Reconsidering No-Go Theorems from a Practical Perspective

Reconsidering No-Go Theorems from a Practical Perspective Abstract I argue that our judgements regarding the locally causal models that are compatible with a given constraint implicitly depend, in part, on the context of inquiry. It follows from this that certain quantum no-go theorems, which are particularly striking in the traditional foundational context, have no force when the context switches to a discussion of the physical systems we are capable of building with the aim of classically reproducing quantum statistics. I close with a general discussion of the possible implications of this for our understanding of the limits of classical description, and for our understanding of the fundamental aim of physical investigation. 1 Introduction 2 No-Go Results   2.1 The CHSH inequality   2.2 The GHZ equality 3 Classically Simulating Quantum Statistics   3.1 GHZ statistics   3.2 Singlet statistics 4 What Is a Classical Computer Simulation? 5 Comparing the All-or-Nothing GHZ with Statistical (In)equalities 6 General Discussion 7 Conclusion 1 Introduction Bell’s inequalities, and results like them, are often presented as ‘no-go theorems’ in the philosophical and physical literature. They are portrayed as demonstrating, for instance, that no ‘locally causal’ (Bell [2004b], Section 7) description of the empirically confirmed predictions of quantum mechanics is possible, that one cannot ascribe ‘elements of reality’, in Einstein, Podolsky, and Rosen's (EPR's) sense (Einstein et al. [1935]), to quantum systems consistently with such predictions, and so on. As I will argue below, however, thinking of Bell’s and related results in this way—without further qualification—can be misleading, for they are not no-go theorems per se. Rather, they are best understood as establishing certain general constraints on locally causal descriptions of joint measurement outcomes. In order to turn Bell’s and similar results into no-go theorems, one must do more than just state them. One must also interpret them in the context of the question one takes them to be informing. Associated with a given context are a number of additional constraints, according to which we judge a locally causal description to be either plausible or implausible in that context. Normally, these ‘plausibility constraints’ are left unstated and this is mostly harmless most of the time, for the relevant context is usually understood to be what I will below be calling the ‘theoretical’. Here, the question we take Bell’s and related results to be answering is the question of whether there is an alternative locally causal description of the natural world that can recover the confirmed predictions of quantum mechanics. The set of plausibility constraints on locally causal descriptions—or anyway, the general character of such constraints—in the context of this question is implicitly understood by all. Bell’s and related results are relevant to other contexts having to do with physical systems as well, however. In particular, in what I will below be calling the ‘practical’ context, we are not concerned with alternative theories of the natural world, classical or otherwise. What we are rather concerned with are the kinds of physical system that we can build in order to recover a particular probability distribution or set of measurement outcomes. As I will describe in more detail below, the plausibility constraints we impose on locally causal descriptions in this context are importantly different from those we impose in the theoretical context. Thus the kinds of locally causal description we think are worth considering as alternatives to quantum description in the former context will be different from the kinds of locally causal description we think are worth considering in the latter. This has interesting and important implications. Previously, I have argued that distinguishing between these two contexts can help us to understand the way certain physical resources are taken advantage of in the science of quantum computation (Cuffaro [2017]). The present article builds upon that work.1 In this article, I will argue that making this distinction can help us to better understand the relative power and scope of no-go theorems, and provide us with an alternative point of view from which to consider the fundamental differences between classical and quantum systems in general. Consider, in particular, Greenberger et al.'s ([1989]) so-called ‘all-or-nothing’ equality. It is typical to see this equality referred to in the philosophical and foundational literature as a more powerful refutation of local causality than Bell’s own statistical inequality. The reason for this is that while a violation of the Greenberger–Horne–Zeilinger (GHZ) equality can in principle be shown with a single quantum experiment, a violation of Bell’s requires repeated quantum experiments to demonstrate (and even then only with increasing, but never absolute, confidence). Greenberger et al.([1990]), p. 1131), for example, write: ‘This incompatibility with quantum mechanics is stronger than the one previously revealed for two-particle systems by Bell’s inequality, where no contradiction arises at the level of perfect correlations’. Clifton et al.([1991], p. 174) likewise write of their improvement on the original GHZ proof: ‘One difference between Bell’s theorem and ours is that his yields a statistical contradiction, whereas ours leads to an algebraic one. So in one respect, our conclusion is stronger’.2 Mermin, for whom the GHZ proof is ‘spectacular’ ([1993], p. 810), enthusiastically agrees: ‘This is an altogether more powerful refutation of the existence of elements of reality than the one provided by Bell’s theorem’ ([1990], p. 11). And Maudlin ([2011], p. 26) concurs: ‘the GHZ scheme brings home the problem for locality all the more sharply’. So, also, does Vaidman ([1999], p. 615): ‘The GHZ proof is the most clear and persuasive proof of nonexistence of local hidden variables’. Indeed, for Vaidman ([1999], p. 616), the significance of the GHZ proof is quite profound: ‘analysis of the GHZ work led me to accept the bizarre picture of quantum reality given by the many worlds interpretation’. Vaidman is not alone in taking the GHZ equality as a starting point from which to consider the question of the interpretation of quantum mechanics (Hemmo and Pitowsky [2003]). There is nothing inappropriate about this kind of talk, so long as it is confined (as it presumably is for the above authors) to what I have called the theoretical context. Interestingly, however, when we move from the theoretical to the practical context, then the above statements are false. As I will elaborate upon in more detail below, in the practical context, the all-or-nothing GHZ theorem loses its force. In the practical context, in fact, it is only statistical inequalities that can legitimately be turned into no-go theorems (although, also interestingly, one must indeed go beyond Bell’s theorem and its associated bipartite entangled states to entangled states of multipartite systems to show this). From this we may say that from the ‘absolute’ point of view—that is when considering the limits of plausible classical physical description as such—it is statistical inequalities, rather than the all-or-nothing GHZ, that are more powerful in the sense alluded to by the authors above, for statistical inequalities—and not the all-or-nothing GHZ—allow us to see that there are quantum statistics that even we can’t plausibly build classical systems to reproduce. The article proceeds as follows: In Section 2, I review Clauser, Horne, Shimony, and Holt's (CHSH's) variant of Bell’s inequality as well as the GHZ equality. In Section 3, I describe some schemes by which one could simulate, using a classical computer, some of the statistics associated with Bell and GHZ states. In Section 4, the meaning of the term ‘classical computer simulation’ is reflected upon. The implications of these reflections for our understanding of the relative strength of statistical no-go theorems vis á vis the all-or-nothing GHZ theorem are considered in Section 5. Finally, their general implications for our understanding of the limits of classical description, and for our understanding of the fundamental aim of physical investigation, are considered in Section 6. 2 No-Go Results 2.1 The CHSH inequality Consider the singlet state |Ψ−⟩=1/2(|↑⟩|↓⟩−|↓⟩|↑⟩), of two spin-½ particles. For a system in this state, the expectation value for joint experiments on its two subsystems is given by the following expression: ⟨σm⊗σn⟩=−m^·n^=−cos θ. (1) Here, σmandσn represent spin-m and spin-n experiments on the first (Alice’s) and second (Bob’s) subsystem, respectively, with m^andn^ the unit vectors representing the orientations of their two experimental devices, and θ the difference in these orientations. It is well known that it is not possible to provide an alternative theory accounting for the predictions associated with this state if that theory makes the very reasonable assumption that the probabilities for local experimental outcomes on Alice’s (and likewise Bob’s) subsystem are completely determined by her local experimental setup, together with a shared hidden variable, λ, taken on by both subsystems at the time the joint state is prepared (that is, while A and B are still physically interacting). For given such a theory, we will have that ⟨σm⊗σn⟩λ=⟨Aλ(m^)×Bλ(n^)⟩, (2) where Aλ(m^)∈{±1}andBλ(n^)∈{±1} represent the results of spin experiments on Alice's and Bob's subsystems, given a specification of the hidden variable λ. But consider the following expression relating the expectation values of different combined spin experiments on Alice’s and Bob’s subsystems for arbitrary directions m^,m^′,n^,andn^′: CHSH: |⟨σm⊗σn⟩+⟨σm⊗σn′⟩|+|⟨σm′⊗σn⟩−⟨σm′⊗σn′⟩|. When Equation (2) holds, one can show that CHSH≤2. (3) The ‘CHSH inequality’ (Clauser et al. [1969]) is one of a family of similar expressions, known collectively as the ‘Bell inequalities’,3 which must be satisfied by locally causal theories aiming to account for the statistics associated with combined spin measurements. The statistical predictions of quantum mechanics violate the CHSH inequality for some experimental configurations. For example, let the system be in the singlet state and let the unit vectors m^,m^′,n^,andn^′ (taken to lie in the same plane) have the orientations 0,π/2,π/4,and−π/4, respectively. The differences, θ, between the different orientations (that is, m^−n^,m^−n^′,m^′−n^, and m^′−n^′) will all be in multiples of π/4 and from Equation (1), we will have: |⟨σm⊗σn⟩+⟨σm⊗σn′⟩|+|⟨σm′⊗σn⟩−⟨σm′⊗σn′⟩|=22≰2. We normally conclude from this that the predictions of quantum mechanics for arbitrary orientations m^,m^′,n^, and n^′ cannot be reproduced by any alternative theory in which all of the correlations between subsystems are due to a common parameter endowed to them at state preparation. But note that these predictions can be reproduced by such a hidden variables theory for certain special cases. In particular, the inequality is satisfied by quantum mechanics when m^ and n^,m^ and n^′,m^′ and n^, and m^′ and n^′ are all oriented at angles with respect to one another that are given in multiples of π/2. Note that these are the cases for which Equation (1) predicts perfect correlation ( θ=π), perfect anti-correlation (θ = 0), or no correlation at all ( θ=π/2). For example, let λ determine Alice’s and Bob’s measurement results in the following way: Aλ(m^)=sign(m^·λ^), Bλ(n^)=−sign(n^·λ^), (4) where sign(x) is a function that returns the sign (+ or −) of its argument. The reader can verify that Equation (4) will recover all of the statistical predictions associated with the singlet state in Equation (1) just so long as the difference in orientation between m^ and n^ is some multiple of π/2 (Bell [2004c], p. 16). 2.2 The GHZ equality We have just seen that the statistics associated with the singlet state for measurement angles differing in proportion to π/2 are reproducible in a local hidden variables theory, such as in Equation (4). But this is not true for every entangled state. In particular, for a system of three spin-½ particles in the state |GHZ⟩=|↑⟩a|↑⟩b|↑⟩c−|↓⟩a|↓⟩b|↓⟩c2, (5) one can, by considering measurements of Pauli observables (σx, σy, σz),4 demonstrate a conflict between the predictions of quantum mechanics and those of a suitably local hidden variables theory. Note that the respective orientations of different ends of an experimental apparatus set up to conduct an experiment involving Pauli observables on a combined system will never differ by anything other than an angle proportional to π/2. To see how this conflict arises, note that in the GHZ state,5 the eigenvalues associated with σx and σy observables on individual subsystems are (as always) ±1, while each of the tripartite observables, σxa⊗σyb⊗σyc, σya⊗σxb⊗σyc, σya⊗σyb⊗σxc (6) (which, as the reader can verify, are compatible) takes the eigenvalue 1; that is, v(σxa⊗σyb⊗σyc)=v(σya⊗σxb⊗σyc)=v(σya⊗σyb⊗σxc)=1. (7) Thus preparing a tripartite system in the state represented by Equation (5) will yield correlations, expressed by Equation (7), between the results of certain measurements at sites a, b, and c. As a, b, and c may in general be quite distant from one another, it is reasonable to assume, if one is reasoning classically, that the subsystems measured at those sites became correlated while they were still in physical interaction with one another (that is, at state preparation) by way of some shared common cause (represented by the variable λ). Thus, at the time of measurement, nothing further should influence an individual outcome aside from the local properties of the experimental setup at a site and of the particle being measured there. Thus the results of the combined measurements in Equation (7) will be factorizable given λ; that is, v(σxa⊗σyb⊗σyc)=1=v(σxa)×v(σyb)×v(σyc), v(σya⊗σxb⊗σyc)=1=v(σya)×v(σxb)×v(σyc), v(σya⊗σyb⊗σxc)=1=v(σya)×v(σyb)×v(σxc). (8) But this cannot be. Multiplying the right-hand sides of Equation (8), and using the fact that v(σya)2=v(σyb)2=v(σyc)2=1, we have it that v(σxa)×v(σxb)×v(σxc)=1, which in turn (given factorizability) implies that v(σxa⊗σxb⊗σxc)=1. (9) Quantum mechanically, however, if we take the product (which we can since they are compatible) of the observables in Equation (6), then because σxσy=iσz=−σyσx, σxσz=−iσy=−σzσx,σyσz=iσx=−σzσy,σxσx=σyσy=σzσz=I, this must yield: (σxa⊗σyb⊗σyc)(σya⊗σxb⊗σyc)(σya⊗σyb⊗σxc) =−σxa⊗σxb⊗σxc. (10) Since each of the observables in Equation (6) takes the eigenvalue 1, this implies that v(σxa⊗σxb⊗σxc)=−1. (11) Thus Equation (9), which we were led to through the assumption of factorizability in Equation (8), contradicts the (empirically confirmed) predictions of quantum mechanics in Equation (11). We conclude, therefore, that the correlations expressed by Equation (7) do not admit of a locally causal description. 3 Classically Simulating Quantum Statistics The results just described have unquestionably deepened our understanding of the implications of our experience with quantum phenomena. Yet the proper analysis and precise significance of these implications for our understanding of nature has, unsurprisingly, been hotly debated both by philosophers and physicists.6 I will not engage directly in those debates here. I will rather ask a different (but as we will see later, not unrelated) question: could one build a classical machine to simulate the quantum measurement statistics described above (and if so, how)? If the answer is yes, then are there any quantum correlations that cannot be so simulated, or is it the case that one can build classical systems to reproduce every possible quantum mechanical correlational effect? As it turns out, this is, in fact, an active area of research, and as we will see shortly, we can indeed give an affirmative answer to the first question. Later, in Section 4, we will begin to consider the philosophical significance of this. In particular, we will see (in Section 5) that it forces us to qualify some of the conclusions we made in Section 2. In the process, we will answer the second question. 3.1 GHZ statistics Consider, first, the case of GHZ correlations.7Table 1 depicts a scheme for reproducing all of the Pauli measurement statistics associated with the state in Equation (5). Table 1. A scheme for reproducing all of the Pauli measurement statistics associated with the tripartite GHZ state a b c σx −R2R3 R2 R3 σy −iR1R2R3 iR1R2 iR1R3 σz R1 R1 R1 I 1 1 1 a b c σx −R2R3 R2 R3 σy −iR1R2R3 iR1R2 iR1R3 σz R1 R1 R1 I 1 1 1 Table 1. A scheme for reproducing all of the Pauli measurement statistics associated with the tripartite GHZ state a b c σx −R2R3 R2 R3 σy −iR1R2R3 iR1R2 iR1R3 σz R1 R1 R1 I 1 1 1 a b c σx −R2R3 R2 R3 σy −iR1R2R3 iR1R2 iR1R3 σz R1 R1 R1 I 1 1 1 In the table, a, b, and c refer to the three subsystems of the system. σx, σy, σz, and I are the Pauli spin operators. Shared variables R1, R2, and R3 are values of ±1 that are assigned to the various subsystems at state preparation through some sequence of local interactions. It is assumed that these variables possess determinate values prior to measurement, but that these values are completely ‘hidden’ in the sense that they can only be revealed by measurement, or what amounts to the same thing: two identical state preparations will, in general, yield (with equal likelihood) different values of R1 (likewise for R2 and R3).8 To determine the outcome of a combined Pauli measurement on the system, one multiplies the entries of the table corresponding to the measurements performed on each subsystem, discarding any remaining unsquared instances of i. For example, a measurement of σy on subsystem a is equivalent to σy⊗I⊗I. The result is given by −iR1R2R3, which, dropping the unsquared i, yields −R1R2R3=±1 with equal likelihood. Measuring σy⊗σx⊗σy, on the other hand, will yield v(σy⊗σx⊗σy)=−iR1R2R3R2iR1R3=−i2R12R22R32=1 (12) with certainty. And so on. It is easy to verify that Table 1’s predictions for every combined Pauli measurement match up with those of quantum mechanics. And like those of quantum mechanics, Table 1’s predictions for combined Pauli measurements on the GHZ state are not factorizable, as, for example, v(σy⊗I⊗I)×v(I⊗σx⊗I)×v(I⊗I⊗σy)=−R1R2R3R2R1R3=−1, (13) in contradiction with Equation (12). The instances of i in the table are the analogue of quantum mechanical ‘non-local’ influences. With only very minor tweaking, however, one can build a model similar to the one represented in Table 1, in which the resulting correlations are both factorizable and consistent with all of the predictions of quantum mechanics for these measurements. Our tweak will be that we will allow the parties to classically signal to one another as follows: Bob (measuring b) and Alice (measuring a) will agree that he will send her a single classical bit indicating whether or not he performed a σy measurement on his subsystem. Upon receipt of this bit, Alice should flip the sign of her local outcome if either she or Bob (or if both of them) measured σy. Thus instead of Equation (13), we will have v(σy⊗I⊗I)×v(I⊗σx⊗I)×v(I⊗I⊗σy)=R1R2R3R2R1R3=1, (14) in agreement with Equation (12). Note that this result is consistent with each of the three measurements, σy⊗I⊗I,I⊗σx⊗I,andI⊗I⊗σy, for each of them separately will yield a value of ±1 with equal probability. This small addition—a single bit—suffices to make the outcome of every joint measurement of Pauli observables factorizable in the model. The scheme generalizes. For an n-partite system in the GHZ state 1/2(|↑⟩⊗n±|↓⟩⊗n), only n – 2 bits are required. The same is true, moreover, not only for n-partite GHZ states, but also for any n-partite state in which each superposition term is expressible as a product of Pauli eigenstates (details are given in Tessier [2004]). One can imagine building a classical computational system, made up of smaller, spatially separated computational subsystems, utilizing a small amount of classical communication as described above, to instantiate this scheme. This is the sense in which one may say that the correlations between the results of Pauli measurements on systems in the GHZ state are ‘efficiently classically simulable’. I will elaborate upon this in more detail in Section 4. But let us first consider how to simulate measurement statistics on a system in the singlet state. 3.2 Singlet statistics As we saw earlier in Equation (4), Bell himself provided a local hidden variables description to account for Pauli measurement statistics in the singlet state.9 Presumably, we could build a classical computer to instantiate that description. Thus Pauli measurements on systems in the singlet state are efficiently classically simulable in this sense. In this case, no communication is needed, in fact. This is evident from Equation (4). It also follows from the fact that Pauli measurement statistics for any n-partite state, whose superposition terms are expressible as products of Pauli eigenstates, can be simulated using n – 2 bits. The singlet state is one such state for which n = 2. Thus for the singlet state, Pauli measurements can be simulated using 2−2=0 bits of communication. It turns out that for the singlet state, we can simulate far more than just Pauli measurements if we allow ourselves only a single additional bit. Astoundingly, we can actually recover the statistics associated with arbitrary projective measurements by using the following method (Toner and Bacon [2003]): Randomly choose, independently, two unit vectors, λ^1 and λ^2. At state preparation, share them with Alice and Bob, and instruct them to take their particles with them to separate, distant locations. Once there, have Alice measure her particle along direction m^, and output the result A=−sign(m^·λ^1). Have her then send a single classical bit, c=sign(m^·λ^1)×sign(m^·λ^2)=±1, to Bob, who finally measures along n^ and outputs the result B=sign[n^·(λ^1+cλ^2)]. 4 What Is a Classical Computer Simulation? We have just seen how to classically simulate the quantum measurement statistics described in Section 2 using only a few additional resources. We saw, specifically, that only a single bit is needed to recover the statistics associated with arbitrary projective measurements on the singlet state, and that only n – 2 bits are needed to recover the statistics associated with Pauli measurements on n-partite GHZ states. Let us now reflect on this. To begin with, let us try and explicate somewhat more precisely what is meant by the statement that a particular set of quantum correlational statistics has been ‘classically simulated’. What is a classical computer simulation, then? Most obviously, perhaps, we may say, to start with, that a classical computer simulation is something performable by a classical computer. Let us consider what is meant by the latter. A classical computer, whatever else it is, is a classical physical system, that is, a system that can be given a classical physical description. Some of the characteristics of classical description are the following: First, a complete classical description of a system, which in general will consist of many subsystems, is always separable into complete descriptions of those individual subsystems. Second, a classical description of the interactions between a system’s subsystems will not violate classical physical law. For instance, the speed by which such interactions propagate should be less than or equal to the speed of light.10 Third, and importantly, the behaviour of classical systems is always describable in principle in terms of local causes and effects, in the metaphysically deflationary sense explicated by Bell ([2004b]). Thus, while a classical system might evince certain correlations between experimental results on its various subsystems—which, in general, can be quite distant from one another—because the system is classical, these correlations will always be describable as arising from a common source situated in the intersection of the prior light cones of the associated measurement events. That is, once we take into account their common causes, these correlations are always factorizable. Classical systems do not violate Bell (or GHZ, or similar) inequalities. Thus classical computer simulations, which are run on classical computers, which are instantiated by classical physical systems, do not either. This is perfectly obvious. And yet it might still strike one as strange. The results of, say, a Bell experiment seem to be evidence for some sort of influence (even if only benign) between the spatially separated subsystems of entangled systems, and we usually take this to demonstrate that quantum systems are ‘non-local’ (or perhaps ‘non-separable’) in some sense (Maudlin [2011]); that is, we take such influences to demonstrate that no locally causal description—no ‘local hidden variables theory’—of the outcomes of these experiments is possible. But the influence of one spatially separated subsystem on another is precisely what we find in the protocols designed to reproduce these outcomes, described in Section 3. And yet these are locally causal descriptions. How can this be? A way of resolving this tension is provided in (Cuffaro [2017]). I will not reproduce that discussion in its entirety here, but rather, in the remainder of this section, I will only summarize the main points emerging from it that are relevant to our own investigation, whose main conclusions will then be elaborated upon in the sequel (for a full elaboration and defence of the points summarized in this section, see the aforementioned article). In my earlier article, I argued that when we make a judgement to the effect that no local hidden variables theory is capable of recovering the quantum mechanical predictions associated with a Bell experiment, implicit in this judgement is a set of constraints upon the kinds of local hidden variables theory that we are willing to entertain. To put it another way: with a little imagination, it is always possible to conceive of loopholes to the Bell inequalities, but only some of these loopholes—the plausible ones—are worth the bother of trying to close. This means, however, that a conclusion such as ‘no (plausible) hidden variables theory of type X is able to reproduce the experimental predictions of quantum mechanics with respect to preparation procedure Y’ cannot really be forced on the basis of Bell’s (or a similar) (in)equality alone. Strictly speaking, to force such a conclusion, what one means by ‘plausible’ must be spelled out first. Usually, of course, we do not really need to be explicit. There is a set of criteria by which we mark a hidden variables theory as plausible that is implicitly understood by everyone. This set includes such things as consistency with our other theories of physics, and with the body of our experiential knowledge in general; we expect a candidate local hidden variables theory not to be conspiratorial, and so on. These and other plausibility constraints are associated with what I called the ‘theoretical’ context (Cuffaro [2017]), wherein we take Bell’s and related results to enlighten questions such as, ‘is there an alternative (locally causal) description of the natural world that can recover the confirmed predictions of quantum mechanics?’. There are other contexts, besides the theoretical, to which Bell’s and related (in)equalities are applicable. In particular, in (Cuffaro [2017]), I described the ‘practical’ context, wherein our concern is not with alternative theories of the natural world, but rather with what we are capable of building with the aim of (classically) reproducing the statistical predictions of quantum mechanics. In the practical context, in other words, we are concerned not with ‘local hidden variables’ descriptions of the natural world, but with ‘practical classical’ descriptions of machines that are possible for us to build to achieve particular ends (in the sequel, I will be using the generic term ‘locally causal description’ to refer to both). Now consider the two classical schemes for simulating quantum statistics described in Section 3. Either of these can be thought of as an alternative locally causal description of the observation of a given set of measurement statistics in the following way: Begin by asking what we really need to recover in an alternative locally causal description of a combined measurement outcome. The answer, invariably, will be just the actual observation of the combined outcome itself. What I mean is this: Although quantum mechanics posits its own particular dynamics—involving, for example, the ‘state collapse’ of a subsystem of a Bell pair upon the measurement of the other subsystem11—an alternative description of the events leading up to the observation of a combined measurement outcome can, in principle, disagree entirely with the quantum mechanical dynamical description. All that an alternative locally causal description needs to be faithful to, in principle, is our actual observation of the combined result. Importantly, in order to be in a position to assert that one has observed a combined measurement outcome, one must first have combined the individual measurement outcomes; Alice and Bob must report their results to Candice (or to each other) over tea or by telephone or in writing or by using some other physical means. This is absolutely necessary and there is simply no escaping it. While these results are being brought together, however, Alice and Bob have time to (conspiratorially) exchange—at subluminal velocity—a series of finite signals with one another, and thus coordinate—and, if necessary, ‘correct’—their individual measurement outcomes in the manner described in Section 3. But notice now that with respect to the observation of the combined measurement outcome—which, to reiterate, is ultimately what we are required to explain—all of this conspiratorial signalling will have occurred in its past light cone. It is thus part of a posited state preparation—a common cause—for that combined outcome and it thus comprises part of a classical, locally causal, alternative description of that combined outcome (Figure 1). Figure 1. View largeDownload slide A conceivable explanation of the observation of the result of a joint experimental outcome, AB, that posits local (conspiratorial) influences interacting within region C. This figure is adapted from (Cuffaro [2017]). Figure 1. View largeDownload slide A conceivable explanation of the observation of the result of a joint experimental outcome, AB, that posits local (conspiratorial) influences interacting within region C. This figure is adapted from (Cuffaro [2017]). In the theoretical context, of course, such a locally causal description would be ruled out as wildly implausible. It would simply be, for this author at any rate, too conspiratorial a loophole to take seriously, let alone to spend any of one’s research funds to try and close. But as I argue in (Cuffaro [2017]), in the practical context, we will not fret if our alternative characterization of a set of measurement statistics is conspiratorial or overly ad hoc from the theoretical point of view, for in the practical context, it is always assumed that the system under consideration has been purposely designed to recover those statistics.12 What we will rule out as implausible, rather, are only descriptions of classical systems that would be too ‘hard’ for someone to build, where ‘hard’ and likewise ‘easy’ are defined naturalistically—that is, by appealing to our best scientific theory of the subject: complexity theory. In complexity-theoretic terms, classically simulating a quantum system is too hard if it requires an ‘enormous’ number of additional resources as compared with those used by the quantum system, where by ‘enormous’ one means a number of additional resources that grows exponentially with respect to the size of the input to the problem, n (that is, on the order of kn). In contrast, ‘a few’ resources are at most polynomial with respect to n (that is, on the order of nk).13 Now if we consider a multi-partite quantum system in the GHZ state, 1/2(|↑⟩⊗n±|↓⟩⊗n), we know, from Section 3.1, that as the number of subsystems, n, grows larger, the number of additional classical bits required to simulate the all-or-nothing GHZ correlations associated with the system does not grow very fast at all (we require only n – 2 additional bits). In this sense, we can say that no matter how large n gets, an alternative classical description of the GHZ correlations present in the system is ‘no harder’ to realize, in a complexity-theoretic sense, than the quantum description. Thus if we imagine being presented with a large ‘black box’ whose inner workings are opaque to us, but which has been constructed to evince CHSH or n-partite GHZ correlations, we can in that case legitimately be sceptical if it is claimed that this box is quantum. For in response to that claim, we can always produce a description of a classical system capable of generating those correlations just as ‘easily’. We can produce, in other words, an alternative locally causal description of the box that is plausible in the practical context. This is what both protocols described in Section 3 amount to.14 5 Comparing the All-or-Nothing GHZ with Statistical (In)equalities We saw in Section 3.2 that if we would like to classically simulate the measurement statistics associated with Pauli measurements on a system in the singlet state, we can do so without the aid of any classical communication. To recover the statistics for arbitrary measurements, on the other hand, we require a single classical bit propagated at subluminal speed. In Section 3.1, we saw that recovering the statistics associated with Pauli measurements on a system in a GHZ state requires only a number of additional resources that are linear in the number of systems, n. All of these measurement statistics are plausibly recoverable, therefore, in the practical context. What if we would like to recover the statistics associated with arbitrary measurements on a system in a GHZ state? Unfortunately, unlike the singlet and other Bell states, it does not seem likely that we could plausibly do so. Tessier ([2004], p. 116) notes that the number of classical bits required to model the quantum mechanical predictions associated with arbitrary projective measurements on n-partite states (for n≥3) in a model like the one pictured in Table 1 is likely to be unbounded. This is consistent with other results due to Jozsa and Linden ([2003]) and Abbott ([2012]), who show (using different methods) that for systems in pure states, an exponential speed-up of quantum over classical computation is possible only when one has available multipartite entanglement with a number of systems n≥3. Returning now to the GHZ argument that I discussed in Section 2.2. As is evident from some of the quotations surveyed in Section 1, it is hard to overestimate the impact that the GHZ proof has had upon the physical and philosophical communities. To quote Mermin ([1990], p. 11) at length: This is an altogether more powerful refutation of the existence of elements of reality than the one provided by Bell’s theorem for the two-particle EPR experiment. Bell showed that the elements of reality inferred from one group of measurements are incompatible with the statistics produced by a second group of measurements. Such a refutation cannot be accomplished in a single run, but is built up with increasing confidence as the number of runs increases […] In the GHZ experiment, on the other hand, the elements of reality require a class of outcomes to occur all of the time, while quantum mechanics never allows them to occur. As we saw earlier, however, before one can properly make statements like this one concerning the relative power of the Bell and GHZ (in)equalities, one must first be clear on the context in which such statements are being made. From a theoretical point of view, it may be that the GHZ argument is more powerful than Bell’s in the above sense. Thus I take no issue with Mermin’s statement as long as it is qualified in this way, as I assume it implicitly is for Mermin. This said, in the practical context, Mermin’s statement is actually false. For from a practical point of view, both the Bell and GHZ (in)equalities are equally weak, in the sense that plausible alternative locally causal descriptions can be provided to account for the associated statistics in both cases. The contrast being drawn in the quote above, however, and in the statements of the other philosophers and physicists cited in Section 1 is not specifically between the results of Bell and GHZ. It is rather between the GHZ equality insofar as it is an all-or-nothing result, and Bell’s inequality insofar as it is a statistical result. But as we have seen, it is quite easy to recover the statistics associated with Pauli measurements on the GHZ state—the measurements for which an all-or-nothing equality can be proved—in the practical context. In this context, in order to use the GHZ state as the basis for a bona fide no-go theorem, one must consider measurements outside of the Pauli group, for the statistics associated with these measurements cannot be plausibly recovered in a locally causal model. Once one does so, however, the ensuing correlations between individual measurement results will no longer be strict but probabilistic. Thus it is statistical results, rather than the all-or-nothing GHZ, that are more powerful in the practical context. 6 General Discussion We have chosen to call classical computer simulations of quantum mechanical correlational statistics ‘locally causal alternative descriptions’ of those correlational statistics. This is not the way it is normally put in the quantum information literature. What is normally claimed for these schemes is rather that they quantify—in terms of the number of classical bits required to reproduce them—the departure from classicality of quantum correlations (besides the work of Tessier, Tessier et al., and Toner and Bacon mentioned above, see also, for example, Brassard et al. [1999]; Rosset et al. [2013]). This way of framing their significance is certainly not incorrect. It is certainly, moreover, very useful.15 From a more philosophical point of view, however, it is preferable to call a spade a spade, so to speak. As the example above involving the opaque black box illustrates, a plausible description of a locally causal computer simulation of a class of quantum phenomena is of the same general kind as a plausible locally causal theoretical description of a class of quantum phenomena. Both kinds of description represent alternative plausible stories one can put forward if one is sceptical of the quantum mechanical description of some set of observations. The essential difference between the theoretical and practical contexts lies in the different presuppositions we make concerning the origin of the physical systems under consideration in each case. In the practical context, we presuppose that a particular system has been designed and built by some rational agent for a particular purpose. We do not presuppose nature to be purposeful in this way in the theoretical context.16 Additionally, calling a spade a spade helps us to understand one reason why the practical context is—or anyway should be—interesting from the traditional philosopher of physics’ point of view. Indeed, such a reader may have been wondering why she should care at all about the practical context—why she should care for anything but the correct theory of the natural world and for its deeper consequences for our fundamental understanding of the ontology of that world. To help appreciate why one should care, imagine if it were possible to easily build classical machines to reproduce the statistics associated with every quantum mechanical measurement (regardless of state). If this were possible, it would be hard to imagine it not having some bearing on our interpretation of the quantum formalism, or at least on our general metaphysical world view. From this and the conclusions of (Pitowsky [1996], pp. 175–6), for instance, it would follow that the metaphysical difference between the indeterminism present in classical as opposed to quantum physics is merely one of degree (in other words, nothing at all). There is no need to engage in such speculation, however, for as we saw in the previous section, it turns out that building such universal simulators is not plausible. And yet it is still quite interesting for a traditional philosopher of physics to know that one could plausibly build classical physical systems to fully reproduce the observable correlational behaviour of systems in Bell states. It is also interesting to know that other quantum correlational phenomena cannot be so reproduced. The reason all of this is interesting is that in both the practical and theoretical contexts, we are speaking about physical systems, after all. Considering the practical context can thus help us to understand the very limits of classical description. It can help us to understand, that is, just how far quantum physical description as such outstrips classical physical description as such. Physics is traditionally conceived, to my mind rightly so, as a primarily theoretical activity, in the sense that it is the general goal of physics to tell us, even if only indirectly, what the world is like independently of ourselves (Fuchs [unpublished (a)], pp. 5–6, [unpublished (b)], pp. 22–3). It is worth noting, however, that this is not the case with every science. We have already discussed computer science. Chemistry is, arguably, a further example of a more practically oriented discipline (Bensaude-Vincent [2009]). But although chemistry (for example) may be a practical science in this sense, there is obviously no question as to whether theoretical knowledge of the systems she is manipulating is relevant to the chemist. Of course it is. Likewise, I want to claim, considering the practical aspect of the systems he is considering can be illuminating for the theoretical investigations of the physicist, in the ways I suggested above, and perhaps in further, as yet unimagined, ways. Explicitly noting the important differences in the presuppositions associated with the practical and theoretical contexts, on the other hand, encourages us to be wary of inappropriately conflating them. We saw, in Section 5, that a practical investigator should take care not to assume that certain no-go results, which are valid in the theoretical context, will constrain what is possible for her to accomplish in the practical context. Additionally, as I have argued in (Cuffaro [2017]), practical investigators attempting to isolate and/or quantify the computational resources provided by physical systems may be in danger of conceptual confusion if they are not cognizant of the differences between the two contexts. Theoretical investigators should also be careful not to conflate the two contexts inappropriately. For instance, Jeffrey Bub’s ([2004]) claim that the fundamental aim of physics consists in the representation and manipulation of information seems to me to be in danger of doing so. Bub’s argument for this conclusion is based on the fact that within the abstract C* algebraic framework, one can characterize quantum theory purely in terms of a set of information theoretic constraints (specifically, no signalling, no broadcasting, and no unconditionally secure bit commitment),17 and that given these constraints, any alternative mechanical theory that aims to solve the measurement problem faces an in-principle problem of underdetermination. This means, for Bub ([2004], p. 243), that ‘our measuring instruments ultimately remain black boxes at some level’. It would take us too far distant from the primary concern of this article to consider Bub’s arguments for this last claim in detail. Let me just say here that even if this claim is true18—which, for different reasons, is a claim that I am generally sympathetic towards and which, to be fair, actually constitutes the main thrust of Bub’s article—it does not obviously follow from it that ‘the appropriate aim of physics at the fundamental level [is] the representation and manipulation of information’ (Bub [2004], p. 242), or that ‘an entangled state should be thought of as a nonclassical communication channel that we have discovered to exist in our quantum universe, i.e., as a new sort of nonclassical “wire”’ (Bub [2004], p. 262). All that would seem prima facie to follow from the claim that our mechanical description of nature must run short is that we cannot descend below the level of the language of information theory when characterizing quantum phenomena. But one could argue that the proper lesson to draw from this is merely that as a result, we must be all the more careful to be precise about what it is that we are saying when we use informational language—careful to distinguish, that is, the circumstances in which we are discussing the ways in which we can use physical systems from the circumstances in which we are describing the nature of the physical world itself, as it exists in itself (that is, without our intervention). The idea that we can and do make such distinctions lies at the very basis of our foregoing discussion. The analysis in Section 4 began from the fact that our conceptions of ‘classical system buildable by a human being’ and of ‘classical system existing naturally’ are—at least pre-theoretically—very different conceptions. I then explicated these pre-theoretic conceptions more precisely in terms of differing sets of plausibility constraints associated with each. What would seem to be asserted by one sympathetic to the view expressed above, however, is that our pre-theoretic intuitions are false—that really, after all, there is only one context worth considering: what I have above referred to as the practical context. It is not incoherent to hold such a view. But if it really is the case that quantum theory is ultimately about nothing other than representing and processing information—that (as seems to be implied by this) the practical context should be our primary concern vis á vis foundational investigation—then it is hard to see why GHZ correlations, in particular, should surprise anyone. For from the practical point of view, GHZ correlations are no more inexplicable than the correlations between the colours of Professor Bertlemann’s socks (if you happen to notice that one of these is pink, you can rest assured that the other is not; see Bell [2004a]). In other words, a plausible locally causal explanation, in the sense of Section 3, is readily available for both of these phenomena in the practical context. But, in fact, we are surprised by GHZ correlations (recall the survey of the reactions to the equality in Section 1), and it is incumbent upon one sympathetic to a view like Bub’s to explain to us why this is so. Before closing this section, let me note that the foregoing considerations are not intended as an argument against a view like Bub’s. Rather, the sketch of an argument contained in the foregoing few paragraphs should be regarded as an invitation to anyone sympathetic to the view that quantum theory just is about representing and manipulating information—a much stronger and deeper claim than the statement that the description of our measuring instruments must necessarily remain opaque at some level—to clarify this position in light of the points that have been made in this article. I will also note that in the foregoing few paragraphs I have characterized informational language as belonging to the practical, rather than the theoretical, context. This is true within a view like Bub’s. However, it may not be true for one who interprets information differently. In particular, it may not be true for one who thinks of information as a kind of physical stuff (Landauer [1996]). There are independent reasons to be sceptical of such a view (see, for instance, Timpson [2013], Section 3.7.1), which I will not discuss here. I will instead close this section by noting that there are obviously interconnections between our discussion of the distinction between the practical and theoretical contexts and the debate over the interpretation of the concept of information. However, clarifying these interconnections would be a project in itself, which I, unfortunately, do not have the space to pursue here (for more on the debate over the interpretation of information, see Lombardi et al. [2015]). 7 Conclusion I reviewed the Bell and GHZ (in)equalities in Section 2, and then in Section 3, I discussed schemes by which one could simulate, using a classical computer, the statistics associated with those (in)equalities. In Section 4, I argued that classical computer simulations are locally causal descriptions in Bell’s sense and hence do not violate the Bell or GHZ (in)equalities, and I argued that this fact is not in tension with our normal judgements regarding the significance of these (in)equalities, as long as one understands that our normal judgements are not valid in the context of a discussion of machines designed to achieve a particular purpose. I then argued, in Section 5, that this has implications for our understanding of the relative strength of the GHZ all-or-nothing equality vis-á-vis statistical inequalities, and I made the perhaps surprising observation that the former actually has no force in the practical context, despite being a quite remarkable result in the theoretical context. I ended, in Section 6, by discussing the general implications of all of the foregoing for our understanding of the limits of classical description, and for our understanding of the fundamental aim of physical investigation. Both in the practical and in the theoretical contexts, one’s concern is, ultimately, very concrete: one is in both cases concerned with plausible descriptions of actual physical systems existing in the world. The difference between the two contexts enters essentially in the different presuppositions we make concerning the origin of the physical systems under consideration. In the practical context, we presuppose that a particular system has been designed and built by some rational agent for a particular purpose. We do not presuppose this in the theoretical context. But because both contexts are concerned with describing physical systems, both contexts are relevant to our understanding of physical systems. And the practical context is particularly illuminating in that it allows us to investigate more thoroughly the possibilities inherent in physical systems. It helps us to probe further and to understand better just what the limits of classical description are, and just how far quantum description outstrips it. At the same time, we must remain on guard not to be misled into thinking that the practical context can supersede the theoretical context for the purposes of our foundational and philosophical investigations into the nature of the (quantum) world. Both of these contexts are rather complementary modes of investigation, aimed at the world and our experience. Acknowledgements Section 6 of this paper was inspired by my conversations with William Demopoulos in the summer of 2013. The paper also benefited from the helpful comments of Samuel Fletcher and two anonymous referees at BJPS. Thanks also to Radin Dardashti, Ryan Samaroo, and to my audiences at the universities of Bristol, Florence, Helsinki, and Oxford for fruitful discussion. The bulk of the research for this project was conducted during my stay at the Munich Center for Mathematical Philosophy, and was funded by the Alexander von Humboldt Foundation. Footnotes 1 The general idea of considering the differences between physical systems from the point of view of a practical context is also present in the work of Pitowsky. For example, in (Pitowsky [1990]), two different ‘practical versus theoretical’ distinctions are mentioned. The first is between computational devices and their natural physical ‘analogues’ (for example, consider a device to determine the secondary structure of a protein versus its physical analogue: the natural process of protein synthesis). The second distinction is not conceptual but pragmatic; for example, for a given NP-complete computational problem, one can ask the ‘practical question’ regarding whether many or most instances of it are effectively efficiently solvable. In (Pitowsky [2002]), the practical question posed regards the number of computational resources required to transform a given system into a state from which a solution to a computational problem can be achieved in polynomial time with a certain probability. 2 Note that, as this quote hints at, Clifton et al. do not argue that the GHZ proof is stronger than Bell’s in every respect. 3 These are named after Bell, the discoverer of the first such result (Bell [2004c]). 4 I have omitted the trivial identity operator I from this list. 5 The following exposition is based on (Mermin [1990]). 6 See, for instance, (Cushing and McMullin [1989]). 7 What follows is a summary of the model given in more detail in (Tessier [2004]; Tessier et al. ([2005])). 8 Note that no such interpretation of R1, R2, and R3 is given in Tessier ([2004]; Tessier et al. [2005]), but it is implicit. 9 The theory given in Equation (4) applies not just to Pauli measurements, of course, but more generally, as we saw, to any combined spin measurement in which the angles at the respective ends of the experimental apparatus differ in proportion to π/2. 10 I am taking ‘classical’ in the sense in which it is typically used in discussions of quantum mechanics; that is, I take classical theory to include special and general relativity. 11 What I am referring to here is merely the standard wave packet reduction postulate assumed in ordinary quantum mechanics. I do not mean to imply that standard quantum mechanics includes an agreed-upon detailed theory of state collapse. 12 Kent ([2005]) actually describes an alternative to quantum theory (his is not a practical investigation) that exploits a similar loophole to the one described above. Kent does not consider his own alternative theory to be as wildly implausible as I think the model described above would be in the theoretical context (as Kent’s is a theoretical investigation, more effort is made to construct a theory that is not excessively ad hoc). That said, Kent ([2005], p. 6) describes a number of strong plausibility considerations (though for him, not conclusive) for ruling out his theory. 13 See (Arora and Barak [2009]). 14 It must not be forgotten, of course, that at least in the case of GHZ states, it is only the outcomes associated with Pauli measurements that are efficiently classically simulable in the sense just described. We will return to this point shortly. 15 Indeed, the general subject of classical simulations of quantum systems is an important and burgeoning area of modern physics (Lee and Thomas [2002]; Borges et al. [2010]). 16 I am not expressing a theological opinion with this statement. I mean only that we do not presuppose this—not in this century, at any rate—for the purposes of scientific investigation. 17 This is proved in (Clifton et al. [2003]). For an illuminating discussion of why one should be sceptical about the significance claimed for this characterization result, see (Myrvold [2010]). 18 For a criticism, see (Timpson [2013], Section 8.3). References Abbott A. A. [ 2012 ]: ‘The Deutsch–Jozsa Problem: De-quantisation and Entanglement ’, Natural Computing , 11 , pp. 3 – 11 . Google Scholar CrossRef Search ADS Arora S. , Barak B. [ 2009 ]: Computational Complexity: A Modern Approach , Cambridge : Cambridge University Press . Google Scholar CrossRef Search ADS Bell J. S. [ 2004a ]: ‘‘Bertlmann’s Socks and the Nature of Reality’, in his Speakable and Unspeakable in Quantum Mechanics , Cambridge : Cambridge University Press , pp. 139 – 58 . Google Scholar CrossRef Search ADS Bell J. S. 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Reconsidering No-Go Theorems from a Practical Perspective

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Abstract

Abstract I argue that our judgements regarding the locally causal models that are compatible with a given constraint implicitly depend, in part, on the context of inquiry. It follows from this that certain quantum no-go theorems, which are particularly striking in the traditional foundational context, have no force when the context switches to a discussion of the physical systems we are capable of building with the aim of classically reproducing quantum statistics. I close with a general discussion of the possible implications of this for our understanding of the limits of classical description, and for our understanding of the fundamental aim of physical investigation. 1 Introduction 2 No-Go Results   2.1 The CHSH inequality   2.2 The GHZ equality 3 Classically Simulating Quantum Statistics   3.1 GHZ statistics   3.2 Singlet statistics 4 What Is a Classical Computer Simulation? 5 Comparing the All-or-Nothing GHZ with Statistical (In)equalities 6 General Discussion 7 Conclusion 1 Introduction Bell’s inequalities, and results like them, are often presented as ‘no-go theorems’ in the philosophical and physical literature. They are portrayed as demonstrating, for instance, that no ‘locally causal’ (Bell [2004b], Section 7) description of the empirically confirmed predictions of quantum mechanics is possible, that one cannot ascribe ‘elements of reality’, in Einstein, Podolsky, and Rosen's (EPR's) sense (Einstein et al. [1935]), to quantum systems consistently with such predictions, and so on. As I will argue below, however, thinking of Bell’s and related results in this way—without further qualification—can be misleading, for they are not no-go theorems per se. Rather, they are best understood as establishing certain general constraints on locally causal descriptions of joint measurement outcomes. In order to turn Bell’s and similar results into no-go theorems, one must do more than just state them. One must also interpret them in the context of the question one takes them to be informing. Associated with a given context are a number of additional constraints, according to which we judge a locally causal description to be either plausible or implausible in that context. Normally, these ‘plausibility constraints’ are left unstated and this is mostly harmless most of the time, for the relevant context is usually understood to be what I will below be calling the ‘theoretical’. Here, the question we take Bell’s and related results to be answering is the question of whether there is an alternative locally causal description of the natural world that can recover the confirmed predictions of quantum mechanics. The set of plausibility constraints on locally causal descriptions—or anyway, the general character of such constraints—in the context of this question is implicitly understood by all. Bell’s and related results are relevant to other contexts having to do with physical systems as well, however. In particular, in what I will below be calling the ‘practical’ context, we are not concerned with alternative theories of the natural world, classical or otherwise. What we are rather concerned with are the kinds of physical system that we can build in order to recover a particular probability distribution or set of measurement outcomes. As I will describe in more detail below, the plausibility constraints we impose on locally causal descriptions in this context are importantly different from those we impose in the theoretical context. Thus the kinds of locally causal description we think are worth considering as alternatives to quantum description in the former context will be different from the kinds of locally causal description we think are worth considering in the latter. This has interesting and important implications. Previously, I have argued that distinguishing between these two contexts can help us to understand the way certain physical resources are taken advantage of in the science of quantum computation (Cuffaro [2017]). The present article builds upon that work.1 In this article, I will argue that making this distinction can help us to better understand the relative power and scope of no-go theorems, and provide us with an alternative point of view from which to consider the fundamental differences between classical and quantum systems in general. Consider, in particular, Greenberger et al.'s ([1989]) so-called ‘all-or-nothing’ equality. It is typical to see this equality referred to in the philosophical and foundational literature as a more powerful refutation of local causality than Bell’s own statistical inequality. The reason for this is that while a violation of the Greenberger–Horne–Zeilinger (GHZ) equality can in principle be shown with a single quantum experiment, a violation of Bell’s requires repeated quantum experiments to demonstrate (and even then only with increasing, but never absolute, confidence). Greenberger et al.([1990]), p. 1131), for example, write: ‘This incompatibility with quantum mechanics is stronger than the one previously revealed for two-particle systems by Bell’s inequality, where no contradiction arises at the level of perfect correlations’. Clifton et al.([1991], p. 174) likewise write of their improvement on the original GHZ proof: ‘One difference between Bell’s theorem and ours is that his yields a statistical contradiction, whereas ours leads to an algebraic one. So in one respect, our conclusion is stronger’.2 Mermin, for whom the GHZ proof is ‘spectacular’ ([1993], p. 810), enthusiastically agrees: ‘This is an altogether more powerful refutation of the existence of elements of reality than the one provided by Bell’s theorem’ ([1990], p. 11). And Maudlin ([2011], p. 26) concurs: ‘the GHZ scheme brings home the problem for locality all the more sharply’. So, also, does Vaidman ([1999], p. 615): ‘The GHZ proof is the most clear and persuasive proof of nonexistence of local hidden variables’. Indeed, for Vaidman ([1999], p. 616), the significance of the GHZ proof is quite profound: ‘analysis of the GHZ work led me to accept the bizarre picture of quantum reality given by the many worlds interpretation’. Vaidman is not alone in taking the GHZ equality as a starting point from which to consider the question of the interpretation of quantum mechanics (Hemmo and Pitowsky [2003]). There is nothing inappropriate about this kind of talk, so long as it is confined (as it presumably is for the above authors) to what I have called the theoretical context. Interestingly, however, when we move from the theoretical to the practical context, then the above statements are false. As I will elaborate upon in more detail below, in the practical context, the all-or-nothing GHZ theorem loses its force. In the practical context, in fact, it is only statistical inequalities that can legitimately be turned into no-go theorems (although, also interestingly, one must indeed go beyond Bell’s theorem and its associated bipartite entangled states to entangled states of multipartite systems to show this). From this we may say that from the ‘absolute’ point of view—that is when considering the limits of plausible classical physical description as such—it is statistical inequalities, rather than the all-or-nothing GHZ, that are more powerful in the sense alluded to by the authors above, for statistical inequalities—and not the all-or-nothing GHZ—allow us to see that there are quantum statistics that even we can’t plausibly build classical systems to reproduce. The article proceeds as follows: In Section 2, I review Clauser, Horne, Shimony, and Holt's (CHSH's) variant of Bell’s inequality as well as the GHZ equality. In Section 3, I describe some schemes by which one could simulate, using a classical computer, some of the statistics associated with Bell and GHZ states. In Section 4, the meaning of the term ‘classical computer simulation’ is reflected upon. The implications of these reflections for our understanding of the relative strength of statistical no-go theorems vis á vis the all-or-nothing GHZ theorem are considered in Section 5. Finally, their general implications for our understanding of the limits of classical description, and for our understanding of the fundamental aim of physical investigation, are considered in Section 6. 2 No-Go Results 2.1 The CHSH inequality Consider the singlet state |Ψ−⟩=1/2(|↑⟩|↓⟩−|↓⟩|↑⟩), of two spin-½ particles. For a system in this state, the expectation value for joint experiments on its two subsystems is given by the following expression: ⟨σm⊗σn⟩=−m^·n^=−cos θ. (1) Here, σmandσn represent spin-m and spin-n experiments on the first (Alice’s) and second (Bob’s) subsystem, respectively, with m^andn^ the unit vectors representing the orientations of their two experimental devices, and θ the difference in these orientations. It is well known that it is not possible to provide an alternative theory accounting for the predictions associated with this state if that theory makes the very reasonable assumption that the probabilities for local experimental outcomes on Alice’s (and likewise Bob’s) subsystem are completely determined by her local experimental setup, together with a shared hidden variable, λ, taken on by both subsystems at the time the joint state is prepared (that is, while A and B are still physically interacting). For given such a theory, we will have that ⟨σm⊗σn⟩λ=⟨Aλ(m^)×Bλ(n^)⟩, (2) where Aλ(m^)∈{±1}andBλ(n^)∈{±1} represent the results of spin experiments on Alice's and Bob's subsystems, given a specification of the hidden variable λ. But consider the following expression relating the expectation values of different combined spin experiments on Alice’s and Bob’s subsystems for arbitrary directions m^,m^′,n^,andn^′: CHSH: |⟨σm⊗σn⟩+⟨σm⊗σn′⟩|+|⟨σm′⊗σn⟩−⟨σm′⊗σn′⟩|. When Equation (2) holds, one can show that CHSH≤2. (3) The ‘CHSH inequality’ (Clauser et al. [1969]) is one of a family of similar expressions, known collectively as the ‘Bell inequalities’,3 which must be satisfied by locally causal theories aiming to account for the statistics associated with combined spin measurements. The statistical predictions of quantum mechanics violate the CHSH inequality for some experimental configurations. For example, let the system be in the singlet state and let the unit vectors m^,m^′,n^,andn^′ (taken to lie in the same plane) have the orientations 0,π/2,π/4,and−π/4, respectively. The differences, θ, between the different orientations (that is, m^−n^,m^−n^′,m^′−n^, and m^′−n^′) will all be in multiples of π/4 and from Equation (1), we will have: |⟨σm⊗σn⟩+⟨σm⊗σn′⟩|+|⟨σm′⊗σn⟩−⟨σm′⊗σn′⟩|=22≰2. We normally conclude from this that the predictions of quantum mechanics for arbitrary orientations m^,m^′,n^, and n^′ cannot be reproduced by any alternative theory in which all of the correlations between subsystems are due to a common parameter endowed to them at state preparation. But note that these predictions can be reproduced by such a hidden variables theory for certain special cases. In particular, the inequality is satisfied by quantum mechanics when m^ and n^,m^ and n^′,m^′ and n^, and m^′ and n^′ are all oriented at angles with respect to one another that are given in multiples of π/2. Note that these are the cases for which Equation (1) predicts perfect correlation ( θ=π), perfect anti-correlation (θ = 0), or no correlation at all ( θ=π/2). For example, let λ determine Alice’s and Bob’s measurement results in the following way: Aλ(m^)=sign(m^·λ^), Bλ(n^)=−sign(n^·λ^), (4) where sign(x) is a function that returns the sign (+ or −) of its argument. The reader can verify that Equation (4) will recover all of the statistical predictions associated with the singlet state in Equation (1) just so long as the difference in orientation between m^ and n^ is some multiple of π/2 (Bell [2004c], p. 16). 2.2 The GHZ equality We have just seen that the statistics associated with the singlet state for measurement angles differing in proportion to π/2 are reproducible in a local hidden variables theory, such as in Equation (4). But this is not true for every entangled state. In particular, for a system of three spin-½ particles in the state |GHZ⟩=|↑⟩a|↑⟩b|↑⟩c−|↓⟩a|↓⟩b|↓⟩c2, (5) one can, by considering measurements of Pauli observables (σx, σy, σz),4 demonstrate a conflict between the predictions of quantum mechanics and those of a suitably local hidden variables theory. Note that the respective orientations of different ends of an experimental apparatus set up to conduct an experiment involving Pauli observables on a combined system will never differ by anything other than an angle proportional to π/2. To see how this conflict arises, note that in the GHZ state,5 the eigenvalues associated with σx and σy observables on individual subsystems are (as always) ±1, while each of the tripartite observables, σxa⊗σyb⊗σyc, σya⊗σxb⊗σyc, σya⊗σyb⊗σxc (6) (which, as the reader can verify, are compatible) takes the eigenvalue 1; that is, v(σxa⊗σyb⊗σyc)=v(σya⊗σxb⊗σyc)=v(σya⊗σyb⊗σxc)=1. (7) Thus preparing a tripartite system in the state represented by Equation (5) will yield correlations, expressed by Equation (7), between the results of certain measurements at sites a, b, and c. As a, b, and c may in general be quite distant from one another, it is reasonable to assume, if one is reasoning classically, that the subsystems measured at those sites became correlated while they were still in physical interaction with one another (that is, at state preparation) by way of some shared common cause (represented by the variable λ). Thus, at the time of measurement, nothing further should influence an individual outcome aside from the local properties of the experimental setup at a site and of the particle being measured there. Thus the results of the combined measurements in Equation (7) will be factorizable given λ; that is, v(σxa⊗σyb⊗σyc)=1=v(σxa)×v(σyb)×v(σyc), v(σya⊗σxb⊗σyc)=1=v(σya)×v(σxb)×v(σyc), v(σya⊗σyb⊗σxc)=1=v(σya)×v(σyb)×v(σxc). (8) But this cannot be. Multiplying the right-hand sides of Equation (8), and using the fact that v(σya)2=v(σyb)2=v(σyc)2=1, we have it that v(σxa)×v(σxb)×v(σxc)=1, which in turn (given factorizability) implies that v(σxa⊗σxb⊗σxc)=1. (9) Quantum mechanically, however, if we take the product (which we can since they are compatible) of the observables in Equation (6), then because σxσy=iσz=−σyσx, σxσz=−iσy=−σzσx,σyσz=iσx=−σzσy,σxσx=σyσy=σzσz=I, this must yield: (σxa⊗σyb⊗σyc)(σya⊗σxb⊗σyc)(σya⊗σyb⊗σxc) =−σxa⊗σxb⊗σxc. (10) Since each of the observables in Equation (6) takes the eigenvalue 1, this implies that v(σxa⊗σxb⊗σxc)=−1. (11) Thus Equation (9), which we were led to through the assumption of factorizability in Equation (8), contradicts the (empirically confirmed) predictions of quantum mechanics in Equation (11). We conclude, therefore, that the correlations expressed by Equation (7) do not admit of a locally causal description. 3 Classically Simulating Quantum Statistics The results just described have unquestionably deepened our understanding of the implications of our experience with quantum phenomena. Yet the proper analysis and precise significance of these implications for our understanding of nature has, unsurprisingly, been hotly debated both by philosophers and physicists.6 I will not engage directly in those debates here. I will rather ask a different (but as we will see later, not unrelated) question: could one build a classical machine to simulate the quantum measurement statistics described above (and if so, how)? If the answer is yes, then are there any quantum correlations that cannot be so simulated, or is it the case that one can build classical systems to reproduce every possible quantum mechanical correlational effect? As it turns out, this is, in fact, an active area of research, and as we will see shortly, we can indeed give an affirmative answer to the first question. Later, in Section 4, we will begin to consider the philosophical significance of this. In particular, we will see (in Section 5) that it forces us to qualify some of the conclusions we made in Section 2. In the process, we will answer the second question. 3.1 GHZ statistics Consider, first, the case of GHZ correlations.7Table 1 depicts a scheme for reproducing all of the Pauli measurement statistics associated with the state in Equation (5). Table 1. A scheme for reproducing all of the Pauli measurement statistics associated with the tripartite GHZ state a b c σx −R2R3 R2 R3 σy −iR1R2R3 iR1R2 iR1R3 σz R1 R1 R1 I 1 1 1 a b c σx −R2R3 R2 R3 σy −iR1R2R3 iR1R2 iR1R3 σz R1 R1 R1 I 1 1 1 Table 1. A scheme for reproducing all of the Pauli measurement statistics associated with the tripartite GHZ state a b c σx −R2R3 R2 R3 σy −iR1R2R3 iR1R2 iR1R3 σz R1 R1 R1 I 1 1 1 a b c σx −R2R3 R2 R3 σy −iR1R2R3 iR1R2 iR1R3 σz R1 R1 R1 I 1 1 1 In the table, a, b, and c refer to the three subsystems of the system. σx, σy, σz, and I are the Pauli spin operators. Shared variables R1, R2, and R3 are values of ±1 that are assigned to the various subsystems at state preparation through some sequence of local interactions. It is assumed that these variables possess determinate values prior to measurement, but that these values are completely ‘hidden’ in the sense that they can only be revealed by measurement, or what amounts to the same thing: two identical state preparations will, in general, yield (with equal likelihood) different values of R1 (likewise for R2 and R3).8 To determine the outcome of a combined Pauli measurement on the system, one multiplies the entries of the table corresponding to the measurements performed on each subsystem, discarding any remaining unsquared instances of i. For example, a measurement of σy on subsystem a is equivalent to σy⊗I⊗I. The result is given by −iR1R2R3, which, dropping the unsquared i, yields −R1R2R3=±1 with equal likelihood. Measuring σy⊗σx⊗σy, on the other hand, will yield v(σy⊗σx⊗σy)=−iR1R2R3R2iR1R3=−i2R12R22R32=1 (12) with certainty. And so on. It is easy to verify that Table 1’s predictions for every combined Pauli measurement match up with those of quantum mechanics. And like those of quantum mechanics, Table 1’s predictions for combined Pauli measurements on the GHZ state are not factorizable, as, for example, v(σy⊗I⊗I)×v(I⊗σx⊗I)×v(I⊗I⊗σy)=−R1R2R3R2R1R3=−1, (13) in contradiction with Equation (12). The instances of i in the table are the analogue of quantum mechanical ‘non-local’ influences. With only very minor tweaking, however, one can build a model similar to the one represented in Table 1, in which the resulting correlations are both factorizable and consistent with all of the predictions of quantum mechanics for these measurements. Our tweak will be that we will allow the parties to classically signal to one another as follows: Bob (measuring b) and Alice (measuring a) will agree that he will send her a single classical bit indicating whether or not he performed a σy measurement on his subsystem. Upon receipt of this bit, Alice should flip the sign of her local outcome if either she or Bob (or if both of them) measured σy. Thus instead of Equation (13), we will have v(σy⊗I⊗I)×v(I⊗σx⊗I)×v(I⊗I⊗σy)=R1R2R3R2R1R3=1, (14) in agreement with Equation (12). Note that this result is consistent with each of the three measurements, σy⊗I⊗I,I⊗σx⊗I,andI⊗I⊗σy, for each of them separately will yield a value of ±1 with equal probability. This small addition—a single bit—suffices to make the outcome of every joint measurement of Pauli observables factorizable in the model. The scheme generalizes. For an n-partite system in the GHZ state 1/2(|↑⟩⊗n±|↓⟩⊗n), only n – 2 bits are required. The same is true, moreover, not only for n-partite GHZ states, but also for any n-partite state in which each superposition term is expressible as a product of Pauli eigenstates (details are given in Tessier [2004]). One can imagine building a classical computational system, made up of smaller, spatially separated computational subsystems, utilizing a small amount of classical communication as described above, to instantiate this scheme. This is the sense in which one may say that the correlations between the results of Pauli measurements on systems in the GHZ state are ‘efficiently classically simulable’. I will elaborate upon this in more detail in Section 4. But let us first consider how to simulate measurement statistics on a system in the singlet state. 3.2 Singlet statistics As we saw earlier in Equation (4), Bell himself provided a local hidden variables description to account for Pauli measurement statistics in the singlet state.9 Presumably, we could build a classical computer to instantiate that description. Thus Pauli measurements on systems in the singlet state are efficiently classically simulable in this sense. In this case, no communication is needed, in fact. This is evident from Equation (4). It also follows from the fact that Pauli measurement statistics for any n-partite state, whose superposition terms are expressible as products of Pauli eigenstates, can be simulated using n – 2 bits. The singlet state is one such state for which n = 2. Thus for the singlet state, Pauli measurements can be simulated using 2−2=0 bits of communication. It turns out that for the singlet state, we can simulate far more than just Pauli measurements if we allow ourselves only a single additional bit. Astoundingly, we can actually recover the statistics associated with arbitrary projective measurements by using the following method (Toner and Bacon [2003]): Randomly choose, independently, two unit vectors, λ^1 and λ^2. At state preparation, share them with Alice and Bob, and instruct them to take their particles with them to separate, distant locations. Once there, have Alice measure her particle along direction m^, and output the result A=−sign(m^·λ^1). Have her then send a single classical bit, c=sign(m^·λ^1)×sign(m^·λ^2)=±1, to Bob, who finally measures along n^ and outputs the result B=sign[n^·(λ^1+cλ^2)]. 4 What Is a Classical Computer Simulation? We have just seen how to classically simulate the quantum measurement statistics described in Section 2 using only a few additional resources. We saw, specifically, that only a single bit is needed to recover the statistics associated with arbitrary projective measurements on the singlet state, and that only n – 2 bits are needed to recover the statistics associated with Pauli measurements on n-partite GHZ states. Let us now reflect on this. To begin with, let us try and explicate somewhat more precisely what is meant by the statement that a particular set of quantum correlational statistics has been ‘classically simulated’. What is a classical computer simulation, then? Most obviously, perhaps, we may say, to start with, that a classical computer simulation is something performable by a classical computer. Let us consider what is meant by the latter. A classical computer, whatever else it is, is a classical physical system, that is, a system that can be given a classical physical description. Some of the characteristics of classical description are the following: First, a complete classical description of a system, which in general will consist of many subsystems, is always separable into complete descriptions of those individual subsystems. Second, a classical description of the interactions between a system’s subsystems will not violate classical physical law. For instance, the speed by which such interactions propagate should be less than or equal to the speed of light.10 Third, and importantly, the behaviour of classical systems is always describable in principle in terms of local causes and effects, in the metaphysically deflationary sense explicated by Bell ([2004b]). Thus, while a classical system might evince certain correlations between experimental results on its various subsystems—which, in general, can be quite distant from one another—because the system is classical, these correlations will always be describable as arising from a common source situated in the intersection of the prior light cones of the associated measurement events. That is, once we take into account their common causes, these correlations are always factorizable. Classical systems do not violate Bell (or GHZ, or similar) inequalities. Thus classical computer simulations, which are run on classical computers, which are instantiated by classical physical systems, do not either. This is perfectly obvious. And yet it might still strike one as strange. The results of, say, a Bell experiment seem to be evidence for some sort of influence (even if only benign) between the spatially separated subsystems of entangled systems, and we usually take this to demonstrate that quantum systems are ‘non-local’ (or perhaps ‘non-separable’) in some sense (Maudlin [2011]); that is, we take such influences to demonstrate that no locally causal description—no ‘local hidden variables theory’—of the outcomes of these experiments is possible. But the influence of one spatially separated subsystem on another is precisely what we find in the protocols designed to reproduce these outcomes, described in Section 3. And yet these are locally causal descriptions. How can this be? A way of resolving this tension is provided in (Cuffaro [2017]). I will not reproduce that discussion in its entirety here, but rather, in the remainder of this section, I will only summarize the main points emerging from it that are relevant to our own investigation, whose main conclusions will then be elaborated upon in the sequel (for a full elaboration and defence of the points summarized in this section, see the aforementioned article). In my earlier article, I argued that when we make a judgement to the effect that no local hidden variables theory is capable of recovering the quantum mechanical predictions associated with a Bell experiment, implicit in this judgement is a set of constraints upon the kinds of local hidden variables theory that we are willing to entertain. To put it another way: with a little imagination, it is always possible to conceive of loopholes to the Bell inequalities, but only some of these loopholes—the plausible ones—are worth the bother of trying to close. This means, however, that a conclusion such as ‘no (plausible) hidden variables theory of type X is able to reproduce the experimental predictions of quantum mechanics with respect to preparation procedure Y’ cannot really be forced on the basis of Bell’s (or a similar) (in)equality alone. Strictly speaking, to force such a conclusion, what one means by ‘plausible’ must be spelled out first. Usually, of course, we do not really need to be explicit. There is a set of criteria by which we mark a hidden variables theory as plausible that is implicitly understood by everyone. This set includes such things as consistency with our other theories of physics, and with the body of our experiential knowledge in general; we expect a candidate local hidden variables theory not to be conspiratorial, and so on. These and other plausibility constraints are associated with what I called the ‘theoretical’ context (Cuffaro [2017]), wherein we take Bell’s and related results to enlighten questions such as, ‘is there an alternative (locally causal) description of the natural world that can recover the confirmed predictions of quantum mechanics?’. There are other contexts, besides the theoretical, to which Bell’s and related (in)equalities are applicable. In particular, in (Cuffaro [2017]), I described the ‘practical’ context, wherein our concern is not with alternative theories of the natural world, but rather with what we are capable of building with the aim of (classically) reproducing the statistical predictions of quantum mechanics. In the practical context, in other words, we are concerned not with ‘local hidden variables’ descriptions of the natural world, but with ‘practical classical’ descriptions of machines that are possible for us to build to achieve particular ends (in the sequel, I will be using the generic term ‘locally causal description’ to refer to both). Now consider the two classical schemes for simulating quantum statistics described in Section 3. Either of these can be thought of as an alternative locally causal description of the observation of a given set of measurement statistics in the following way: Begin by asking what we really need to recover in an alternative locally causal description of a combined measurement outcome. The answer, invariably, will be just the actual observation of the combined outcome itself. What I mean is this: Although quantum mechanics posits its own particular dynamics—involving, for example, the ‘state collapse’ of a subsystem of a Bell pair upon the measurement of the other subsystem11—an alternative description of the events leading up to the observation of a combined measurement outcome can, in principle, disagree entirely with the quantum mechanical dynamical description. All that an alternative locally causal description needs to be faithful to, in principle, is our actual observation of the combined result. Importantly, in order to be in a position to assert that one has observed a combined measurement outcome, one must first have combined the individual measurement outcomes; Alice and Bob must report their results to Candice (or to each other) over tea or by telephone or in writing or by using some other physical means. This is absolutely necessary and there is simply no escaping it. While these results are being brought together, however, Alice and Bob have time to (conspiratorially) exchange—at subluminal velocity—a series of finite signals with one another, and thus coordinate—and, if necessary, ‘correct’—their individual measurement outcomes in the manner described in Section 3. But notice now that with respect to the observation of the combined measurement outcome—which, to reiterate, is ultimately what we are required to explain—all of this conspiratorial signalling will have occurred in its past light cone. It is thus part of a posited state preparation—a common cause—for that combined outcome and it thus comprises part of a classical, locally causal, alternative description of that combined outcome (Figure 1). Figure 1. View largeDownload slide A conceivable explanation of the observation of the result of a joint experimental outcome, AB, that posits local (conspiratorial) influences interacting within region C. This figure is adapted from (Cuffaro [2017]). Figure 1. View largeDownload slide A conceivable explanation of the observation of the result of a joint experimental outcome, AB, that posits local (conspiratorial) influences interacting within region C. This figure is adapted from (Cuffaro [2017]). In the theoretical context, of course, such a locally causal description would be ruled out as wildly implausible. It would simply be, for this author at any rate, too conspiratorial a loophole to take seriously, let alone to spend any of one’s research funds to try and close. But as I argue in (Cuffaro [2017]), in the practical context, we will not fret if our alternative characterization of a set of measurement statistics is conspiratorial or overly ad hoc from the theoretical point of view, for in the practical context, it is always assumed that the system under consideration has been purposely designed to recover those statistics.12 What we will rule out as implausible, rather, are only descriptions of classical systems that would be too ‘hard’ for someone to build, where ‘hard’ and likewise ‘easy’ are defined naturalistically—that is, by appealing to our best scientific theory of the subject: complexity theory. In complexity-theoretic terms, classically simulating a quantum system is too hard if it requires an ‘enormous’ number of additional resources as compared with those used by the quantum system, where by ‘enormous’ one means a number of additional resources that grows exponentially with respect to the size of the input to the problem, n (that is, on the order of kn). In contrast, ‘a few’ resources are at most polynomial with respect to n (that is, on the order of nk).13 Now if we consider a multi-partite quantum system in the GHZ state, 1/2(|↑⟩⊗n±|↓⟩⊗n), we know, from Section 3.1, that as the number of subsystems, n, grows larger, the number of additional classical bits required to simulate the all-or-nothing GHZ correlations associated with the system does not grow very fast at all (we require only n – 2 additional bits). In this sense, we can say that no matter how large n gets, an alternative classical description of the GHZ correlations present in the system is ‘no harder’ to realize, in a complexity-theoretic sense, than the quantum description. Thus if we imagine being presented with a large ‘black box’ whose inner workings are opaque to us, but which has been constructed to evince CHSH or n-partite GHZ correlations, we can in that case legitimately be sceptical if it is claimed that this box is quantum. For in response to that claim, we can always produce a description of a classical system capable of generating those correlations just as ‘easily’. We can produce, in other words, an alternative locally causal description of the box that is plausible in the practical context. This is what both protocols described in Section 3 amount to.14 5 Comparing the All-or-Nothing GHZ with Statistical (In)equalities We saw in Section 3.2 that if we would like to classically simulate the measurement statistics associated with Pauli measurements on a system in the singlet state, we can do so without the aid of any classical communication. To recover the statistics for arbitrary measurements, on the other hand, we require a single classical bit propagated at subluminal speed. In Section 3.1, we saw that recovering the statistics associated with Pauli measurements on a system in a GHZ state requires only a number of additional resources that are linear in the number of systems, n. All of these measurement statistics are plausibly recoverable, therefore, in the practical context. What if we would like to recover the statistics associated with arbitrary measurements on a system in a GHZ state? Unfortunately, unlike the singlet and other Bell states, it does not seem likely that we could plausibly do so. Tessier ([2004], p. 116) notes that the number of classical bits required to model the quantum mechanical predictions associated with arbitrary projective measurements on n-partite states (for n≥3) in a model like the one pictured in Table 1 is likely to be unbounded. This is consistent with other results due to Jozsa and Linden ([2003]) and Abbott ([2012]), who show (using different methods) that for systems in pure states, an exponential speed-up of quantum over classical computation is possible only when one has available multipartite entanglement with a number of systems n≥3. Returning now to the GHZ argument that I discussed in Section 2.2. As is evident from some of the quotations surveyed in Section 1, it is hard to overestimate the impact that the GHZ proof has had upon the physical and philosophical communities. To quote Mermin ([1990], p. 11) at length: This is an altogether more powerful refutation of the existence of elements of reality than the one provided by Bell’s theorem for the two-particle EPR experiment. Bell showed that the elements of reality inferred from one group of measurements are incompatible with the statistics produced by a second group of measurements. Such a refutation cannot be accomplished in a single run, but is built up with increasing confidence as the number of runs increases […] In the GHZ experiment, on the other hand, the elements of reality require a class of outcomes to occur all of the time, while quantum mechanics never allows them to occur. As we saw earlier, however, before one can properly make statements like this one concerning the relative power of the Bell and GHZ (in)equalities, one must first be clear on the context in which such statements are being made. From a theoretical point of view, it may be that the GHZ argument is more powerful than Bell’s in the above sense. Thus I take no issue with Mermin’s statement as long as it is qualified in this way, as I assume it implicitly is for Mermin. This said, in the practical context, Mermin’s statement is actually false. For from a practical point of view, both the Bell and GHZ (in)equalities are equally weak, in the sense that plausible alternative locally causal descriptions can be provided to account for the associated statistics in both cases. The contrast being drawn in the quote above, however, and in the statements of the other philosophers and physicists cited in Section 1 is not specifically between the results of Bell and GHZ. It is rather between the GHZ equality insofar as it is an all-or-nothing result, and Bell’s inequality insofar as it is a statistical result. But as we have seen, it is quite easy to recover the statistics associated with Pauli measurements on the GHZ state—the measurements for which an all-or-nothing equality can be proved—in the practical context. In this context, in order to use the GHZ state as the basis for a bona fide no-go theorem, one must consider measurements outside of the Pauli group, for the statistics associated with these measurements cannot be plausibly recovered in a locally causal model. Once one does so, however, the ensuing correlations between individual measurement results will no longer be strict but probabilistic. Thus it is statistical results, rather than the all-or-nothing GHZ, that are more powerful in the practical context. 6 General Discussion We have chosen to call classical computer simulations of quantum mechanical correlational statistics ‘locally causal alternative descriptions’ of those correlational statistics. This is not the way it is normally put in the quantum information literature. What is normally claimed for these schemes is rather that they quantify—in terms of the number of classical bits required to reproduce them—the departure from classicality of quantum correlations (besides the work of Tessier, Tessier et al., and Toner and Bacon mentioned above, see also, for example, Brassard et al. [1999]; Rosset et al. [2013]). This way of framing their significance is certainly not incorrect. It is certainly, moreover, very useful.15 From a more philosophical point of view, however, it is preferable to call a spade a spade, so to speak. As the example above involving the opaque black box illustrates, a plausible description of a locally causal computer simulation of a class of quantum phenomena is of the same general kind as a plausible locally causal theoretical description of a class of quantum phenomena. Both kinds of description represent alternative plausible stories one can put forward if one is sceptical of the quantum mechanical description of some set of observations. The essential difference between the theoretical and practical contexts lies in the different presuppositions we make concerning the origin of the physical systems under consideration in each case. In the practical context, we presuppose that a particular system has been designed and built by some rational agent for a particular purpose. We do not presuppose nature to be purposeful in this way in the theoretical context.16 Additionally, calling a spade a spade helps us to understand one reason why the practical context is—or anyway should be—interesting from the traditional philosopher of physics’ point of view. Indeed, such a reader may have been wondering why she should care at all about the practical context—why she should care for anything but the correct theory of the natural world and for its deeper consequences for our fundamental understanding of the ontology of that world. To help appreciate why one should care, imagine if it were possible to easily build classical machines to reproduce the statistics associated with every quantum mechanical measurement (regardless of state). If this were possible, it would be hard to imagine it not having some bearing on our interpretation of the quantum formalism, or at least on our general metaphysical world view. From this and the conclusions of (Pitowsky [1996], pp. 175–6), for instance, it would follow that the metaphysical difference between the indeterminism present in classical as opposed to quantum physics is merely one of degree (in other words, nothing at all). There is no need to engage in such speculation, however, for as we saw in the previous section, it turns out that building such universal simulators is not plausible. And yet it is still quite interesting for a traditional philosopher of physics to know that one could plausibly build classical physical systems to fully reproduce the observable correlational behaviour of systems in Bell states. It is also interesting to know that other quantum correlational phenomena cannot be so reproduced. The reason all of this is interesting is that in both the practical and theoretical contexts, we are speaking about physical systems, after all. Considering the practical context can thus help us to understand the very limits of classical description. It can help us to understand, that is, just how far quantum physical description as such outstrips classical physical description as such. Physics is traditionally conceived, to my mind rightly so, as a primarily theoretical activity, in the sense that it is the general goal of physics to tell us, even if only indirectly, what the world is like independently of ourselves (Fuchs [unpublished (a)], pp. 5–6, [unpublished (b)], pp. 22–3). It is worth noting, however, that this is not the case with every science. We have already discussed computer science. Chemistry is, arguably, a further example of a more practically oriented discipline (Bensaude-Vincent [2009]). But although chemistry (for example) may be a practical science in this sense, there is obviously no question as to whether theoretical knowledge of the systems she is manipulating is relevant to the chemist. Of course it is. Likewise, I want to claim, considering the practical aspect of the systems he is considering can be illuminating for the theoretical investigations of the physicist, in the ways I suggested above, and perhaps in further, as yet unimagined, ways. Explicitly noting the important differences in the presuppositions associated with the practical and theoretical contexts, on the other hand, encourages us to be wary of inappropriately conflating them. We saw, in Section 5, that a practical investigator should take care not to assume that certain no-go results, which are valid in the theoretical context, will constrain what is possible for her to accomplish in the practical context. Additionally, as I have argued in (Cuffaro [2017]), practical investigators attempting to isolate and/or quantify the computational resources provided by physical systems may be in danger of conceptual confusion if they are not cognizant of the differences between the two contexts. Theoretical investigators should also be careful not to conflate the two contexts inappropriately. For instance, Jeffrey Bub’s ([2004]) claim that the fundamental aim of physics consists in the representation and manipulation of information seems to me to be in danger of doing so. Bub’s argument for this conclusion is based on the fact that within the abstract C* algebraic framework, one can characterize quantum theory purely in terms of a set of information theoretic constraints (specifically, no signalling, no broadcasting, and no unconditionally secure bit commitment),17 and that given these constraints, any alternative mechanical theory that aims to solve the measurement problem faces an in-principle problem of underdetermination. This means, for Bub ([2004], p. 243), that ‘our measuring instruments ultimately remain black boxes at some level’. It would take us too far distant from the primary concern of this article to consider Bub’s arguments for this last claim in detail. Let me just say here that even if this claim is true18—which, for different reasons, is a claim that I am generally sympathetic towards and which, to be fair, actually constitutes the main thrust of Bub’s article—it does not obviously follow from it that ‘the appropriate aim of physics at the fundamental level [is] the representation and manipulation of information’ (Bub [2004], p. 242), or that ‘an entangled state should be thought of as a nonclassical communication channel that we have discovered to exist in our quantum universe, i.e., as a new sort of nonclassical “wire”’ (Bub [2004], p. 262). All that would seem prima facie to follow from the claim that our mechanical description of nature must run short is that we cannot descend below the level of the language of information theory when characterizing quantum phenomena. But one could argue that the proper lesson to draw from this is merely that as a result, we must be all the more careful to be precise about what it is that we are saying when we use informational language—careful to distinguish, that is, the circumstances in which we are discussing the ways in which we can use physical systems from the circumstances in which we are describing the nature of the physical world itself, as it exists in itself (that is, without our intervention). The idea that we can and do make such distinctions lies at the very basis of our foregoing discussion. The analysis in Section 4 began from the fact that our conceptions of ‘classical system buildable by a human being’ and of ‘classical system existing naturally’ are—at least pre-theoretically—very different conceptions. I then explicated these pre-theoretic conceptions more precisely in terms of differing sets of plausibility constraints associated with each. What would seem to be asserted by one sympathetic to the view expressed above, however, is that our pre-theoretic intuitions are false—that really, after all, there is only one context worth considering: what I have above referred to as the practical context. It is not incoherent to hold such a view. But if it really is the case that quantum theory is ultimately about nothing other than representing and processing information—that (as seems to be implied by this) the practical context should be our primary concern vis á vis foundational investigation—then it is hard to see why GHZ correlations, in particular, should surprise anyone. For from the practical point of view, GHZ correlations are no more inexplicable than the correlations between the colours of Professor Bertlemann’s socks (if you happen to notice that one of these is pink, you can rest assured that the other is not; see Bell [2004a]). In other words, a plausible locally causal explanation, in the sense of Section 3, is readily available for both of these phenomena in the practical context. But, in fact, we are surprised by GHZ correlations (recall the survey of the reactions to the equality in Section 1), and it is incumbent upon one sympathetic to a view like Bub’s to explain to us why this is so. Before closing this section, let me note that the foregoing considerations are not intended as an argument against a view like Bub’s. Rather, the sketch of an argument contained in the foregoing few paragraphs should be regarded as an invitation to anyone sympathetic to the view that quantum theory just is about representing and manipulating information—a much stronger and deeper claim than the statement that the description of our measuring instruments must necessarily remain opaque at some level—to clarify this position in light of the points that have been made in this article. I will also note that in the foregoing few paragraphs I have characterized informational language as belonging to the practical, rather than the theoretical, context. This is true within a view like Bub’s. However, it may not be true for one who interprets information differently. In particular, it may not be true for one who thinks of information as a kind of physical stuff (Landauer [1996]). There are independent reasons to be sceptical of such a view (see, for instance, Timpson [2013], Section 3.7.1), which I will not discuss here. I will instead close this section by noting that there are obviously interconnections between our discussion of the distinction between the practical and theoretical contexts and the debate over the interpretation of the concept of information. However, clarifying these interconnections would be a project in itself, which I, unfortunately, do not have the space to pursue here (for more on the debate over the interpretation of information, see Lombardi et al. [2015]). 7 Conclusion I reviewed the Bell and GHZ (in)equalities in Section 2, and then in Section 3, I discussed schemes by which one could simulate, using a classical computer, the statistics associated with those (in)equalities. In Section 4, I argued that classical computer simulations are locally causal descriptions in Bell’s sense and hence do not violate the Bell or GHZ (in)equalities, and I argued that this fact is not in tension with our normal judgements regarding the significance of these (in)equalities, as long as one understands that our normal judgements are not valid in the context of a discussion of machines designed to achieve a particular purpose. I then argued, in Section 5, that this has implications for our understanding of the relative strength of the GHZ all-or-nothing equality vis-á-vis statistical inequalities, and I made the perhaps surprising observation that the former actually has no force in the practical context, despite being a quite remarkable result in the theoretical context. I ended, in Section 6, by discussing the general implications of all of the foregoing for our understanding of the limits of classical description, and for our understanding of the fundamental aim of physical investigation. Both in the practical and in the theoretical contexts, one’s concern is, ultimately, very concrete: one is in both cases concerned with plausible descriptions of actual physical systems existing in the world. The difference between the two contexts enters essentially in the different presuppositions we make concerning the origin of the physical systems under consideration. In the practical context, we presuppose that a particular system has been designed and built by some rational agent for a particular purpose. We do not presuppose this in the theoretical context. But because both contexts are concerned with describing physical systems, both contexts are relevant to our understanding of physical systems. And the practical context is particularly illuminating in that it allows us to investigate more thoroughly the possibilities inherent in physical systems. It helps us to probe further and to understand better just what the limits of classical description are, and just how far quantum description outstrips it. At the same time, we must remain on guard not to be misled into thinking that the practical context can supersede the theoretical context for the purposes of our foundational and philosophical investigations into the nature of the (quantum) world. Both of these contexts are rather complementary modes of investigation, aimed at the world and our experience. Acknowledgements Section 6 of this paper was inspired by my conversations with William Demopoulos in the summer of 2013. The paper also benefited from the helpful comments of Samuel Fletcher and two anonymous referees at BJPS. Thanks also to Radin Dardashti, Ryan Samaroo, and to my audiences at the universities of Bristol, Florence, Helsinki, and Oxford for fruitful discussion. The bulk of the research for this project was conducted during my stay at the Munich Center for Mathematical Philosophy, and was funded by the Alexander von Humboldt Foundation. Footnotes 1 The general idea of considering the differences between physical systems from the point of view of a practical context is also present in the work of Pitowsky. For example, in (Pitowsky [1990]), two different ‘practical versus theoretical’ distinctions are mentioned. The first is between computational devices and their natural physical ‘analogues’ (for example, consider a device to determine the secondary structure of a protein versus its physical analogue: the natural process of protein synthesis). The second distinction is not conceptual but pragmatic; for example, for a given NP-complete computational problem, one can ask the ‘practical question’ regarding whether many or most instances of it are effectively efficiently solvable. In (Pitowsky [2002]), the practical question posed regards the number of computational resources required to transform a given system into a state from which a solution to a computational problem can be achieved in polynomial time with a certain probability. 2 Note that, as this quote hints at, Clifton et al. do not argue that the GHZ proof is stronger than Bell’s in every respect. 3 These are named after Bell, the discoverer of the first such result (Bell [2004c]). 4 I have omitted the trivial identity operator I from this list. 5 The following exposition is based on (Mermin [1990]). 6 See, for instance, (Cushing and McMullin [1989]). 7 What follows is a summary of the model given in more detail in (Tessier [2004]; Tessier et al. ([2005])). 8 Note that no such interpretation of R1, R2, and R3 is given in Tessier ([2004]; Tessier et al. [2005]), but it is implicit. 9 The theory given in Equation (4) applies not just to Pauli measurements, of course, but more generally, as we saw, to any combined spin measurement in which the angles at the respective ends of the experimental apparatus differ in proportion to π/2. 10 I am taking ‘classical’ in the sense in which it is typically used in discussions of quantum mechanics; that is, I take classical theory to include special and general relativity. 11 What I am referring to here is merely the standard wave packet reduction postulate assumed in ordinary quantum mechanics. I do not mean to imply that standard quantum mechanics includes an agreed-upon detailed theory of state collapse. 12 Kent ([2005]) actually describes an alternative to quantum theory (his is not a practical investigation) that exploits a similar loophole to the one described above. Kent does not consider his own alternative theory to be as wildly implausible as I think the model described above would be in the theoretical context (as Kent’s is a theoretical investigation, more effort is made to construct a theory that is not excessively ad hoc). That said, Kent ([2005], p. 6) describes a number of strong plausibility considerations (though for him, not conclusive) for ruling out his theory. 13 See (Arora and Barak [2009]). 14 It must not be forgotten, of course, that at least in the case of GHZ states, it is only the outcomes associated with Pauli measurements that are efficiently classically simulable in the sense just described. We will return to this point shortly. 15 Indeed, the general subject of classical simulations of quantum systems is an important and burgeoning area of modern physics (Lee and Thomas [2002]; Borges et al. [2010]). 16 I am not expressing a theological opinion with this statement. 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