Bell’s theorem has been called “the most profound discovery of science”. However, there have been controversies on the deep implications of the theorem. This online workshop aims to highlight the existing debates and address the controversies. Read More
Workshop Date: Thursday, December 18, 2014 to Friday, January 16, 2015
Organizers: International Journal of Quantum Foundations
The Assumptions of Bell’s Proof
This topic contains 28 replies, has 5 voices, and was last updated by Roderich Tumulka 5 years, 5 months ago.

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December 21, 2014 at 9:12 am #1672
While it is widely agreed that Bell’s theorem is an important result in the foundations of quantum physics, there is much disagreement about what exactly Bell’s theorem shows. It is agreed that Bell derived a contradiction with experimental facts from some list of assumptions, thus showing that at least one of the assumptions must be wrong; but there is disagreement about what the assumptions were that went into the argument. In this paper, I make a few points in order to help clarify the situation.
December 21, 2014 at 2:44 pm #1674It is often amazing to see how far definitions of the same term may differ when used by different physicists. Roderich Tumulka offers four definitions of reality, and he argues that the Many Words interpretation is in conflict already with the weakest of them (his R4). Accordingly, it would violate the reality assumption most strongly. I completely disagree. (Note that his definition refers to local properties already, and thus may presume the locality of ´´beables´´!) Everett clearly assumed the wave function to be a complete and ontic concept.
Everett’s Many Worlds interpretation is ´´realistic´´ in the sense that – in contrast to complementarity or QBism, for example – it is using a unique and consistent kinematical and dynamical conception to describe an observerindependent empirical world (that is, all observations performed by observers who can communicate and can in principle be described as parts of this world). There is no mention of space or locality yet. In fact, Everett’s specific reality is defined by a wave function in configuration space (including spin variables etc.), which is thus nonlocal in threedimensional space. (This holds even for collapse theories.)
The concept of locality is here used only but essentially dynamically (sometimes called ´´relativistic causality´´), which is still possible if this Hilbert space does have a local basis. This dynamical locality has the consequence that reasonable observer systems have to be local systems, which can have definite states in the quantum formalism only in Everett’s autonomous branches (defined by means of decoherence).
We may all have very different opinions about quantum theory at this stage, but we should not decide between them just by constructing our own definitions. In my opinion, the wave function, if universal and taken seriously (which leads to Everett), tells us that the world may be assumed to be real but has to be nonlocal. So it is remarkable that Tumulka and I come to the same verbal conclusion about locality being the culprit and being ruled out in spite of using quite different definitions of ´´realism´´.
December 22, 2014 at 2:56 am #1677I would just note that the assertion that decoherence can define a basis in the Everettian approach is far from settled science, and has been contested in particular in http://arxiv.org/abs/1406.4126 (Kastner 2014, SHPMP 48, 5658)
Given a universal wave function and nothing else, we get the Everettian picture. But more generally, taking the wave function as ontic does not limit us to the Everettian picture. The transactional interpretation takes the wave function as ontic and includes absorber response, leading to real collapse and determinate spacetime events. The wave function and its responses are nonlocal objects, in accordance with Bell’s theorem, but the results of collapse are local.
December 25, 2014 at 4:02 pm #1701Ruth, I don’t intend here to discuss the transactional interpretation, but the ´´result of a collapse´´ in the usual sense is a new global wave function, and hence still kinematically nonlocal. Only certain macroscopic aspects of this state may be approximately local, and even this is so just by construction: the collapse is usually defined to mimic what can well be understood by decoherence. Or do you know any example of an apparent quantum jump (into an apparent ensemble of final states) that cannot be understood by decoherence in principle?
Surely, decoherence requires a drastic nonequilibrium – like many other arrows of time. This requires a very special (such as symmetric or homogeneous) initial condition at the big bang, for example. (I have written a whole book about that.) But what is ´´circular´´ about such an explanation (as you say in the abstract of your cited paper)? The evolution of an individual Everett branch is clearly not deterministic, and many of its properties are ´´created by chance´´ during its evolution, often related to symmetry breaking. So the resulting structure cannot be presumed.
December 31, 2014 at 12:30 am #1737Roderick,
Thank you for clarifying a number of issues that are often confused.
I think further clarification is still needed on three points (see my contribution to this forum).
1. Your formulation of Locality appeals to a notion of influence you allow may be symmetric in a way that a causal relation cannot be. What could this amount to beyond mutual counterfactual dependence? Without further clarification I can’t see why failure of the principle as you formulate it should be thought to have anything to do with the intuitive idea of locality that Bell was trying to capture, as formulated in the initial intuitive statement of a principle of local causality in La Nouvelle Cuisine which he formulated in explicitly causal terminology.
2. In your rendering of the EPR argument (following Bell) you say this:
“Anyway, Bob’s outcome was already fixed before his experiment.” This would be a reasonable thing for Alice to say, and a reasonable thing for Bob to say in a Newtonian world. But it would not be a reasonable thing for Bob to say in a relativistic world, where fixity or certainty must be relativized to a spacetime point (or region), since this is required to define the past as it backward lightcone.
3. Point 2 extends to “the” probability of Bob’s outcome: this must also be relativized to a spacetime point or region. Alice and Bob correctly assign different probabilities to Bob’s outcome at the same time in (say) the lab. frame, after Alice’s measurement but before Bob’s (assumed spacelike separated). Bell did not notice the need for this relativization, thus rendering his more formal probabilistic statement of Local Causality unclear. I agree that each probability is evidentially related to a relative frequency (cf. your response to Werner): but the relative frequencies also differ since they are defined over different reference classes.January 9, 2015 at 1:02 am #17758 January 2015
Dear Richard, regarding your 1737,
I see no reason why in a relativistic world probabilities need teo be relativized to a particular spacetime point. That may be convenient for some purposes, but why is it needed in general? In a relativistic world, just as in a nonrelativistic world, you need to make clear what you are talking about, and if the probabilities are conditional it is helpful to state the conditions. But I confess I don’t see what you are getting at.
Bob Griffiths
January 9, 2015 at 1:06 am #17768 January 2015
Dear Roderich,
I am pleased to see that someone besides me thinks that Kolmogorov probabilities are not to be abandoned when discussing quantum mysteries. I have made careful use of them in demonstrating that quantum mechanics is local in a fairly precise sense of that term in an article you do not seem to be aware of, or at least it was not listed in your references, namely Found. Phys. 41 (2011) 705; arXiv:0908.2914. I commend this to you, as I regard it as a solid proof that Einstein was right: I prove a form of Einstein locality. True, one should be careful what one means by this, and I give there a definition in precise quantum terms and would be interested in what you think of it. I have thought of offering a substantial prize to the first person who can find a serious error in it; one reason I am pretty confident no one will collect is that the referee was David Mermin who made a thoroughgoing but unsuccessful attempt to poke holes in it.
You should, I think, take Werner’s comments, your [21], more seriously, for even if he has not done particularly well in replying to Maudlin, he has a quite sound intuition that there is something different about the quantum world which Maudlin has missed. An example which I spotted in your article is in your discussion of a spinhalf particle prepared by Carol with a particular spin orientation c, which is then passed to Donald who is unable to determine c by an experiment. There your mistake is in supposing that a spinhalf particle prepared in this manner somehow “carries” the information c, as a “matter of fact in nature”. But this is not so. A spin half particle can carry at most one bit of information, so it cannot carry the value of c. (In the dense coding protocol two bits can be transmitted, but that is done using entangled states between two qubits, so again no more than one bit per qubit. If you are interested in or know something about quantum information I refer you to my article Phys. Rev. A 66 (2002) 012311. quantph/0203058)
A final remark has to do with your reference to counterfactuals when dealing with a stochastic theory. You will find a discussion of how I think it should be done in Ch. 19 of my book, and an application to demolishing the nonlocality claim based on Hardy’s paradox in Ch. 25. More recently Henry Stapp has challenged my locality argument using a counterfactual approach, and we exchanged a pair of papers in Found. Phys. 42 (2012) pp. 647655; arXiv:1111.5364; Found. Phys. 42 (2012) 674684; arXiv:1201.0255. If you are interested, take a look and see who you think comes out ahead.
To summarize, I would argue that Bell’s theorem tells us that quantum results contradict CLASSICAL hidden variables, and the best way to see this is to employ standard (Kolmogorov) probability theory to the corrlelation problem while paying attention to the fact that the quantum Hilbert space is not a classical phase space.
Bob Griffiths
January 11, 2015 at 3:53 pm #1792To Bob at 1775,
I gave the reason in section 3 of my paper and illustrated it in the spacetime diagrams of figures 2 and 3. Please explain what you don’t understand so I can help you see the force of this reason.
January 11, 2015 at 3:53 pm #1791To Bob at 1775,
I gave the reason in section 3 of my paper and illustrated it in the spacetime diagrams of figures 2 and 3. Please explain what you don’t understand so I can help you see the force of this reason.
January 18, 2015 at 3:22 am #1835Thank you to all of the commentators for your remarks!
To Dieter at 1674:
You may be surprised but I completely agree with you that Everett’s manyworlds proposal is a realist theory. In my paper, I considered the conditions (R1) through (R4) for their potential relevance to Bell’s theorem, not for categorizing interpretations of quantum mechanics as realist ones or others. Also, I do not imply that “realism” is a good name for any of (R1) through (R4); I formulate these conditions as possible interpretations of what people might mean when mentioning realism as an assumption of Bell’s proof.
On the question whether the manyworlds picture is local or nonlocal I have commented in Brit.J.Phil.Sci. 62:1 (arxiv:0903.2211), Section 5. Maybe we agree more than you think.
To Richard at 1737:
ad 1. Mutual counterfactual dependence is one possibility, but not the only one. Mutual stochastic dependence is another. Think of the flashes in GRWf, for example; they occur in a fundamentally stochastic way with a nonlocal joint distribution.
ad 2. and 3. I am happy to change my statement to “Bob’s outcome was already fixed on some spacelike hypersurface before his experiment.” Would that take care of your concern?
To Bob at 1776:
Thank you for pointing to your article. I think that your definition of Einstein locality does not capture Einstein’s, or Bell’s, or my idea of locality. I am happy with your formulation “Objective properties of isolated individual systems do not change when something is done to another noninteracting system,” but only if “property” is understood in the ordinary English meaning of the word as something that a system either has or does not have, not if “property” means projection operator. That is because locality, as I understand it and as I believe Einstein and Bell understood it, refers to reality, to “the real factual situation” (Einstein), to the variables that do have values.
Also, I disagree with your objection to my CarolDonald example. The fact that Carol knows the quantum state of each of the particles she prepared (and can prove this by predicting the outcome with certainty for the corresponding direction in 3space) shows that there is a matter of fact in nature about the quantum state of each of these particles. I guess our disagreement is related to the previous point: You need to think in terms of the real factual situation, not in terms of projection operators.
Best regards,
RoderichJanuary 19, 2015 at 3:16 pm #1851Dear Roderich,
I find your reply to my comment interesting, but I wonder if you could elaborate on the following. My use of a subspace for a quantum property goes back to von Neumann, and I employ it at the microscopic level or spin half and at the macroscopic level, because I believe that quantum mechanics applies at all length scales, “from the quarks to the quasars”. Do you share this faith, or do you think that quantum theory breaks down someplace between the quarks and the quasars, and if so, where? If you share my belief that the physics of macroscopic objects is fundamentally quantum mechanical, how within quantum mechanics, as distinct from hidden variable theories, do you represent macroscopic properties? I.e., if not by projectors or quantum subspaces, what do you propose? Or are you a believer in classical hidden variables?
Regarding the CarolDonald example, you need to maintain a distinction between Alice’s preparation procedure, which was recorded in her notebook, and the information which is carried by the particle in isolation from Alice’s notebook. The analogy I like to use is the student who flunks the examination despite the fact that he has sat faithfully in my lectures and thus has been “prepared”. You should think carefully before claiming that there are facts about the world which are even in principle inaccessible to investigation; all sorts of nonsense can proceed under that guise.
Bob Griffiths
January 19, 2015 at 3:44 pm #1852Roderich,
In response to your 1835 you ask (re my 2,3)
“I am happy to change my statement to “Bob’s outcome was already fixed on some spacelike hypersurface before his experiment.” Would that take care of your concern?”
No, this does not address my concern. In a relativistic spacetime fixity, like chance, should not be relativized to a spacelike hypersurface but to the backward light cone of a spacetime region (ideally, a point) representing the momentary location of a hypothetical optimally informed agent. Then whether Bob’s outcome was fixed depends on what that backward light cone is, in such a way that differently located Alice and Bob could both be right even when they disagree about whether Bob’s outcome was “already” fixed.
January 20, 2015 at 3:34 pm #1857Dear Bob,
Let me first address the CarolDonald issue. I was, in fact, paying attention to the distinction you emphasize; let me elaborate some more. Let n_k be the unit vector in the direction in which Carol has prepared the spin of particle k. She predicts that if one were to make a SternGerlach experiment on particle k in the direction n_k, the result will be “up.” According to the rules of quantum mechanics, her prediction is correct in 100% of the cases. According to the same rules, the prediction would be falsified at least sometimes if she had named other directions. Therefore, nature must, in order to produce results in agreement with the rules of quantum mechanics, remember for each particle k what the direction n_k was. Therefore, nature must remember for each particle in this experiment its quantum state up to a phase factor.
On the use of projections: Like you, I guess that the predictions of quantum mechanics are correct from the quarks to the quasars. Projections are a mathematical tool for computing these predictions. However, projections and properties are not the same concept. When contemplating Bell’s proof, we need to consider, besides observables and outcomes of experiments, possibilities for what the reality might be like, and to this end it is necessary to consider properties in the ordinary English meaning, as distinct from projections.
Best regards,
RoderichJanuary 20, 2015 at 3:36 pm #1858Dear Richard,
Indeed one might suspect that statements about probability, or even fixity, must be relativized in the way you suggest. So let me go through the reasoning in more detail. The stateofaffairs (i.e., the “real, factual situation,” the “beables”) on that hypersurface includes Alice’s outcome, so it predetermines Bob’s outcome (as Bob will choose the z direction). Now, if we assume locality then nonlocal beables and the local beables outside Bob’s past light cone cannot have any influence on Bob’s outcome. Therefore, Bob’s outcome is predetermined by the local beables on the intersection of that hypersurface with Bob’s past light cone. I see no way around this conclusion.
Best, Roderich
January 20, 2015 at 4:43 pm #1859Indeed I do believe (not merely suspect) that statements about probability and fixity must be relativized in the way I suggested. I question your use of the word “predetermines”. Certainly Alice’s outcome (ideally) lies on a spacelike hypersurface whose future domain of dependence includes Bob’s outcome. But points on that hypersurface outside Bob’s outcome’s backward light cone do not lie in its absolute past. The relevant notion of predetermination (also fixity, also chance) is determination by the contents of Bob’s outcome’s backward light cone. Since Alice’s outcome lies outside Bob’s backward light cone it can play no role in predetermining Alice’s outcome. This is not merely a semantic issue. Just as we ordinarily take the past to be fixed in a Newtonian spacetime because we assume that in principle it is accessible to our present observations, so also we should consider the absolute past of Bob’s outcome as fixed in a relativistic spacetime because it is in principle accessible from that event. Events like Alice’s outcome are just as inaccessible from the spacetime location of Bob’s outcome as events in its absolute future, so (by parity of reasoning with the Newtonian case) they are not fixed, relative to Bob’s outcome and so not candidates for “predetermining” that outcome.
Of course, if we had some independent reason for thinking Alice’s outcome is in principle accessible from Bob’s outcome that would also be a reason to reevaluate that reasoning. I believe quantum theory gives us no such reason.January 20, 2015 at 7:05 pm #1860Dear Roderich,
Thank you for #1857. Let me take up the second topic first: properties of macroscopic systems. Since you believe that quantum theory applies at the macroscopic level, you need some way to represent properties on a Hilbert space, assuming you are not using hidden variables. I use projectors, because these are the ways properties are represented at the microscopic level. For example, for a harmonic oscillator the property that E=(n+1/2) omega in units hbar=1 is the projector [n]=n><n. The property that the energy is less than 2 omega is [0]+[1] in this notation, and so forth; this is what I tell my students, and, as I say, it goes back to von Neumann. So I am asking what you use for the property “energy of this object is between 5 and 6 Joules” at the macroscopic level if you do not employ, as I do, a projector.
Bob Griffiths
January 20, 2015 at 7:35 pm #1861Dear Roderich,
Again, relative to your #1857. Many people have the mistaken notion that spin half can contain large amounts of information because they visualize the quantum state as a classical arrow with a precise direction, or a classical spinning top with a precise axis of rotation. While such pictures are useful, they can mislead. In particular Sx and Sy do not commute with Sz, and, furthermore the square of the total spin, in units with hbar=1, is 3/4, not 1/4, which would be the case if the spin had a precise direction. So what I tell my students is that when we try and visualize a spin half particle as a classical object, we should think of, say, z+>, as a top whose spin axis is in a random direction except for the fact that its z component of angular momentum is positive. This agrees with the idea that a measurement of Sx or Sy on such an object will typically yield a random value, whereas a measurement of Sz has a probability of 100% of giving the right sign (+ or ) for the angular momentum. So if Carol tells Donald the direction in which to do the measurement he will get the right answer with 100% probability, but not otherwise. Of course all of this is a classical picture, so bound to mislead in some way, but I think a lot less misleading than the picture of a precisely defined axis of rotation.
Of course in the quantum case one wants to apply consistent quantum ideas, and the notion of “information” in the quantum mechanics is not altogether straightforward. However, there are ways of looking at it which I think are fairly compelling. The first is the approach of the physical chemists, who in calculating the entropy of a gas have to take account of the internal degrees of freedom if they are going to get the right answer. The spin of a free spin half particle contributes log 2 to the entropy, and this is confirmed by comparison with experiment. The second approach is one you will find in my “Nature and location of quantum information,” Phys. Rev. A 66 (2002) 012311; quantph/0203058, where I argue that log 2 per qubit gives the correct results for to take an example, dense coding. So far as I know the notion of ‘1 bit per qubit’ is now widely accepted by the practioners of quantum information, though not necessarily among the quantum foundations people who write on the subject. I had a little quarrel with Scott Aaronson about this some years back, but I think he has probably begun to see why my perspective is sensible, though I am not entirely sure.
Best wishes, Bob Griffiths
January 24, 2015 at 5:56 pm #1894Dear Roderich,
In section 7 of your paper you lay out an argument as to why (L) implies (R1). Here’s why that argument fails to establish its intended conclusion.
I quote the crucial steps from your paper:
“Assume locality. Alice’s experiment takes place in a spacetime region A and Bob’s in B at spacelike separation. There is a Lorentz frame in which A is finished before B begins; thus, in this frame, there is a time at which Alice’s experiment already has a definite outcome. She can therefore predict Bob’s outcome with certainty, although she cannot transmit this information to Bob before Bob carries out his experiment. Anyway, Bob’s outcome was already fixed on some spacelike hypersurface before his experiment.”
What that last sentence means depends on the meaning of ‘fixed’. Two possible meanings are relevant:
1. An event e may be said to be fixed relative to events on a spacelike hypersurface whose future domain of dependence includes e.
2. An event e may be said to be fixed relative to events in the past light cone of e.
Since the argument assumed locality, only events in its past light cone can influence Bob’s outcome. So anyone who accepts (L) should adopt meaning 2 as the appropriate understanding of fixity. Based on that understanding, what can it mean to say e is (or is not) fixed relative to events on a spacelike hypersurface whose future domain of dependence includes e? I see two possible ways to understand this:
1a. e necessarily occurs if certain events occur somewhere on this spacelike hypersurface.
1b. e necessarily occurs if certain events occur on this spacelike hypersurface in e’s past light cone.
But (L) renders events outside Bob’s outcome’s past light cone unable to influence it. So 1b is the right way to understand 1. Since Alice’s outcome lies outside the past light cone of Bob’s, it follows that Bob’s outcome is not fixed relative to events on a spacelike hypersurface that includes B within its future domain of dependence. The state of affairs inside the past light cone of B, but before B itself, did not include a fact about the value Bz that Bob will obtain if he carries out a quantum measurement of I x sigmaz.
This objection to your version of the EPR argument generalizes, to become an objection to Bell’s derivation of his condition on probabilities from his statement of local causality (in La Nouvelle Cuisine, 2004 p. 243). I’ve explained this in greater detail in my paper “Local causality, probability and explanation” posted on the IJQF John Bell workshop 2014: see especially section 3.Best,
RichardJanuary 24, 2015 at 5:56 pm #1895Dear Roderich,
In section 7 of your paper you lay out an argument as to why (L) implies (R1). Here’s why that argument fails to establish its intended conclusion.
I quote the crucial steps from your paper:
“Assume locality. Alice’s experiment takes place in a spacetime region A and Bob’s in B at spacelike separation. There is a Lorentz frame in which A is finished before B begins; thus, in this frame, there is a time at which Alice’s experiment already has a definite outcome. She can therefore predict Bob’s outcome with certainty, although she cannot transmit this information to Bob before Bob carries out his experiment. Anyway, Bob’s outcome was already fixed on some spacelike hypersurface before his experiment.”
What that last sentence means depends on the meaning of ‘fixed’. Two possible meanings are relevant:
1. An event e may be said to be fixed relative to events on a spacelike hypersurface whose future domain of dependence includes e.
2. An event e may be said to be fixed relative to events in the past light cone of e.
Since the argument assumed locality, only events in its past light cone can influence Bob’s outcome. So anyone who accepts (L) should adopt meaning 2 as the appropriate understanding of fixity. Based on that understanding, what can it mean to say e is (or is not) fixed relative to events on a spacelike hypersurface whose future domain of dependence includes e? I see two possible ways to understand this:
1a. e necessarily occurs if certain events occur somewhere on this spacelike hypersurface.
1b. e necessarily occurs if certain events occur on this spacelike hypersurface in e’s past light cone.
But (L) renders events outside Bob’s outcome’s past light cone unable to influence it. So 1b is the right way to understand 1. Since Alice’s outcome lies outside the past light cone of Bob’s, it follows that Bob’s outcome is not fixed relative to events on a spacelike hypersurface that includes B within its future domain of dependence. The state of affairs inside the past light cone of B, but before B itself, did not include a fact about the value Bz that Bob will obtain if he carries out a quantum measurement of I x sigmaz.
This objection to your version of the EPR argument generalizes, to become an objection to Bell’s derivation of his condition on probabilities from his statement of local causality (in La Nouvelle Cuisine, 2004 p. 243). I’ve explained this in greater detail in my paper “Local causality, probability and explanation” posted on the IJQF John Bell workshop 2014: see especially section 3.Best,
RichardJanuary 28, 2015 at 5:47 pm #1898Dear Richard (at #1859 and 1894),
I like that you point specifically to the step in the reasoning that you are objecting to. That helps for a good discussion.
I see that the word “predetermines” has connotations that are irrelevant to the argument. For my purposes, “x predetermines y” just means “y is a function of x.” It is not necessary to refer to your definitions 1, 2, 1a, or 1b. So let me go through the reasoning again.
Let R_A and R_B be the spacetime regions in which Alice’s and Bob’s experiment, respectively, are carried out, let A and B be Alice’s and Bob’s outcomes, respectively, let H be a spacelike hypersurface after R_A and before R_B, let P(R_B) denote the relativistic past of R_B (interior of the past light cone), and HP(R_B) the intersection of H and P(R_B). Let S(H) be the stateofaffairs on H. (People often find it difficult to understand what that means, we may talk more about this later.) Then B=f(S(H)) with some function f. Now assume locality. Then B cannot depend on local beables outside P(R_B) (nor on nonlocal beables involving both sides), so B=g(S(HP(R_B))) with some function g. That is what I concluded.
Maybe these remarks provide some clarification.
Best, Roderich
January 28, 2015 at 5:48 pm #1899Dear Bob,
Thank you for your replies #1860 and 1861. There are two relevant meanings of the word “property”: let me call them “English property” (the ordinary meaning of “property” in English) and “quantum property.” An English property of a system is something that a system either has or has not, while a “quantum property” of a system is a subspace of the system’s Hilbert space (or, equivalently, a projector); of course, a system may be in a superposition of a subspace S and its orthogonal complement, in which case one is neither justified in saying that the system has the quantum property S nor that it does not have the quantum property S. For a harmonic oscillator, “the energy is less than 2 omega” is a quantum property. For a macroscopic system such as Schroedinger’s cat, “alive” is a quantum property, and may also be an English property depending on the chosen solution to the quantum measurement problem. In fact, the measurement problem can be phrased as saying that for the theory to make sense we need that “alive” is also an English property of cats (except in a manyworlds framework). In quantum physics we usually focus on quantum properties, but for investigating locality we need to consider English properties.
Concerning classical pictures of spin, I would not tell my students they should think of z+> as a top whose spin axis is random with some constraints, exactly because this picture is bound to mislead. So I am perhaps actually more inclined against classical pictures than you.
But the salient point is that my reasoning does not make use of any classical picture of spin. It only makes use of the 2dimensional Hilbert space of a spin1/2, of the quantum rules for prediction, and of the mathematical fact that the 1dimensional subspaces in that Hilbert space are in a onetoone correspondence to the unit vectors in 3dimensional physical space. I label vectors in Hilbert space by directions in 3space, but do not assume spinning classical tops. Now you need to read my CarolDonald argument again. By the way, I agree that entropy is the log of the dimension of the subspace corresponding to a given macrostate, and that does not affect my CarolDonald argument.
Best, Roderich
January 28, 2015 at 9:54 pm #1900Dear Roderich,
In response to your #1899. Thanks for your clarification, but I am still unsure where you stand. Let us start with ‘English’ properties, by which I think you mean macroscopic properties. Will you allow me to assign things like “pointer is directed at the symbol L on the box” to a Hilbert subspace (of necessarily enormous dimension)? Same with “cat is alive” and “cat is dead”, to (necessarily orthogonal) Hilbert subspaces, call them A and D? May we also assign what arises unitarily from Schrodinger’s experiment (and some people think is somehow a cat, though I don’t) to a Hilbert subspace S? Do you agree that if these symbols stand for projectors, then AD=DA=0, whereas [A,S] and [D,S] are nonzero (and not trivially small)? For spin half, I believe we agree that x+> and z+> are distinct as mathematical objects, and that Carol can produce either of these reliably with her apparatus. Would you agree that Donald, if not given additional information, has no reliable way to distinguish them by some sort of measurement? (He is only given one try; if we send him a thousand particles identically prepared in one of the two states he can, of course, do a lot better.) If you don’t agree, what experiment do you propose? If you do agree, how do you understand his inability? Please excuse me if you have already answered some of these in preceding posts; my memory is none too good. My hope is that we might at least agree about what we disagree about, but that is still not clear to me.
January 28, 2015 at 11:41 pm #1901Roderich, (in response to your 1898)
We can’t decide whether B is function of events on all or part of H without applying some theory. Simply observing relative frequencies of 1 in a sequence of supposedly similar sets of events can’t exclude failure of a corresponding functional relation in unobserved sets.
If we apply quantum theory, we see that Prob(B/S(H))=1 but Prob(B/S(HP(R_B)))=½. I’ll use the word ‘certain’ to refer to probability 1. Then B is certain conditional on S(H), but B is not certain conditional on S(HP(R_B)). Your formulation of locality implies that events outside P(R_B) cannot influence events in R_B: I take it that one thing this is intended to exclude is influencing whether or not events in R_B are certain. I claim there is no such influence here: specifying the outcome at R_A or the state on H outside (P(R_B) does not alter the fact that B is certain conditional on S(H), but B is not certain conditional on S(HP(R_B)). That Prob(B/S(H))=1 but Prob(B/S(HP(R_B)))=½ does not establish failure of locality, as you have defined it.February 2, 2015 at 2:59 pm #1943Dear Bob, (at #1900)
Yes, it would good to find out what exactly we disagree about. So let me answer your questions about my position. By “English properties” I did not mean macroscopic properties. In your example, I assume that S is 1dimensional. I agree that with “cat is alive” is associated a (highdimensional) subspace A of the Hilbert space of the relevant particles, and similarly for D, and that AD=DA=0, whereas [A,S] and [D,S] are significantly nonzero. I also agree that x+> and z+> are distinct as mathematical objects, and that Carol can produce either of these reliably with her apparatus. I also agree that Donald, if not given additional information, has no reliable way to distinguish them by some sort of measurement. How do I understand this inability? I conclude that, in this example, nature knows the actual quantum state of every particle (as does Carol), and the laws of nature are such that Donald has no access to this information. Nature can keep a secret. There are limitations to knowledge. There is a fact about the quantum state of this particular particle, and Donald has no way of finding out. That may be shocking, but we better get used to it. There are things we cannot measure. And once we think of it, that seems obvious: You may not remember what you had for breakfast on February 1 ten years ago, but there was a definite fact about what you had. But now there is no experiment that could answer what you had. So it may seem very natural that there are things we cannot measure.
Best, Roderich
February 2, 2015 at 2:59 pm #1944Dear Richard, (in response to your #1901)
I agree that “we can’t decide whether B is a function of events on all or part of H without applying some theory” if “theory” means laws governing a possible reality (i.e., what I called a “scenario” in my paper). Quantum theory is often understood as not talking about reality but only about empirical predictions; but if we understand “quantum theory” as saying that, for an EPR pair of particles, there are no further variables in addition to the wave function (i.e., what I called “Bohr’s scenario” in my paper), then I agree that “Prob(B/S(H))=1 but Prob(B/S(HP(R_B)))=1/2.” These relations show that Bohr’s scenario is nonlocal (because they show that the stateofaffairs on H outside P(R_B) has an influence on B). I agree in particular that, in this scenario, B is certain conditional on S(H) and not certain conditional on S(HP(R_B)). I further agree that my formulation of locality implies that events outside P(R_B) cannot influence events in R_B. However, I disagree with your last statement, and maintain that the conjunction of Prob(B/S(H))=1 and Prob(B/S(HP(R_B)))=1/2 entails a failure of locality.
Your statement before seems not relevant. To explain, let me use the notation Y for S(HP(R_B)) and X for the remainder of S(H). The condition that X has no influence on B can be expressed as Prob(B/X,Y)=Prob(B/Y). I agree with your remark that specifying X does not alter the fact that Prob(B/X,Y)=1 but Prob(B/Y)=1/2; but the relevant fact for whether X has an influence on B is that specifying X alters the conditional probability of B; that is, that Prob(B/X,Y)=Prob(B/Y).
Best, Roderich
February 4, 2015 at 1:58 pm #1945Dear Roderich,
Thank you for your explanation. I think your “English property” would be what I call a “quasiclassical property/projector” using the language of GellMann and Hartle. On the other stuff I think we will just have to agree to disagree. I once told d’Espagnat that the simplest explanation for why those mysterious superluminal influences are unable to carry any information is that they don’t exist.
Best, Bob Griffiths
February 4, 2015 at 2:08 pm #1946Dear Bob,
Yes, let’s agree to disagree. Thank you for the discussion.
All the best, Roderich
February 6, 2015 at 10:31 pm #1947I disagree that the condition that X has no influence on B can be expressed as Prob(B/X,Y)=Prob(B/Y): that condition merely expresses the inequality of two general probabilities, each of which may be used to infer a (different) chance of an outcome event being of type B (there is no unique chance of an event’s being of type B in this case—see my paper). To connect probabilities and chances to Bell’s intuitive statement of local causality one needs at least to show that the first of these chances could have resulted from a hypothetical intervention that brought about an Xtype outcome (rather than a different X’type outcome). But this is not true: if one accepts quantum theory one thereby accepts that an Xtype(as opposed to an X’type) outcome could not have been brought about as a result of a hypothetical intervention. In that sense, quantum theory implies both that Prob(B/X,Y)/=Prob(B/Y) and that an Xtype outcome has no influence on whether the other outcome is or is not of type B.
 This reply was modified 5 years, 6 months ago by Richard Healey. Reason: should say is not equal to
March 2, 2015 at 8:53 pm #1986Dear Richard,
Perhaps I understand better now the root of our disagreement. It seems to have something to do with whether the definition of locality refers to actions of an agent on one side having consequences on the other side, or whether it refers to events on one side having consequences on the other side, where events may include random events not controlled by an agent. When I use the term “locality,” I mean the latter, and so I did not mention any agent when formulating the locality condition (L) in my paper, while I called the variant with an agent “control locality” (CL). As I say near the end of my paper, CL alone is not sufficient to imply Bell’s inequality; in fact, it’s not clear what CL would mean for a stochastic theory. In the passage from Bell’s 1964 paper that I quoted in the last section of my paper, he describes the assumption of EPR’s argument as CL, and I’d say Bell is inaccurate in this passage; the EPR argument requires L. Maybe you mean CL by “locality,” and therefore find that determinism can’t be concluded from it.
Best, Roderich

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