Guido Bacciagaluppi A Critic

Guido Bacciagaluppi
email: g.bacciagaluppi@abdn.ac.uk

A Critic Looks at QBism

This chapter comments on that by Chris Fuchs on qBism. It presents some mild criticisms of this view, some based on the EPR and Wigner's friend scenarios, and some based on the quantum theory of measurement. A few alternative suggestions for implementing a subjectivist interpretation of probability in quantum mechanics conclude the chapter.

"M. Braque est un jeune homme fort audacieux. [...] Il méprise la forme, réduit tout, sites et figures et maisons, à des schémas géométriques, à des cubes. Ne le raillons point, puisqu'il est de bonne foi. Et attendons." 1 Thus commented the French art critic Louis Vauxcelles on Braque's first oneman show in November 1908, thereby giving cubism its name. Substituting spheres and tetrahedra for cubes might be more appropriate if one wishes to apply the characterisation to qBism -the view of quantum mechanics and the quantum state developed by Chris Fuchs and co-workers (for a general reference see either the paper in this volume, or [START_REF] Fuchs | QBism, the Perimeter of Quantum Bayesianism[END_REF]). In this note, I shall not comment on other possible analogies, nor shall I present an exhaustive critical review of qBism (for an excellent one, see Timpson

Brunism, classical and quantum

First of all, what are subjective probabilites (aka credences), or what are they for in a pragmatist understanding of probability? They are strategies we adopt in order to navigate the world. Some may turn out to be more useful than others in practice, but that does not change the fact that they are subjective constructs.

Take for instance (classical) coin tossing. Subjective probabilities describe our expectations for outcomes of succesive tosses (and guide our betting behaviour). We start off with certain priors; these are updated as we go along, reflecting past performance. But where do our original priors come from? Let us dispel the idea that they stem from some intuitive feel or mystic vision. They might, but they generally will not, and indeed need not. Especially in more complex situations, we can and will adopt a theoretical model to choose our priors. Such a model may involve a thorough analysis of the whole set-up, or be even quite crude and just make use of simple properties of the coin alone (e. g. weight distribution). In any case, it will involve consideration of objective (but non-probabilistic!) properties of the system under consideration.

If the observed frequencies match the ones determined through the model, it may be that the data 'confirm' our theoretical model. This, however, does not mean that our strategy is any less subjective. Indeed, there is no necessary connection between, say, weight distribution and relative frequencies: for any possible sequence of results there is an initial condition that will produce it, irrespective of the details of the weight distribution of the coin. Nor is there for the 'Brunist' any compelling rational justification in the sense of David [START_REF] Lewis | A Subjectivist Guide to Objective Chance[END_REF] for letting our credences depend on the weight distribution, only pragmatic criteria such as simplicity, past performance etc. 'Confirmation' is only a reflection of such past performance, and bears no guarantee of future reliability.

One might object that de Finetti himself showed that if one's priors for sequences of results are exchangeable, the posteriors will converge (with subjective probability 1). This is surely a sign of tracking something 'out there' (or so the objection goes). 3 As long as we are dealing with finite samples, however, any apparent convergence of one's probability assignments (one's own or intersubjectively) again merely reflects past performance. And the expectation of future convergence depends crucially on the subjective assumption of exchangeability. Exchangeability is an assumption about the structure of one's subjective priors, and is entirely independent of any objective behaviour of the system under consideration.

Even this more technical objection, thus, does not alter the fact that according to de Finetti there is no sense in which our probability judgements are right or wrong. As de Finetti very graphically expresses, PROBABILITIES DO NOT EXIST.

The same is true in qBism. Quantum probabilities according to qBism simply are subjective probabilities in the sense of de Finetti. They are strategies that we adopt in navigating an (admittedly) unexpectedly strange world. They may be impressively successful in terms of calculational power, past performance etc., but that makes them no less subjective than the theoretical models we might adopt for coin tossing (or indeed in classical statistical mechanics!).

What may strike one as strange, or at least unfamiliar, is applying de Finetti's ideas to a case which is generally thought to be genuinely indeterministic. There may be a price to pay if one does so, because if probabilities are not objective, then any use of them in describing the observed regularities in the world cannot be thought of as expressing law-like behaviour in any necessary sense. But if one is happy with a broadly Humean view of laws, this will not strike one as a disadvantage of the application of de Finetti's views to an indeterministic context.

Another point worth making explicitly (because it is not normally included in presentations of qBism), is that even in the quantum case one may adopt theoretical models for fixing one's priors. For instance, one might use techniques for selecting a Hamiltonian and calculating its ground state, and set one's subjective probabilities according to that. The data might then seem to 'confirm' some particular choice of Hamiltonian. But according to the qBist there will be no necessity or rationally compelling reasons for using it to fix our credences, only pragmatic ones such as calculational power, past performance etc.

Fuchs generally talks of quantum states themselves as subjective (and this leads him also to a view of Hamiltonians as subjective). What I am saying instead is that one can take Hamiltonians to be objective properties of quantum systems, just like the weight distribution in a coin, and further as non-probabilistic properties, again just like weight distribution. Indeed, one can even think of quantum states (!) as objective properties of a system, stripping them of their customary probabilistic elements, and thinking for instance of the ground state 'merely' as a state of definite energy. (After all, Heisenberg, Pauli, Schrödinger and others all had notions of quantum states before Born introduced his 'statistical' interpretation of the wave function.4 ) The probabilistic association would attach to such a state exclusively in the way we use it to fix our credences. We shall return to the idea of objective non-probabilistic quantum states in Section 3.

Worries

2.1 EPR and Wigner's friend One worry I wish to air about qBism relates to one of its reputed major selling points, namely its 'local' account of 'nonocal' EPR correlations. The qBist story goes like this. Consider Alice's credences about Bob's electron. Her initial credences might be equal to 1/2 for the result of any spin experiment she might perform on Bob's side. But if she performs first a spin measurement on her own side -say in direction x with result 'up' -, she will then update her credences about further measurements on Bob's electron. In particular she will then believe with all her heart (credence 1) that if she performs a spin-x experiment on Bob's electron, this will give the result 'down'. What has changed, however, are simply her own credences. Similarly, Bob has credences about Alice's electron, which also change in an analogous manner, but although they are about Alice's electron, they are his credences, and (insofar as beliefs can be said to be located anywhere5 ) they are in his head. These two autonomous points of view can be then married together to form a composite picture of the EPR pair. (Thus, indeed, bringing out an analogy between qBism and cubism!)

The worry is about this marriage. QBism, just like 'classical Brunism', presupposes that different agents be able to share data, an assumption that underlies the intersubjective agreement between different agents derivable from de Finetti's theorem. Different agents may become aware at different times of different pieces of data, but they can inspect each others' data and pool them together. Indeed, suppose Alice and Bob both perform spin measurements in direction x, and then later meet to compare results. From Alice's point of view, her asking Bob his result is her own measurement of Bob's electron (which in the meantime has interacted with this further system called Bob), and her own measurement results are timelike related. But if she wishes to take Bob's own report seriously as providing her with data that at the time of his measurement were available to him but not yet to her, then the mystery of perfectly correlated spacelike separated events returns.6

Wigner's friend (another case for which qBism claims to have a ready explanation) can be seen as a variant of the above. Bob performs an experiment in a lab that is isolated from Alice (perhaps because Alice and Bob are spacelike separated at the time). Alice can later perform a measurement on the content of Bob's lab (either repeating Bob's measurement or asking him for a report). If she does repeat Bob's measurement, the result she observes coincides with the result of the earlier measurement as reported by Bob, again suggesting that she should take the report seriously as describing objective data that were not yet available to her. Unlike the EPR case, this is not particularly puzzling. But in the Wigner's friend scenario, we are invited to consider also the case in which Alice performs instead an interference experiment on the entire contents of Bob's lab, and thereby 'quantum erases' Bob's result. In qBist terms, this could be understood merely as Alice performing some manipulation that leads her to change her own credences about the results of her asking Bob what he has seen. She now expects from Bob not some or other report of a definite result, but a definite report of not having performed the experiment. But this description misses out on the fact that Alice's manipulation has in fact obliterated also Bob's piece of data and any memory that Bob had of it (unless, that is, we assume that Bob did not really possess any such piece of data in the first place).

Thus, if we believe that data obtained by different agents are equally objective, thus understanding 'pooling of data' literally, we have problems. There are no such problems once the data have been pooled together, but we have two puzzling cases in situations where Bob's data are not yet available to Alice. In the EPR case, qBism remains silent on why Alice and Bob's data should be correlated, and in the Wigner's friend case, it remains silent on how Alice can erase Bob's data. The choice for qBists seems to be between: (a) providing us with a further story about data and/or agents themselves, rather than just strategies for how agents update their credences in the face of new data; and (b) some kind of solipsism or radical relativism, in which we care only about single individuals' credences, and not about whether and how they ought to mesh.

Hidden constraints on probability assignments

My main worry, however, runs deeper in the conceptual foundations of qBism. A central idea of qBism is that the view is not a modification of but an addition to Bayesian coherence. This addition is equally normative, but rather than being rational in origin, it is empirically motivated. It is essentially contained in the formula

Q(D j ) = (d + 1) d 2 i=1 P (H i )P (D j |H i ) -1 , (1)
which constrains the relation between probabilities in certain pairs of actual and counterfactual situations. The actual situation is a measurement of the family of projections D j , and the formula compares the probabilities Q(D j ) with the probabilities d 2 i=1 P (H i )P (D j |H i ) that would have been obtained if a previous measurement of the 'fiducial SIC' with effects H i had been performed (a generalisation of this formula holds if the D j are effects).7 This is, indeed, a situation on which Bayesian coherence is silent. The law of total probability

Q(D j ) = d 2 i=1 P (H i )P (D j |H i ) (2)
is clearly a prescription for relating the probabilities of two actual measurements.

However, I claim that formula (1) already presupposes very strong constraints on our subjective probability assignments. Indeed, the very idea of well-defined probabilities Q(D j ) for projections (or more generally effects) D j already embodies such strong constraints, even before we start comparing our probability assignments for actual measurements to our probability assignments for counterfactual measurements.

In order to substantiate this claim, let me recall some standard material.

It is nowadays customary in quantum mechanics and quantum information to describe (general) measurements using the concepts of operations and of POVM s (positive-operator-valued measures).

Operations are families of transformations on the quantum states (thought of as transformations induced by 'measurements'). Such transformations could for instance be of the form ('pure operation')

ρ → A i ρA * i (3)
(with the right-hand side suitably renormalised), and each such transformation takes place with probability

Tr(A i ρA * i ) = Tr(ρA * i A i ) . (4)
For this expression to indeed define a probability distribution over the various possible transformations we must have:

i D i := i A * i A i = 1 (5)
(with 1 the identity operator). The thus defined operators D i are so-called effects, i. e. they are positive (self-adjoint with positive spectrum) and with spectrum contained in the interval [0, 1]. The mapping from the indices i (or sets thereof) to the associated D i (or sums thereof) is thus an effect-valued measure, also called positive-operator-valued measure (POVM).

One should note crucially that the probabilities associated with a transformation are fixed just by the corresponding POVM.

Further, one easily sees that a pure operation such as the above can always be implemented by coupling the system to an ancillary system,

|ψ ⊗ |ϕ 0 → i A i |ψ ⊗ |ϕ i (6)
for some orthonormal family |ϕ i , and then collapsing onto the latter. Note that such coupling is indeed unitary because of ( 5). (This result, suitably generalised to all operations, is known as the Naimark dilation theorem.)

Here is a very familiar example.

Example 1 (von Neumann measurement):

Let A i := |ψ i ψ i | for some orthonormal basis. We implement it via i α i |ψ i ⊗ |ϕ 0 → i α i |ψ i ⊗ |ϕ i , (7)
and we have

A * i A i = P i := |ψ i ψ i | , (8)
so the correspondiong POVM is projection-valued.

A von Neumann measurement, however, is not the only experimental procedure for measuring a projection-valued measure, as the following example shows.

Example 2 ('measurement of the second kind'):

Let the |ψ i be as above, and let

B i = |ψ ′ i ψ i | for some arbitrary unit vectors |ψ ′ i . We have i α i |ψ i ⊗ |ϕ 0 → i α i |ψ ′ i ⊗ |ϕ i . (9)
Note that

B * i B i = |ψ i ψ ′ i |ψ ′ i ψ i | = P i , (10)
and the transformation is indeed associated to the same projection-valued measure as in Example 1. The difference is that the 'collapsed' state of the system after the measurement is no longer an eigenstate of the measured observable (in the traditional sense of a self-adjoint operator with spectral measure defined by the P i ). The measurement is not 'minimally distrurbing'.

How do these examples lead to a worry about qBism? Note that (1) contains probability assignments Q(D j ) referring to measurements of POVMs irrespective of which transformations are used to implement them. Thus, it presupposes that we assign the same probabilities to the results of the two transformations in Examples 1 and 2, even though they correspond to different lab procedures. QBism is currently silent on why we have this constraint on our probability assignments. Of course, we can say it is empirically wellestablished, and we can derive it theoretically from the quantum mechanical theory of measurement if we apply the usual Born rule to the ancillary system in the model. But if (1) is meant to be a simple axiom embodying one of the main modifications of Bayesian coherence theory that are supposed to lead to quantum mechanics, it appears that it already presupposes a lot of the structure it is trying to explain.

The point can be made even more strikingly using a further example.

Example 3 (sequential von Neumann measaurements):

Concatenating two operations yields a further operation, e. g.

ρ → P i ρP i → Q j P i ρP i Q j . (11)
Indeed, defining

A ij := Q j P i , (12)
we obtain a corresponding POVM:

ij A * ij A ij = ij P i Q j P i = i P i = 1 . ( 13
)
This POVM can be measured via two sequential von Neumann measurements. For instance, if the P i and Q j project onto spin-1/2 eigenstates in directions x and y, we can implement the composite POVM by letting an electron pass in sequence two Stern-Gerlach magnets at right angles to each other (two sequential interactions between the spin and spatial degrees of freedom of the electron) and then measuring on which quadrant of the screen the electron impinges.

But we can also implement it using a single interaction with an ancillary system, e. g. by defining

B ij := P i Q j P i . (14)
Indeed,

B * ij B ij = ( P i Q j P i ) 2 = P i Q j P i = A * ij A ij , (15)
and the corresponding POVM is the same as in ( 13).

This is again a case of two totally different laboratory procedures that allow one to measure the same POVM, this time an effect-valued rather than projection-valued one.

We can again note that the qBist formula relating the probabilities we assign to results of (actual) measurements of effect-valued measures to the probabilities in the corresponding counterfactual situations (a fairly straightforward generalisation of ( 1)) presupposes that we assign the same probabilities also to these two procedures.

This presupposition, however, is even more suspect than in the case of Examples 1 and 2, because the procedure defined by ( 12) already implicitly contains the collapse of the state (or at least the effective collapse through decoherence of the spin state by the spatial degree of freedom along the first measured direction x), which even more strongly suggests that the strategy currently pursued within qBism for axiomatising and/or understanding quantum mechanics implicitly presupposes a lot of the structure it is trying to explain.

Other pragmatist alternatives

Whether or not qBism will turn out to provide a fully successful new framework for understanding quantum mechanics, it has already shattered a taboo: that of using a subjectivist approach to probabilities in quantum mechanics. The traditional view of course is that quantum probabilities are the paradigm of objective probabilities. The idea that a radical subjectivist approach à la de Finetti might be applied to the quantum case (which at least prima facie is truly indeterministic) used to be inconceivable.

The inconceivable having now been conceived, I wish to suggest in this section that in fact a subjectivist/pragmatist approach to probability can be applied in the context of just about any approach to quantum mechanics.

In particular, it can be applied also to approaches that adopt an ontic view of the quantum state. As pointed out in Section 1, an ontic view of the quantum state might be adopted also within qBism, as long as the quantum state itself is not seen as a probabilistic entity (although this is not part of the usual presentations of qBism8 ). Other approaches to the foundations of quantum mechanics such as the Bohm theory, GRW theories, or the Everett theory explicitly take an ontic view of the quantum state (thus at least in principle also providing an answer to the question of the ontology of data and agents, cf. above Section 2.1).

I shall now sketch very briefly in turn how each of these can adopt a radical subjectivist view of probabilities, thus taking the quantum state as ontic but as non-probabilistic (whether or not it is customary for them to do so!).

Bohm

It is easiest to think of subjectivism about probabilities in the case of the Bohm theory. Indeed, the Bohm theory is a deterministic theory, and we are familiar with applying the classic de Finetti analysis to deterministic cases.

The Bohm theory (or de Broglie-Bohm theory, or pilot-wave theory) describes 'classical' configurations evolving deterministically in a way fixed by the quantum wave function of the total system. The usual statistical predictions of quantum mechanics are recovered if one assumes that the configurations in an ensemble are distributed according to the usual quantum mechanical distribution (a condition which is preserved over time). The situation is very similar to that of classical statistical mechanics (with the difference that now the 'equilibrium' distributions are time-dependent), and this analogy has been developed in considerable detail.9

Note that the quantum states in the Bohm theory are both ontic and nonprobabilistic. They are the 'pilot waves' of the theory. They acquire a probabilistic significance only if we adopt a strategy of choosing our subjective probabilities to be equal to the modulus squared of the wave function. While such a strategy is highly successful,10 the quantum wave functions provide no fundamental constraint on initial positions, nor indeed on particle distributions in ensembles. This is evident from the fact that non-equilibrium pilot-wave theory is equally intelligible and may even have very interesting applications [START_REF] Valentini | Inflationary Cosmology as a Probe of Primordial Quantum Mechanics[END_REF].

Just as in the case of the weight distribution of a coin -which turns out to be a reliable indicator of the statistical behaviour under repeated tossing only under certain 'typical' conditions -, so the wave function describing an ensemble of particles (more precisely, the so-called effective wave function, i. e. the component of the wave function responsible for the motion) is a reliable indicator of the statistical behaviour of the particles only on the assumption of typicality. Thus in both cases a physical but non-probabilistic property associated with the system under consideration (and one that can be modelled theoretically) is used as a pragmatic short-cut for fixing our subjective probabilities for the behaviour of the system.

GRW

It is less familiar to think in terms of subjective probabilities in the case of spontaneous collapse theories, i. e. theories in which the Schrödinger evolution is modified in a way that reproduces the phenomenology of collapse. Such theories were shown to be viable by [START_REF] Ghirardi | Unified Dynamics for Microscopic and Macroscopic Systems[END_REF] and [START_REF] Pearle | Reduction of the State Vector by a Nonlinear Schrödinger Equation[END_REF][START_REF] Pearle | Combining Stochastic Dynamical State-Vector Reduction with Spontaneous Localization[END_REF], and are thus also know as GRW (or GRWP) theories. 11

Since in GRW theories we have genuine indeterministic evolution of the quantum state, they are generally thought of in terms of objective collapse probabilities, much like 'traditional' quantum mechanics. However, the idea that in the case of genuine indeterminism probabilities should be thought of as objective presupposes that a viable account of objective probabilities be given, e. g. in terms of frequencies or propensities, and both of these accounts suffer from more or less severe problems. The third strategy open to objectivists is to apply [START_REF] Lewis | A Subjectivist Guide to Objective Chance[END_REF] 'principal principle', i. e. to argue that there are compelling rational reasons for adopting a particular recipe to fix one's subjective probabilities. Perhaps some version of 'Humean objective chances' can deliver on this, but the step to subjectivism might be very short in that case. 12 The quantum state in GRW theories is clearly ontic, indeed at least prima facie provides the 'stuff' the world is made of. 13 But as regards the probabilistic evolution of the state, we can adopt de Finetti's position, holding that there are no right or wrong probabilities about how the state evolves.

We can take the GRW theory (a theoretical model of these probabilities) as a pragmatic recipe for fixing our subjective probabilities for the dynamical behaviour of the states (much as we take weight distributions to guide our expectations about the behaviour of tossed coins). And we can push the line that de Finetti's position is not only tenable in an indeterministic context, but that it may be even less artificial than others in the context of GRW theories.

11 For an accessible reference, see [START_REF] Ghirardi | Collapse Theories[END_REF].

12 On Humean objective chances, see e. g. [START_REF] Hoefer | The Third Way on Objective Probability: A Sceptic's Guide to Objective Chance[END_REF], and for their application to GRW and for more general discussion of objective probabilities in GRW, see [START_REF] Frigg | Probability in GRW Theory[END_REF].

13 There is a debate about the most natural interpretation of spontaneous collapse theories: whether -in increasing order of resilience against the so-called 'tails' problemit is in terms of wave functions, matter density, or collapse events (so-called 'flashes' or 'hits'). For details see e. g. [START_REF] Ghirardi | Local Measurements of Nonlocal Observables and the Relativistic Reduction Process[END_REF], [START_REF] Allori | On the Common Structure of Bohmian Mechanics and the Ghirardi-Rimini-Weber Theory[END_REF]), Bacciagaluppi (2010).

Everett

The final and most interesting case is that of Everett. 14 As is well known, Everett takes the wave function of the universe seriously as providing the ontology of the theory, and the Schrödinger equation as providing its dynamics. Collapse is explained through the correlational structure of the universal wave function, whereby the quantum state appears to collapse to an internal observer (or whatever other system is 'recording' collapse events). Each outcome of a collapse is equally real, relative to the corresponding component of the observer.

Modern-day Everettians refine Everett's original analysis of the correlational structure of the universal wave function through the use of decoherence theory (cf. e. g. Wallace (2010a), [START_REF] Bacciagaluppi | The Role of Decoherence in Quantum Mechanics[END_REF]). However, there is a generalised perception of a problem in making sense of probabilities in the Everett theory, precisely because, say, in the context of a sequence of measurements on an ensemble of systems, all sequences of outcomes are actualised (in different 'worlds' or 'branches' of the universal wave function). Thus, not only are there no right or wrong probabilities, but probabilities would appear to make no sense at all (at least in the sense that we do not seem to be ignorant of what will be the outcome of a measurement 15 ). A breakthrough on this question was achieved not many years ago by David [START_REF] Deutsch | Quantum Theory of Probability and Decisions[END_REF] and David [START_REF] Wallace | Quantum Probability from Subjective Likelihood: Improving on Deutsch's Proof of the Probability Rule[END_REF]Wallace ( , 2010b)), who adopted the Lewisian strategy sketched above and argued, first, that rational decision theory can be applied to the case of an Everettian agent located before a branching event (see also [START_REF] Greaves | Understanding Deutsch's Probability in a Deterministic Multiverse[END_REF] and [START_REF] Greaves | Everett and Evidence[END_REF]), and, crucially, that rationality constraints on such an agent will force them to adopt the quantum probabilities for the results of the branching. Thus, quantum probabilities (at least insofar as they apply to such a decision-theoretic situation) are objective chances in the sense of Lewis. The approach based on the Deutsch-Wallace theorem appears to command quite a consensus among 14 Cf. also my comments in [START_REF] Bacciagaluppi | The Many Facets of Everett's Many Worlds[END_REF], which reviews the state-of-the-art volume on the Everett theory edited by Saunders et al. (2010). Everett's complete writings on quantum mechanics, together with a wealth of other original material, have been published and annotated by [START_REF] Barrett | The Everett Interpretation of Quantum Mechanics[END_REF].

15 Note that, as clearly shown by [START_REF] Vaidman | On Schizophrenic Experiences of the Neutron or Why We Should Believe in the Many-Worlds Interpretation of Quantum Theory[END_REF], we do have ignorance of the result of a measurement at least after the measurement has occurred and we are not yet aware of the result. After the branching induced by a measurement there is a genuine question about self-location.

modern Everettians, but has also been the object of strong criticism. 16 I wish to suggest that Everettians can avail themselves of an alternative subjectivist strategy, taking 'branch weights' as guides for navigating a branching universe. The choice of branch weight as quantifying probability (or 'typicality') can be pragmatically justified on the basis that it has performed well in the past, and on the basis of its being the 'natural' measure on branches, e. g. because of [START_REF] Gleason | Measures on the Closed Subspaces of a Hilbert Space[END_REF] theorem or the Deutsch-Wallace theorem (this is [START_REF] Greaves | Everett and Evidence[END_REF] take on the latter), or because it is conserved by the dynamics analogously to the measure in classical statistical mechanics (this is in fact the justification proposed by Everett (see e. g. Barrett and Byrne 2012, pp. 274-275)). This reading of probability in Everett needs of course to be developed further, but would provide a particularly striking way of combining an ontic view of the quantum state with a subjectivist view of quantum probability. 17 Such an application of 'Brunism' to the quantum case would of course be much tamer than Chris Fuchs's usual emphasis on quantum states themselves being subjective. Stretching the metaphor, if we allow Everett's universal wave function to explain what agents and outcomes are in the first place, the heroic phase of 'analytical qBism' will give way to a much tamer 'synthetic qBism'. That said -returning to the history of art -I have always preferred the analytic phase of cubism to the synthetic one! 16 For a lively and representative sample of the literature, see the relevant contributions by Albert, Greaves and Myrvold, Kent, Price, Saunders, and Wallace, as well as the transcripts of the discussions, in Saunders et al. (2010). In particular, [START_REF] Price | Decisions, Decisions, Decisions: Can Savage Salvage Everettian Probability?[END_REF] argues that agents may have global reasons on which to base their decisions, i. e. reasons other than the utilities of their successors. Such arguments of course undermine the idea that there should be compelling rational arguments for adopting the usual quantum probabilities in Everett.

17 In further work, in particular with my graduate students, I hope to elaborate both on the analogy between Everett's view of probability in his own theory and in classical statistical mechanics -in particular on how it allows one to make statistical inferences in an Everettian universe -, and on the pragmatist reading of typicality, both in Everett and in classical statistical mechanics.

With the addition of Subsection 2.1, the material in this note was presented at the Bertinoro conference. My thanks go to Maria Carla Galavotti, Roberta Campaner, Beatrice Collina and all the colleagues in the ESF network who organised the conference. But most of all I wish to thank Chris Fuchs for the pleasure of endless discussions of quantum Bayesianism over the years, and for removing the prejudice in my mind that a subjective interpretation of quantum probabilities must be a non-starter.

Such a view seems to be suggested for instance by[START_REF] Greaves | Everett and Evidence[END_REF].

When Schrödinger introduced his wave functions, he clearly understood them as representing physical states of matter. But also Heisenberg and other advocates of matrix mechanics had a notion of stationary state, both preceding and distinct from Schrödinger's wave functions. Cf.Bacciagaluppi and Valentini (2009, Ch. 3) and[START_REF] Bacciagaluppi | The Statistical Interpretation according to Born and Heisenberg[END_REF].

Disregarding notions of extended cognition -which are presumably beside the point here.

The situation is quite analogous to that of collapse on the forward light cone. If tech-

nically feasible, a theory in which the collapse of the quantum state takes place along the forward light cone of a triggering event would be manifestly Lorentz-invariant. However, in the case of an EPR pair, it would leave unexplained how space-like separated collapses are correlated so as to match up on the overlap of their future light cones. Cf. e. g. the discussion inBacciagaluppi (2010).

See below for the definition of an effect.

I cannot resist teasing Chris here by pointing out that pace the remarks inFuchs (2010, pp. 24-25), the results of decoherence just scream out for an ontic interpretation of the quantum state (so much classical-like structure within the quantum state that could be used to explain the classical worldif only we could avail ourselves to an ontic interpretation of the state). Cf.Bacciagaluppi (2012, esp. Section 3.5). The quantum-toclassical transition is here to stay, and qBism ought to incorporate it.

For general references to the Bohm theory, see e. g.[START_REF] Bohm | The Undivided Universe: An Ontological Interpretation of Quantum Theory[END_REF],[START_REF] Goldstein | Bohmian Mechanics[END_REF][START_REF] Holland | The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics[END_REF].

And can be justified using arguments analogous to those employed in classical statistical mechanics (see e. g.Valentini (1991a,b),[START_REF] Dürr | Quantum Equilibrium and the Origin of Absolute Uncertainty[END_REF],[START_REF] Towler | Timescales for Dynamical Relaxation to the Born Rule[END_REF]).


Introduction

The papers in this issue are by participants in a Workshop on Time-symmetry in Quantum Mechanics, which took place at the Centre for Time, University of Sydney, Australia in August, 2005. 1 In this Preface, we attempt to give some of the motivation for the subject of the Workshop and briefly mention a few highlights in the history of the subject.

Motivation

Reasons for examining time-symmetric formulations of physical theories generally and quantum mechanics in particular are discussed more fully in [START_REF] Price | Time's arrow and Archimedes' point: new directions for the physics of time[END_REF]. Here we mention just a few considerations. To remove any doubt, the Workshop was not concerned, at least directly, with the formal topic of time reversal invariance in quantum mechanics. The Workshop was motivated firstly by the puzzle of why the formalism of standard quantum mechanics is almost invariably dealt with in a time asymmetric way. While at least at first sight the Schrödinger equation may seem time-symmetric, the projection postulate is definitely time-asymmetric, so that to the list of puzzles that measurement brings into quantum mechanics one needs to add that of time-asymmetry. 2

According to several workers in the field, the source of the problem may lie even deeper, in the use of the Schrödinger equation itself. Indeed, in the case of classical electromagnetism, the advanced solutions are discarded. In the case of standard (non-relativistic) quantum mechanics, two "Schrödinger" equations emerge as the non-relativistic limit of a relativistic treatment (see, for example, [START_REF] Cramer | The transactional interpretation of quantum mechanics[END_REF]). One equation corresponds to evolution of the quantum state forwards in time and the other (its complex conjugate) corresponds to evolution backwards in time. Many of the discussions of time symmetry in quantum mechanics, including several (but by no means all) examples in the present issue, rely on some version of a "two-vector" formalism, which explicitly includes both "Schrödinger" equations.

A further motivation for considering time-symmetric formulations of quantum mechanics is the possibility that they might throw light back onto the measurement problem and other interpretational issues, such as the EPR paradox and nonlocality. It is at this point that issues of time symmetry in quantum mechanics become explicitly related also to the exploration of backwards causation.

In the face of Bell"s theorem, the usual approach is to accept either action at a distance or uncaused correlations via the concepts of "nonlocality" or "nonseparability" of quantum systems. As pointed out in [START_REF] Price | Time's arrow and Archimedes' point: new directions for the physics of time[END_REF] and several of the papers in this issue, the alternative of using a time-symmetric theory with backwards causation has been very much neglected. If the effect of a measurement setting is allowed to propagate backwards in time from one measurement to the source (and forwards in time to the other measurement), the space-like correlations can in principle be easily explained using time-like causal processes. 2 One needs to distinguish further whether the postulate is taken to be part of the theory in some form (collapse theories) or only effective (no-collapse theories). In the latter case, much more work is needed on the relation between time symmetry and decoherence (for some of the issues involved, see [START_REF] Bacciagaluppi | Probability, arrow of time and decoherence[END_REF]).

Besides nonlocality, some time-symmetric approaches may provide an intuitive explanation of contextuality; for example the probabilities in Aharonov and Vaidman"s (2002) two-vector formalism are contextual (as, by Gleason"s theorem, must any probabilities that are not defined as usual by a single vector or density matrix); see also [START_REF] Miller | A contextual probability measure for real properties of quantum systems[END_REF].

History

There are many references to the desirability of time-symmetric formulations of physical theories. [START_REF] Hokkyo | Variational formulation of transactional and related interpretations of quantum mechanics[END_REF] refers to some of the literature. Here we mention only some of the main contributions which are concerned directly or indirectly with non-relativistic quantum mechanics. Perhaps the first significant step was made in the 1940s, when Richard Feynman began a research program initially concerned with classical electromagnetism but later applied to quantum mechanics and quantum field theory. The program relied on four assumptions (Schweber, 1994, p. 383), of which the third was this: The fundamental (microscopic) phenomena in nature are symmetrical with respect to interchange of past and future. This assumption led Feynman, in joint work with Wheeler, to a theory of electromagnetism in which the above assumption was satisfied by including advanced and retarded potentials with equal status (Wheeler and Feynman, 1945).

As Wheeler and Feynman note, the proposal to accord an equal status to the emission and absorption of radiation develops a suggestion due to [START_REF] Tetrode | Über den Wirkungszusammenhang der Welt. Eine Erweiterung der klassischen Dynamik[END_REF], who had proposed, as Wheeler and Feynman put it (1945, p. 159), "to abandon the conception of electromagnetic radiation as an elementary process and to interpret it as a consequence of an interaction between a source and an absorber". A consequence of this suggestion, as Tetrode himself describes it, would be that "the sun would not radiate if it were alone in space and no other bodies could absorb its radiation"" (Tetrode 1922, p. 325). This is a striking early manifestation of the view that what happens in reality might depend crucially on conditions in the future, as well as in the past.

In his PhD thesis in 1942, Feynman had already noted that the same kind of reasoning has consequences for quantum mechanics : "… the spontaneous radiation of an atom in quantum mechanics also, may not be spontaneous at all, but induced by the interaction with other atoms, …" (Feynman, 1942, p. 4), and this led him ultimately to the path integral formulation. The half-advanced and half-retarded potentials do not appear in the final expression for the action in the path integral formulation [START_REF] Feynman | The development of the space-time view of quantum electrodynamics[END_REF] but conceptually the theory still depends on "the amplitude that the source will emit and the detector receive" (Feynman, 1965, p. 167, emphasis added). One consequence is that, because radiation must be received by other atoms, "an atom alone in empty space would, in fact, not radiate" (Feynman, 1942, p. 4, emphasis in the original).

There have been several attempts over the years to elaborate on the advanced-action aspect of the path integral formulation. In the current volume, Hokkyo considers the two-slit experiment from that point of view and suggests there may be an experimental test.

At about the same time (see footnote 1 in Price"s paper in this issue) as Feynman"s program was underway, Costa de Beauregard (1953,1977,1979) had already proposed a timesymmetric, retrocausal view of quantum mechanics, to address what is now known as nonlocality.

In 1964, Aharonov, Bergmann and Lebowitz (ABL) noted that the quantum theory of measurement introduced an apparent time-asymmetry in quantum mechanics at a fundamental level. They proposed that this was due a bias introduced because quantum mechanics was concerned with calculating probabilities for ensembles of quantum systems "pre-selected" by the preparation procedure. They considered ensembles which were also post-selected, by the outcome of a final measurement in a sequence of measurements and showed that, at the microscopic level, "in time-symmetrically constructed ensembles the laws of probability are also time-symmetric" [START_REF] Aharonov | Time symmetry in the quantum process of measurement[END_REF]. The ABL formalism led to the twovector re-formulation of quantum mechanics [START_REF] Aharonov | The two-state vector formalism of quantum mechanics[END_REF]Aharonov and Tollaksen, 2007), also referred to as time-symmetric quantum mechanics (TSQM). TSQM is open to conflicting interpretations. On one reading of the formalism, the "two-vectors" are the preparation state evolved forward in time and the state selected by a later measurement evolved backwards in time. Seen in this light, TSQM makes the same testable predictions as quantum mechanics and its major advantage is in providing a different point of view which stimulates new ideas or solutions to old problems. On another level, it has been suggested for instance that TSQM allows unmeasured properties to be ascribed to a quantum system between measurements, in which case it leads to a different ontology from standard quantum mechanics. Whether or not that step is justified has been a matter of discussion in the literature and the contribution by Ruth Kastner in the present volume touches on that issue. If TSQM is taken to have ontological significance, it represents an advanced-action theory (for the most recent comments on its ontological status, see Aharonov et al. (2007)) which provides a specific expression for how the hidden variables are partly determined by the next measurement as well as the preparation. The problem of nonlocality over spacelike intervals is dealt with by the backwards evolution, over a guaranteed timelike path, of a subsequent measurement state. As noted above, if the TSQM formalism is also applicable as a type of hidden variable theory, then the contextuality requirement is accommodated automatically in TSQM.

The transactional interpretation of quantum mechanics proposed by John [START_REF] Cramer | The transactional interpretation of quantum mechanics[END_REF] in the 1980s was another major step in the application of the "advanced action" concept in quantum mechanics. The "transaction" is based on the exchange of retarded and advanced waves which are the solutions, respectively, of the Schrödinger equation and its complex conjugate (which both result from the taking the non-relativistic limit of the Dirac eqution).

The emitter-absorber transaction provides a specific explanation of nonlocality and the transactional interpretation expressly proposes a different ontology from standard quantum mechanics. In the current volume, Ruth Kastner deals with the transactional interpretation in relation to the concept of "weak values" in TSQM. Tim [START_REF] Maudlin | Quantum non-locality and relativity[END_REF] has suggested that the transactional interpretation is inconsistent. In the current volume, Jossi Berkovitz discusses this problem in terms of the causal loops which are involved.

The above few examples show that concept of advanced action or backward causation is of interest from the foundations point of view and as a stimulant of new ideas. It is also useful from a practical point of view. Over many years David Pegg has applied the ideas of postselection in quantum engineering applications as well as investigating these ideas from a fundamental viewpoint. In the current volume he briefly reviews the relevant formalism and applies the ideas to some well-known quantum paradoxes.

4

Problems and outlook

There are several general problems which an advanced-action formulation of quantum mechanics must deal with. First, it might be wondered whether it is sensible to propose an advanced-action theory when nature appears to exhibit several "arrows of time" specifying a preferred direction of time evolution and causality. In the present volume, Larry Schulman discusses the thermodynamic and cosmological arrows of time in relation to retrocausality. Apart from dealing with the arrows, he also refers to his earlier proposal that the "grotesque" states (superpositions of macroscopic objects predicted by standard quantum mechanics without the arbitrary collapse postulate) can be avoided by special initial states. He suggests that these "special states" may be the result of boundary conditions in which "there is a substantial lack of entanglement among large objects". A similar condition is relied on by David Miller in his contribution to the present volume where it is proposed that measurement outcomes in the laboratory are ultimately determined by initial and final boundary conditions on a cosmological time scale, rather than directly by the properties of the measured quantum system which are then said to be determined by the preparation and measurement themselves.

Another problem concerns the possibility of closed causal loops. The contribution in the current volume by Jossi Berkovitz provides a very thorough discussion of the types of both deterministic and indeterministic loops which might arise in such models, and whether or not they are inconsistent and therefore pose a problem to advanced-action theories. David Pegg"s contribution shows how closed causal loops can be investigated in the laboratory.

Finally, there remain conceptual and technical challenges associated with the project of implementing these ideas in realistic models. The contributions in this volume make progress with these challenges, but much remains to be done. We trust that these papers will provide a stimulus to further research.

Acknowledgements

We are very grateful to the Australian Research Council and the University of Sydney for financial support for the Workshop, and to Brad Weslake for much administrative assistance. We also wish to thank Geurt Sengers, Hanneke Janssen and Karen Coldwell for their input into editing and producing this issue; and the referees of the papers, for their valuable contributions to the final result.


Guido Bacciagaluppi

Leggett's theorem without inequalities *

We prove a no-go theorem for a class of hidden variables theories that satisfy parameter independence. Specifically, we show that, assuming two conditions, there are no non-trivial hidden variables models of the quantum predictions for product measurements on two systems in any maximally entangled state in a Hilbert space of dimension at least 3 × 3. The two conditions are parameter independence and a condition that we call conditional parameter independence. The result is analogous to the recent no-go theorems based on Leggett's inequalities and their generalisations.

Introduction

The main assumption in the standard derivations of the Bell inequalities is Bell's factorisation condition on the joint probabilities for outcomes of an EPRtype experiment [START_REF] Bell | Introduction to the hidden-variable question[END_REF]. The best-known analysis of the meaning of this condition was given by [START_REF] Jarrett | On the physical significance of the locality conditions in the Bell arguments[END_REF] and by [START_REF] Shimony | Events and processes in the quantum world[END_REF], and interprets factorisation as the conjunction of two conditions, called outcome independence (OI) and parameter independence (PI) by Shimony (or completeness and locality, respectively, by Jarrett). 1 These two conditions in turn capture the idea that:

(OI) given a complete description of the state of the system and the experimental context, the outcomes on the two sides of an EPR experiment should be statistically independent;

(PI) given the complete state of the system, the probabilities for the outcomes on each side of an EPR experiment (say, Bob's) should be independent of the experimental context on the other side (say, Alice's).

The general view is that PI is a locality requirement, since its violation would allow signalling across the wings of the experiment if one could fix the values of the hidden variables, while violations of OI (or at least violations of OI alone) are taken to be compatible with special relativity and indicative rather of some form of holism or non-separability, and in this sense allow for a 'peaceful coexistence' [START_REF] Shimony | Metaphysical problems in the foundations of quantum mechanics[END_REF] between quantum mechanics and special relativity.

According to this view, one might be able to construct non-trivial hidden variables theories that coexist peacefully with special relativity. Recent work, however, has uncovered constraints on the possibility of hidden variables theories that satisfy PI. This work centres around [START_REF] Leggett | Nonlocal hidden-variable theories and quantum mechanics: an incompatibility theorem[END_REF] inequalities and their further theoretical elaborations and experimental tests (cf. [START_REF] Branciard | A simple approach to test Leggett's model of nonlocal quantum correlations[END_REF]. In this paper, we shall prove a theorem analogous to (and in certain ways generalising) this work, and in the process arguably make more explicit some of the assumptions involved.

Specifically, we shall show that any hidden variables theory that makes nontrivial predictions (i. e. improves on the quantum predictions if the hidden variables λ are known) for maximally entangled states in dimension 3 × 3 or higher and that satisfies both PI and a further condition, which we call conditional parameter independence (CPI), is incompatible with the predictions of quantum mechanics.

CPI is closely related to the condition of 'constrained locality' introduced (for a different purpose) by [START_REF] Jones | Against experimental metaphysics[END_REF], and reads informally as:

(CPI) given the complete state of the system, the probabilities for the outcomes on each side of an EPR experiment, conditional on the outcomes on the other side, should be independent of the experimental context on the other side.

Our result shows that, if one holds fast to PI, a hidden variables theory will either have to become trivial in modelling the predictions of quantum mechanics for maximally entagled states, or will have to violate CPI, which in our opinion amounts to a new peculiar form of contextuality.

The paper is structured as follows. We first recall the standard framework of hidden variables theories in section 2. Then we state and prove our main theorem in section 3. The final section 4 spells out the relation to Leggett's theorem.

The framework of hidden variables

We shall presuppose the usual formal framework of hidden variables theories as used in the discussion of the Bell inequalities. In such a theory, the complete state of a system is given at the individual level by a pair

(ρ, λ) , ( 1
)
where ρ is the quantum state and λ is called a hidden variable. At the statistical level, one will have some distribution ρ(λ) over the hidden variables. Typically, ρ(λ) will depend on the quantum state ρ, but not on other factors such as the choice of the measurements to be performed on the system (which is generally assumed to be specifiable independently at the initial time).

Once the full experimental context is given, the complete state of the system determines the probabilities for the outcomes of measurements of all quantum mechanical observables. For present purposes, we can restrict ourselves to observables represented by projection-valued measures. We shall thus take a hidden variables theory to determine all probabilities of the form:

p iP (ρ,λ) (P ) , (2)
for all possible states (ρ, λ), projectors P and measurement contexts i P suitable for measuring P . In the case of measurements of product projections P ⊗ Q, we shall assume that the experimental context can be thought of as separable into local contexts i P and i Q pertaining to the two wings of the experiment.

As to whether one should include the microscopic degrees of freedom of the apparatus, including any 'apparatus hidden variables', in λ or in the experimental contexts, this ambiguity is of no essential importance in the following. For convenience, we shall adopt the former choice, because in this case we can take i P to be in principle completely fixed by the experimenter and need not explicitly consider also probability distributions over the contexts.2

Given the above, it is required of a hidden variables theory that averaging over the hidden variables must reproduce the statistical predictions of quantum mechanics:

p ρ (P ) = p iP (ρ,λ) (P )ρ(λ)dλ , (3)
i. e. averaging over λ with the distribution ρ(λ) yields the quantum mechanical probabilities p ρ (P ) = Tr(ρP ) ,

which are in fact independent of any specific context i P (besides the fact that it is appropriate to measuring P ).

A hidden variables theory, finally, will be said to be trivial (or trivial for some set of states ρ) if it makes the same probabilistic predictions as quantum mechanics even for every single λ, i. e. if the complete states (ρ, λ) do not improve on the statistical predictions of the quantum states ρ.

The theorem

Let us state the precise definitions of the conditions mentioned in the introduction. Outcome independence (OI) is the condition that:

p iP ,iQ (ρ,λ) (P, Q) = p iP ,iQ (ρ,λ) (P )p iP ,iQ (ρ,λ) (Q) . ( 5
)
Parameter independence (PI) requires that:

p iP ,iQ (ρ,λ) (P ) = p iP (ρ,λ) (P ) independently of i Q (6)
(and analogously for Q independently of i P ). Finally, conditional parameter independence (CPI) requires that:

p iP ,iQ (ρ,λ) (P |Q) = p iP (ρ,λ) (P |Q) independently of i Q (7)
(and analogously for Q|P independently of i P ).

We shall now prove the following

Theorem: There is no hidden variables theory satisfying PI and CPI that makes non-trivial predictions for measurements of product projections on two n-dimensional systems, with n finite and greater or equal 3, in a maximally entangled state.

The theorem follows from four separate results. The first is

Proposition 1: Take a hidden variables model of the quantum predictions for a maximally entangled state in dimension 2 × 2 or higher, and impose PI. It follows that the marginal probabilities of the model must be non-contextual.

Indeed, it is a property of maximally entangled states that for any projection P of Alice's system, there is a projection Q of Bob's system, such that P and Q are perfectly correlated. That is,

p Ψ (P ) = p Ψ (P, Q) = p Ψ (Q) . (8)
We shall now fix Ψ and drop the index Ψ (or ρ) from the formulas.

It follows that, in order for the hidden variables theory to reproduce this prediction on average, one must have

p iP ,iQ λ (P ) = p iP ,iQ λ (P, Q) = p iP ,iQ λ (Q) , (9)
also for every individual λ, i P and i Q .

But then, p iP ,iQ λ

(P ) = p iP ,i ′ Q λ (P ) implies p iP ,iQ λ (Q) = p iP ,i ′ Q λ (Q), i.e.
imposing PI enforces independence of the local experimental context. Again because of PI, this holds independently of whether or not P is in fact measured alongside Q.

Proposition 1 is merely a variant of the non-locality arguments for contextual hidden variables theories (taken in the sense of 0-1 valuations on the projections in Hilbert space) that have been discussed several times in the past, notably by Stairs (1983), [START_REF] Heywood | Nonlocality and the Kochen-Specker paradox[END_REF], [START_REF] Brown | Nonlocality and Gleason's lemma. Part I. Deterministic theories[END_REF] and [START_REF] Bacciagaluppi | Critique of Putnam's quantum logic[END_REF]. Indeed, taking the special case of all probabilities having values 0 or 1, and of dimension 3 × 3 or higher, Proposition 1 yields a non-contextual value assignment to all projections on Alice's (or Bob's) side, which is impossible by the Kochen-Specker theorem or the corollary to Gleason's theorem. Therefore, one obtains a proof of non-locality for deterministic hidden variables theories without the use of inequalities.

In our (probabilistic) case, applying the result to dimension 3 × 3 or higher, one can invoke Gleason's theorem and conclude that the 'hidden' marginals must be quantum mechanical in form, i.e. given by density operators in some convex decomposition of the completely mixed state.

In order to obtain incompatibility with quantum mechanics we therefore have to impose some further condition, namely CPI:

p iP ,iQ (ρ,λ) (P |Q) = p iP (ρ,λ) (P |Q) . (10)
Evidently, CPI is similar in form to PI, but the two are logically independent. Indeed, it is easy to imagine examples in which CPI is satisfied but PI is violated (this possibility plays an important role in [START_REF] Jones | Against experimental metaphysics[END_REF]).

Conversely, one can easily imagine examples in which CPI is violated, but the dependence on the nearby context is washed out if one averages over the nearby outcomes, so that PI is satisfied.

Note that, if one assumes OI, then PI and CPI are trivially equivalent. This means in particular that, if one assumes PI, a violation of CPI implies a violation of OI, so that hidden variables theories satisfying PI but violating CPI indeed violate the assumptions of Bell's theorem. We shall now prove the following Proposition 2: Take any probability distribution on the product projections in dimension 2 × 2 or higher having non-contextual marginals. Imposing CPI will then force non-contextuality of the joint distribution.

The proof is trivial, since imposing non-contextuality on the marginal probabilities and on the conditional probabilities obviously enforces that of the joint probabilities. Explicitly, assuming CPI, for any such probability distribution, say, p iP ,iQ λ (P, Q) we have:

p iP ,iQ λ (P, Q) = p iP ,iQ λ (P |Q)p iP ,iQ λ (Q) = p iP λ (P |Q)p λ (Q) , (11)
where the first factor on the right-hand side is independent of λ because of CPI, and the second because of the non-contextuality of the marginals. Similarly,

p iP ,iQ λ (P, Q) = p iP ,iQ λ (Q|P )p iP ,iQ λ (P ) = p iQ λ (Q|P )p λ (P ) . (12)
By ( 11) and ( 12), the joint probabilities are independent of both i P and i Q .

Thus, while in dimension 3 × 3 or higher PI constrains the marginals p iP ,iQ λ (P ) and p iP ,iQ λ (Q) to be non-contextual, it does not constrain in the same way the joint probabilities p iP ,iQ λ (P, Q). The joint probabilities could still depend explicitly on the contexts i P and i Q , since it is obvious that fixing the marginals of a probability distribution does not fix the distribution itself. Different pairs of local contexts (i P , i Q ) could determine different joint distributions, all compatible with the same non-contextual marginals. CPI rules out precisely this peculiar form of contextuality.

We shall see, however, that the only non-contextual probability distribution of this kind that reproduces the perfect correlations of a maximally entangled quantum state is that given by the quantum state itself. The next step in the proof is a rather well-known generalisation of Gleason's theorem for the case of product projections (see e. g. Barnum et al., 2005, esp. section 4, and references therein):

Proposition 3: Take any probability distribution on the product projections on H ⊗ H in dimension 3 × 3 or higher, say p λ (P, Q). Then there is a self-adjoint operator Λ on H ⊗ H, with Tr(Λ) = 1, such that

p λ (P, Q) = Tr(ΛP ⊗ Q) . (13)
Equivalently, there is an affine and positive map Φ Λ from H to itself, with Tr(Φ Λ (1)) = 1, such that

p λ (P, Q) = Tr(P Φ Λ (Q)) . (14)
Intuitively, the positive operator Φ Λ (Q) is the state on Alice's side after Bob has performed a selective measurement of Q; indeed, it is an (unnormalised) density matrix. Note that the operator Λ itself need not be positive (i. e. the expression (13) need not be positive other than for product projections). For the special case of probability distributions corresponding to quantum states, Λ is of course a density matrix, and the conclusion holds also in dimension 2 × 2.

Proposition 3 establishes that the hidden states λ of a model of EPR-type measurements satisfying both PI and CPI must be equivalent to such operators Λ. Of course, if one wished to construct theories of this form, one would need to worry how one could extend (contextually) the probability measures of the form Tr(ΛP ⊗ Q) to the full set of projections while maintaining positivity. The question is moot, however, because for maximally entangled states there are no such non-trivial Λ. This is the content of Proposition 4: Let ρ be a maximally entangled state in dimension 2 × 2 or higher. Then, ρ cannot be decomposed as a convex sum of the form

ρ = ηΛ 1 + (1 -η)Λ 2 , ( 15
)
with 0 < η < 1 and Λ 1 and Λ 2 two self-adjoint operators as in Proposition 3.

For dimension 2 × 2, given that ρ is pure, the claim follows from an old result by Choi, namely that the extremal points in the set of all Λ are the pure quantum states and their partial transposes (see [START_REF] Barnum | Influence-free states on compound quantum systems[END_REF]. The proof of Proposition 4 for the general case will be postponed to the appendix.3

Putting together Propositions 1-4, we obtain our desired result. Indeed, take a maximally entangled state ρ in dimension 3 × 3 or higher, and consider possible hidden variables models for the corresponding quantum probabilities. From Proposition 1, assuming PI, we show that the marginals of the hidden distributions are non-contextual. From Proposition 2, assuming CPI, we show that the joint hidden distributions are non-contextual. From Proposition 3, the hidden states are equivalent to operators of the form Λ, and from Proposition 4, Λ = ρ for all hidden states. That is, there is no non-trivial hidden variables model satisfying PI and CPI for maximally entangled states in dimension 3 × 3 or higher.

Note that in dimension 2×2 the theorem fails. A counterexample is provided by the model of the singlet state probabilities constructed using Popescu-Rohrlich non-local boxes [START_REF] Cerf | Simulating maximal quantum entanglement without communication[END_REF]. The possibility of such a model, however, is limited to dimension 2 × 2. [START_REF] Bell | Speakable and Unspeakable in Quantum Mechanics[END_REF] Leggett's theorem without inequalities [START_REF] Leggett | Nonlocal hidden-variable theories and quantum mechanics: an incompatibility theorem[END_REF] has shown that a certain class of hidden variables theories in dimension 2 × 2 that violate the assumptions of Bell's theorem can be nevertheless ruled out through another inequality, which is satisfied by these theories but violated by quantum mechanics. In what follows we shall base ourselves on the modification and generalisation of Leggett's results given by [START_REF] Branciard | A simple approach to test Leggett's model of nonlocal quantum correlations[END_REF].

The relevant class of hidden variables theories is defined as follows. Each λ is given by a pair of vector states µ, ν, and the corresponding probability distributions for the outcomes of product measurements have the form

p a,b µν (α, β) = 1 4 1 + α aσ µ + β bσ ν + αβC µν (a, b) . (16)
Such distributions return the quantum mechanical marginals for µ and ν, but allow for arbitrary correlation coefficients C µν (a, b). They have no context dependence whatever, so satisfy PI (and in fact CPI), but allow for violations of OI. 4Imposing positivity on ( 16) leads to inequalities of the type

1 3 3 i=1 |C(a i , b i ) + C(a i , b ′ i )| ≤ 2 - 2 3 | sin ϕ 2 | , (17)
for judiciously chosen directions a i , b i , b ′ i , and where ϕ is the angle between the directions b i and b ′ i . The corresponding quantum prediction is

1 3 3 i=1 |C(a i , b i ) + C(a i , b ′ i )| = 2| cos ϕ 2 | , (18)
which violates the inequality for a wide range of angles. This violation has been confirmed experimentally. Similar inequalities that violate quantum mechanics have been derived for the more general class of hidden variables with

p a,b µν (α, β) = 1 4 1 + αη aσ µ + βη bσ ν + αβC µν (a, b) , (19)
where 0 < η ≤ 1. These theories allow for arbitrary correlations and (collectively) for arbitrary non-flat marginals.5

Thus, Branciard et al.'s versions of Leggett's theorem are formulated in 2 × 2 dimensions and show that hidden variables models making non-trivial predictions of a certain kind (pure marginals or pure to degree η) for the singlet or any sufficiently entangled state and satisfying both PI and CPI will be incompatible with the predictions of quantum mechanics. The incompatibility is shown by means of Leggett's inequalities, which lead to direct experimental tests. Our theorem is formulated in 3 × 3 dimensions and shows that any hidden variables model making non-trivial predictions for the singlet state (including models with flat marginals) and satisfying both PI and CPI will be incompatible with the predictions of quantum mechanics. The incompatibility does not rely on inequalities.

The analogy is like that between Bell's original proof of non-locality and those without inequalities by Stairs and others.

If one compares the two theorems (apart from the difference in the dimension of the Hilbert space), the main novelty in our theorem is that it imposes no restriction at all on the form of the marginals. By the same token, however, since the non-triviality assumption is no longer quantified, we lose the possibility of a direct quantitative discrimination between quantum mechanics and hidden variables theories.

While, as in Leggett's case, our theorem yields new constraints on hidden variables theories that violate the assumptions of Bell's theorem, we believe it also provides a simple explicit classification of which hidden variables theories are possible if one is prepared to violate the relevant assumptions. Any hidden variables theory will either have to violate parameter independence (such as de Broglie-Bohm theory), or trivialise for the case of maximally entangled states (such as [START_REF] Beltrametti | A classical extension of quantum mechanics[END_REF] canonical classical extension of quantum mechanics, which trivialises for all pure states), or, finally, violate CPI. A hidden variables theory violating this last assumption would exhibit a new and peculiar form of contextuality.

Appendix: proof of Proposition 4

We begin by stating a very trivial lemma:

Lemma 1: For any pure quantum state |Ψ , consider the corresponding mapping Φ |Ψ Ψ| as in Proposition 3. Then for any 1-dimensional projection Q with non-zero probability p |Ψ (Q) := Tr(|Ψ Ψ|1 ⊗ Q), the positive operator Φ |Ψ Ψ| (Q) is also proportional to a 1-dimensional projection.

Indeed, if Bob measures Q and 'collapses' the state onto 1 ⊗ Q|Ψ , it is wellknown that the (unnormalised) 'collapsed' state on Alice's side is pure.

Next, we prove the lemma that is crucial for the proof:

Lemma 2: Take any pure quantum state |Ψ , and assume

Φ |Ψ Ψ| = ηΦ Λ1 + (1 -η)Φ Λ2 , (20)
for suitable Λ k (k = 1, 2) defined as in Proposition 3. Then, for any 1dimensional projection Q,

Φ Λ k (Q) = c k (Q)Φ |Ψ Ψ| (Q) , (21)
with proportionality factor c k (Q) ≥ 0 that may depend explicitly on Q.

Indeed, because of Lemma 1, if Q is a 1-dimensional projection, so (up to a positive multiple) is Φ |Ψ Ψ| (Q). But then it is possible to satisfy

Φ |Ψ Ψ| (Q) = ηΦ Λ1 (Q) + (1 -η)Φ Λ2 (Q) (22)
only if Φ Λ1 (Q) and Φ Λ2 (Q) are themselves positive multiples of Φ |Ψ Ψ| (Q), because projection operators generate extremal rays in the convex cone of positive operators on H.

With Lemma 2 in hand, we only need to show that c k (Q) = 1 for all Q in order to prove Proposition 4.

Now take a basis of 1-dimensional projections Q i in the space of operators on H, and let Q = i α i Q i be an arbitrary 1-dimensional projection. Then, by Lemmas 1 and 2,

Φ Λ k (Q) = Φ Λ k ( i α i Q i ) = i α i Φ Λ k (Q i ) = = i α i c k (Q i )Φ |Ψ Ψ| (Q i ) . (23)
But we similarly know that

Φ Λ k (Q) = c k (Q)Φ |Ψ Ψ| (Q) = c k (Q)Φ |Ψ Ψ| ( i α i Q i ) = = c k (Q) i α i Φ |Ψ Ψ| (Q i ) . (24)
Therefore,

i α i c k (Q)Φ |Ψ Ψ| (Q i ) = i α i c k (Q i )Φ |Ψ Ψ| (Q i ) . ( 25
)
We now specialise to the case of a maximally entangled |Ψ , and prove our final (

) 28
We can now complete the proof of Proposition 4. Consider again the basis Q i in the space of operators. By Lemma 3, if |Ψ is maximally entangled, the operators Φ |Ψ Ψ| (Q i ) will also form a basis. In this case, therefore, by the uniqueness of basis expansions in (25) it follows that

c k (Q) = c k (Q i ) (29)
for arbitrary 1-dimensional projections Q and all Q i in the expansion of Q. It follows that

c k (Q) = c k , (30)

Lemma 3 : 4

 34 For any maximally entangled quantum state |Ψ , the mapping Φ |Ψ Ψ| , up to a positive factor, is unitary with respect to the Hilbert-Schmidt scalar product Tr(A * B).The proof is as follows. Writing a quantum state in biorthogonal form,|Ψ = i β i |ψ i |ϕ i , we have that Tr(|Ψ Ψ|P ⊗ Q) = Tr i β i |ψ i |ϕ i j β j ψ j | ϕ j | P ⊗ Q = = Tr P ij β i β j |ψ i ψ j | ϕ j |Q|ϕ i , j |ψ i ψ j | ϕ j |Q|ϕ i .(27)Now, if ρ = |Ψ Ψ| is maximally entangled, on can check explicitly that Φ |Ψ Ψ| is unitary up to a positive real multiple. Indeed,Tr Φ |Ψ Ψ| (A * )Φ |Ψ Ψ| (i | ϕ i |A * |ϕ j 1 n 2 mn |ψ n ψ m | ϕ m |B|ϕ n * |ϕ j ϕ j |B|ϕ i = 1 n Tr(A * B) .

  independently of Q. Finally, since Tr(Φ |Ψ Ψ| (1)) = Tr(Φ Λ k (1)) = 1, we have c k = 1. This completes the proof.

* An expanded version of this article will be made available on the PHILSCI archive (http://philsci-archive.pitt.edu/).† Centre for Time, Department of Philosophy, University of Sydney, and Institut d'Histoire et de Philosophie des Sciences et des Techniques (CNRS, Paris 1, ENS), Paris. Address for correspondence: Department of Philosophy, University of Sydney, NSW 2006, Australia (e-mail: guidob@univ-paris1.fr).1 Shimony's and Jarrett's conditions are not identical, but the differences will be largely irrelevant in the following. See the remarks in the next section.

In this case ρ(λ) could in principle include 'conspiratorial' correlations between 'system hidden variables' and 'apparatus hidden variables'. In order to derive the Bell inequalities, these correlations need to be ruled out separately (or one can average over the apparatus hidden variables).

The original proof of this conjecture was suggested to me by HowardBarnum and Alex Wilce (private discussion, College Park, 26 April 2008), to both of whom I am extremely grateful. The proof given in the appendix uses many of the same ingredients (in particular the crucial Lemma 2), but it is more elementary.

In this notation, α and β label the results for measurements in the given (signed) directions a, b. Since these results flip signs if the directions do, there is no dependence on the sign of a and b if the outcomes are labelled by Hilbert-space projections (as done in the rest of this paper).

The hidden variables correspond to pairs of density matrices ηµ+(1-η)1 and ην +(1-η)1.

Acknowledgements

The beginning of these investigations goes back to fruitful discussions with Soazig Le Bihan in 2005-06, and this paper was partly prompted by her work (cf. Le Bihan, 2008). More recent discussions with Harvey Brown have been very helpful in clarifying the issues further. For the proof of Proposition 4, I am very much indebted to suggestions by and correspondence with Howard Barnum and Alex Wilce. I am further very grateful to Matteo Morganti and especially Nicolas Gisin for focussing my attention on Leggett's theorem. I wish to thank Stephen Bartlett, Richard Gill, Owen Maroney, Huw Price, Rafael Sorkin and Hans Westman for useful discussions and criticism, and, finally, Michiel Seevinck for his very perceptive remarks on the final draft. This paper was written during my tenure of a Senior Research Fellowship at the Centre for Time, Department of Philosophy, University of Sydney, and in part during a research visit at the Perimeter Institute for Theoretical Physics, Waterloo, Ontario. Financial support from the Australian Research Council (Discovery Project 0556184) and the PIAF Collaboration is gratefully acknowledged.

 Stairs, A. (1983), 'Quantum logic, realism, and value definiteness', Philosophy of Science 50, 578-602.