[ 55] { | } · · relabelling the indices x, y. These two behaviours saturate the j=1 k=1 first CHSH expression Eq. (2), i.e. ¯ (j) ˜ (k) ¯ (j) ˜ (k) where S1 and S1 correspond to Pab xy xy and Pab xy xy respec- { | } { | } S˜ (1) = + 4 , S˜ (1) = 0 for i = 2, 3, 4 , tively. For this particular choice of distributions, we have 1 i ¯ (j) (2) [51] S1 6 2 (for local behaviours) and Eq. (51) (for PR-boxes), S˜ (2) = 4 , S˜ = 0 for i = 2, 3, 4 , | | 1 − i from which the following bound is obtained

and similarly the remaining six boxes saturate the other three 16 (k) ¯ (j) CHSH expressions Eqs. (3)-(5) (in particular we have S˜ = 0 S1 = pj S1 + 4 q1 4 q2 1 ∑ · · − · for k = 3, ... , 8). In what follows, PR-boxes will be used because j=1 16 *To change signs: (a a) (S S ). To switch between expressions: (x 1 x) → − ⇒ i → − i → − ⇒ 6 2 ∑ pj + 4 q1 6 2 + 2 q1 .[ 56] (S S ), (y 1 y) (S S ) and (x 1 x & y 1 y) (S S ). · · · 1 ↔ 3 → − ⇒ 1 ↔ 2 → − → − ⇒ 1 ↔ 4 j=1

Pawel Blasiak, Emmanuel M. Pothos, James M. Yearsley, Christoph Gallus and Ewa Borsuk 3 of5 Note that in the last inequality we used the normalisation (with x, y M). The decomposition Eq. (11) entails splitting 16 8 ∈ ∑j=1 pj + ∑k=1 qk = 1. Thus, in the decomposition of Eq. (54) of Eq. (6) into two components 1 the coefficient q1 > S1 1. A quick comparison of Eqs. (53) 2 Pab xy = Pab xyλ Pλ + Pab xyλ Pλ xy ,[ 61] − | ∑ | ∑ | | and (54) reveals that such a defined distribution is well-defined, λ Λ · λ Λ · ¯ ∈ F ∈ NF i.e., Pab xy > 0, as long as q1 > 1 p. This is true for p > 1 | − which describe fundamentally different situations with and (4 S1). 2 − without freedom of choice, i.e. corresponding to the hidden Ad. (b). Since both sets of behaviours Pab xy xy and the | variable λ being respectively in the free domain (ΛF) or non- ˜ (1) {¯ } PR-box Pab xy xy are non-signalling, then Pab xy xy defined in free domain (ΛNF). This can be used to write the correlation { | } { | } Eq. (53) is non-signalling too. Therefore, by virtue of Fine’s coefficients in the form theorem [4, 5], it suffices to check that all CHSH inequalities ¯ ab xy = ab xyλ Pλ + ab xyλ Pλ xy ,[ 62] are satisfied Si 6 2. ∑ ∑ | | | h i λ Λ h i · λ Λ h i · By linearity of the CHSH expresions, from Eq. (53) we get ∈ F ∈ NF 1 1 p (1) S¯ = S − S˜ . It follows that where ab xyλ ∑ , ab Pab xyλ is the correlation coefficient for i p i p i h i ≡ a b | − fixed λ. Hence, the expression S defined in Eq. (58) splits into S (1 p) S˜ (1) two parts S¯ = | i − − i | | i | p S1 4 (1 p) S = αxy ab xyλ Pλ + [63] (51) | − − | for i = 1 , ∑ ∑ p λ Λ x,y · h i · = [57] ∈ F  Si 4 S1   | p | 6 −|p | for i = 1 ,  6 αxy ab xyλ Pλ xy .[ 64] ∑ ∑ | λ Λ x,y · h i · where the last inequality for i = 1 is due to Remark 3. Now, ∈ NF   6 1 it is straightforward to check that for p = (4 S ) all four Note that in the first term probability Pλ factors out due to the 2 − 1 CHSH expressions satisfy the local bound, i.e. we have S¯1 = 2 condition in Eq. (11) defining the free domain ΛF. So far, the ¯ | | causal structure of the HV model has not been constrained. and Si 6 2 for i = 1. | | 6 Now, we make the locality assumption Eq. (8) which entails

the following factorisation ab λ = a λ b λ , with a λ = h ixy h ix h iy h ix ∑ a Pa xλ and b yλ = ∑ b Pb yλ. Then the expression in Eq. (63) DIRECT PROOF OF THEOREM3 a | h i b | is bounded by Sloc of Eq. (60), i.e., We start by considering a wide class of upper bounds on loc loc Eq. (63) S Pλ = S Pλ .[ 65] the measure free choice µF. This result will be used to prove 6 ∑ ∑ λ Λ · · λ Λ Theorem 3 with the help of correlations saturating the so called ∈ F ∈ F chained Bell inequalities (for a special choice of measurement For the expression in Eq. (64) we use S# of Eq. (59) to make the settings on a Bell state). following estimate

(67) Upper bounds on measure of free choice µF . Let x, y M Eq. (64) αxy Pλ xy = αxy 1 Pλ 6 ∑ ∑ | ∑ ∑ ∈ x,y | |λ Λ x,y | | · − λ Λ with the choice of settings M unspecified for the time being. ∈ NF  ∈ F  Our analysis will proceed with algebraic expressions which # 6 S 1 ∑ Pλ ,[ 66] take the following form · − λ Λ  ∈ F 

S = α ab ,[ 58] where we use the fact that conditional probabilities Pλ xy are ∑ xy xy | x,y · h i normalised and Λ = Λ Λ (cf. Eq. (11)) , which entails F ∪ NF

where αxy 0, 1 are some arbitrary coefficients and 1 = Pλ xy = Pλ + Pλ xy [67] ∑ | ∑ ∑ | ∈ { ± } λ Λ λ Λ λ Λ ab xy ∑ , ab Pab xy is the usual correlation function. For ∈ ∈ F ∈ NF h i ≡ a b | each such expression S we will look at two auxiliary quantities. for each x, y M. Putting everything together, from The first one counts the number of non-zero terms in Eq. (58), Eqs. (63)/(64) and∈ (65)/(66) we get the inequality i.e., loc # S S Pλ + S (1 Pλ ) , # 6 ∑ ∑ = α .[ 59] · λ Λ · − λ Λ S ∑ xy ∈ F ∈ F x,y | | which provides the following bound on the amount of free The second one is the upper bound of Eq. (58) with the corre- choice in a given LHV model (see Remark 1) lation coefficients replaced by a product, i.e.,† S# S Pλ .[ 68] loc ∑ 6 # − loc S = max αxy ax by .[ 60] λ Λ S S 16 ax 61 ∑ ∈ F − x,y · − 16 by 61 − Since this inequality should be satisfied by any LHV model # loc Note that both S and S depend only on the coefficients αxy reproducing given behaviour Pab xy xy in a Bell experiment { | } (and not on the correlation functions ab ). (irrespective of the distribution of settings P ), then we get h ixy xy In the following, we consider an arbitrary HV model of upper bound on the the measure of free choice defined in

a given set behaviour Pab xy xy obtained in a Bell experiment Eq. (13) { | } # †A closer inspection of Eq. (60) reveals the maximum is attained at one of the extremal points S S a , b = 1 (since the expression ∑ α a b is a linear function for each variable a , b ). µF 6 − .[ 69] x y ± x,y xy · x y x y S# Sloc − 4 of5 Pawel Blasiak, Emmanuel M. Pothos, James M. Yearsley, Christoph Gallus and Ewa Borsuk x, y implements projective measurements in the respective bases ϕ , ϕ ⊥ for Alice and ϕ , ϕ ⊥ for Bob, where | x i | x i | y i | y i ϕ = cos (x 1)θ 0 +sin (x 1)θ 1 ,[ 72] | x iA − | iA − | iA ϕ = cos (y 1 )θ 0 + sin (y 1 )θ 1 ,[ 73] | y iB − 2 | iB − 2 | iB π for x, y M 1, 2, . . . , M and θ = 2M 1 . See Fig.S 2 (on the right)∈ for≡ illustration. { Then} all terms− in Eq. (70) become equal to ab = cos θ except for the last one which is equal h ixy to ab = 1, and hence we get h i1M − S = (2M 1) cos θ + 1 M − 2 ( ) π + & 2M 1 1 2 (2M 1)2 1 .[ 74] − − −   Substituted into Eq. (71), this gives

π2 µF 6 / 0 .[ 75] Fig. S2. Chained Bell inequalities. On the left, there is a schematic illustration of the ( ) M ∞ 4 2M 1 → chained expression in Eq. (70) with each one of 2M terms ab represented by the − h ixy line x y. Solid lines correspond to the ’+’ signs and the wavy line corresponds to the − This justifies our claim about freedom of choice reduced to zero ’ ’ sign. On the right, the xˆ zˆ section of the Bloch sphere concerns measurement − − in any local account of the quantum state Φ+ . Generalisation directions defined in Eqs. (72) and (73). Alice measures along the directions ϕx A | i | i to any maximally entangled two-qubit state is straightforward. (depicted in magenta) and Bob measures along the directions ϕy B (depicted in π blue), with x, y = 1, 2, . . . , M and θ = 2M 1 . − References Crucially, the bound in Eq. (69) holds for any expression S 1. S Popescu, D Rohrlich, Quantum Nonlocality as an Axiom. as defined in Eq. (58). Let us emphasise that the numbers S# Found. Phys. 24, 379 (1994). loc and S are determined solely by the choice of coefficients αxy , 2. V Scarani, Bell Nonlocality. (Oxford University Press), while S itself depends on the observed experimental statis- (2019).

tics Pab xy xy through the correlation coefficients ab xy , see 3. S Pironio, Violations of Bell inequalities as lower bounds | Eqs.{ (58)-(}60). h i on the communication cost of nonlocal correlations. Phys. Rev. A 68, 062102 (2003). Proof of Theorem3. Quantum correlations are stronger than 4. A Fine, Hidden Variables, Joint Probability, and the Bell the classical ones, but much weaker than what one might Inequalities. Phys. Rev. Lett. 48, 291–295 (1982). possibly imagine [1, 6]. For the proof of Theorem 3 is however 5. JJ Halliwell, Two proofs of Fine’s theorem. Phys. Lett. A enough to exclude all freedom of choice for any local account 378, 2945–2950 (2014). of a maximally entangled state when the number of settings 6. S Popescu, Nonlocality beyond quantum mechanics. Nat. x, y M = 1, 2, ... , M goes to infinity M ∞. Phys. 10, 264 (2014). Let∈ us take{ expression} of Eq. (58) in→ a special chained 7. PM Pearle, Hidden-Variable Example Based upon Data form [7, 8] Rejection. Phys. Rev. D 2, 1418 (1970). 8. SL Braunstein, CM Caves, Wringing out better Bell inequal- S = ab + ab + ab + ab + ... M h i11 h i21 h i22 h i32 ities. Ann. Phys. 202, 22 (1990). ... + ab ab ,[ 70] h iMM − h i1M which follows the pattern illustrated in Fig.S 2 (on the left). # From simple term counting we have SM = 2 M. Now, if locality is imposed this leads to the so called chained Bell inequalities: SM 6 2 (M 1), see [7, 8]. It is straightforward to see that − loc ‡ Eq. (60) saturates to SM = 2 (M 1). This means that for the expression Eq. (70) the upper bound− on the amount of free choice in Eq. (69) becomes

2 M SM 1 µF 6 − = M SM .[ 71] 2 M 2 (M 1) − 2 − − Now, we will assume that the correlations in a Bell experi- ment are described by the quantum statistics given by the Bell state Φ 1 ( 0 0 + 1 1 ) and the choice of settings | +i ≡ √2 | iA | iB | iA | iB

‡Locality entails factorisation ab = a b =: a b . Then S = (a + a )b + (a + a )b + h ixy h ix h iy x y M 1 2 1 2 3 2 ... + (a a )b . Suppose a , b = 1, then out of M terms taking values 0, 2 at least one M − 1 M x y ± ± has to be equal to zero (if aM = a1 it is the last one, otherwise it is one of the remaining ones). In such a case S 2 (M 1). This extends to intermediate values 1 a , b +1, since S is M 6 − − 6 x y 6 M a linear function for each of the variables and hence the global maximum is attained at the extremal points a , b = 1. Clearly, taking a = b = 1 we get the maximum, i.e. Sloc = 2 (M 1). x y ± x y M −

Pawel Blasiak, Emmanuel M. Pothos, James M. Yearsley, Christoph Gallus and Ewa Borsuk 5 of5