Einstein, Podolsky, and Rosen

This post is a part of a series on physical theory.

In 1935, Einstein, Podolsky, and Rosen published a paper in the Physical Review entitled “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” The paper constitutes part of the foundation for Einstein’s now well-known skepticism about the entrenched probabilism of quantum mechanics, which is sometimes summarized by the quote often attributed to him that God does not play dice with the universe.

EPR (as they are now called) do not propose experimental results but rather a thought experiment and a philosophical argument. In a particularly vivid variant of their thought experiment due to Bohm (1951)—now sometimes called the EPR experiment—two particles, an electron and a positron, separate at a point of radioactive decay. According to the experiment, they speed off in separate directions in a way where their total spin is zero. Along a chosen axis each of the particles can take one of two real values ( and ). For the observable property of spin along this axis, I will write for short “.”

According to the formalism of quantum mechanics, systems describing these two particles, individually, can be formulated as Hilbert spaces—that is, complex valued inner product spaces—and the representation of the spin observable along the chosen axis is represented in the formalism by a Hermitian (self-adjoint) operator. The observable values for the operator are then the eigenvalues of such operators (and since they are Hermitian, these values will be real numbers). Corresponding to each eigenvalue, there is an eigenspace. As a fact of linear algebra, there is a projection operator which takes any complex valued vector in the space and projects it onto the eigenspace defined by the th eigenvalue. For an observable , I will write the projection operator as “” and its th eigenvalue as “.” Likewise its projection operator for this eigenvalue will be written “.”

States of the positron system and the electron system are given by complex valued vectors. Part the mystery of quantum mechanical systems is that in general there is no certain answer to experimental questions like “Does the electron have value of spin ½ in the direction given that the state is ?” According to the Born rule, that answer is given by a probability value which is obtained from the projection operator on the eigenspace defined by the eigenvalue . So if the state is , then the probability of observable having value its th eigenvalue is given by the following formula:





What notion of probability is involved here? The motivating premise behind this question is that the probabilism about the Born rule should be interpreted as some measure of subjective uncertainty. But there are many problems with this purported resolution. For one thing, subjectivists about probabilistic uncertainty usually say that the rules of probability serve to constrain rational partial belief (through the rules specified by the Kolmogorov axioms) by permitting a family of probability functions. So, strictly speaking, any measure which, for example, assigns and nonnegative values which sum to unity should be legitimate probabilities (i.e rational degrees of belief). However the Born rule and the theory behind it does not merely specify a class of such functions but points to one in particular. Presumably, on the Born rule account, assessing the probability in any other way is then a mistake. In other words, the Born rule specifies a measure of something objective rather than subjective uncertainty.

Consider the example of the Blackjack table, where there is no real question about what will be the case when the dealer turns the card. We take it that there is an absolute (i.e. certain) fact of the matter about the order of the cards in the deck once it has been shuffled. The uncertainty of the card game is generated by your own subjective situation: the order of the deck, by design, has been hidden from you. To use the language of Einstein’s theological metaphor, an all-knowing God would probably find Blackjack totally uninteresting: the order could not possibly be hidden from an omniscient agent. Indeed, you also would probably find Blackjack uninteresting if you knew the order of the cards: there would be no surprises.

The difference in the situation with the positron and electron is that the state vector in the formalism of quantum mechanics is an element of the so-called phase space, which contains the set of all possibilities. That is, the space contains all the possible configurations of reality and so a single element of phase space describes one possible situation completely. Here, then, is a mystery: how can there be uncertainty about a situation when it has been specified completely? That is Einstein’s mystery, and the EPR experiment proposed is meant to show that there is some mistake in thinking that the formalism of quantum mechanics is complete.

The argument Einstein, Podolsky, and Rosen give in the paper is not a purely mathematical one but instead depends upon several metaphysical principles. In particular, they argue for a Sufficiency Criterion of Reality which provides a criterion for what any theory counts as a real element. The EPR argument is these days largely considered outmoded because of a purely mathematical argument from the 1960s due to J.S. Bell. According to Bell and his eponymous theorem, some of EPR’s assumptions about the localization of causal influence purportedly would yield mathematical predictions which do not in fact match those of quantum mechanics. So whatever EPR are arguing against, they must not really be taking quantum mechanics as their target.

Nevertheless, it remains a mystery how quantum mechanics can both have a phase space which describes the world totally and nevertheless yield results which are objectively uncertain. What I would like to do in the following paper is to consider the EPR argument as one concerning metaphysics rather than mathematical physics. In particular, I would like to consider the all-important Sufficiency Criterion for Reality (or just Sufficiency Criterion for short). I would like to argue here that there is a mistake in the underlying metaphysical principles of the argument. In Section 1, I review in more detail the EPR Argument, and I shall argue that it depends on something slightly different than the Sufficiency Criterion which I call the Necessity Criterion. In Section 2, I shall argue that Einstein, Podolsky, and Rosen cannot justify the appeal to the Necessity Criterion because they mistake the probabilities of quantum mechanics for predictions.

The EPR Argument



Einstein, Podolsky, and Rosen begin their discussion by drawing a distinction between the concepts a physical theory introduces and objective physical reality which are independent of any physical theory. The concepts are themselves not part of physical reality but intended to correspond to it. From this distinction, they write that there are two questions regarding the success of a theory:

In attempting to judge the success of a physical theory, we may ask ourselves two questions: (1) “Is the theory correct?” and (2) “Is the description given by the theory complete?” It is only in the case in which positive answers may be given to both of these questions, that the concepts of the theory may be said to be satisfactory.



Philosophers have long debated whether concepts themselves constitute part of reality or whether they are somehow separate from it. The locus classicus of the debate is found in the medieval dispute between the nominalists and realists but arguably it is also found in the more recent debate between those who are descriptivists about referring expressions and the Fregeans.

Here, EPR appear to want to maintain a view weaker than nominalism since they claim that physical concepts need only be separate from objective physical reality (leaving open the possibility that they are real objects of another kind). Further, they appear to have in mind a view that physical theories and the physical concepts of which they are made up represent aspects of physical reality. So, on their view, theories are collections of something like what philosophers would today call propositional entities. If we were interpreting theories as they are understood by logic, the theories would then be sets of propositions embedded within some system of logic (perhaps first order logic).

Answering the first of their two questions is a matter of degree. The degree of correctness is the degree of agreement between the conclusions of the theory and human experience. This is something like a baseline empiricist attitude, but of course it masks many details about what a physical theory must contain. At the very least, it must contain some claims which specify a system of measurement. They say as much when they write that “experience … in physics takes the form of experiment and measurement.”

For Einstein, Podolsky, and Rosen the preceding remarks concern the correctness of a theory insofar as it is the degree to which the theory accurately reflects reality—that is, to what degree are the claims it makes are true with respect to experience understood through measurement. But there is a second question of a theory’s success concerning how much its reflects reality by its breadth rather than its accuracy. That is, there is a separate question of the degree to which the theory’s claims account for all the features of the (physical) world.

Ostensibly, this second question concerns a class of fundamental physical theories. For there are clearly theories which can be developed within physics which address only narrow aspects of reality. For example, we can develop a physical theory of the working of an electrical microchip or the mechanics of an engine. Control theory, likewise, is a well-established subdiscipline of physics and engineering of this kind which develops its own specialized vocabulary and concepts (e.g. feedback, sensor, controller, etc.) to describe a large class of mechanical systems. But it can hardly be criticized for failing to be a theory of everything (or even everything physical).

What EPR seem to have in mind are other theories like the system of Lagrangian mechanics or quantum mechanics, which are thought, at least prima facie, to describe physical reality in full generality. Unlike theories of the working of a train engine, EPR write we may ask of these more general theories whether they totally capture all the elements of reality. That is, we may ask whether they are complete. They say in their paper that what they wish to assess is the second question about quantum mechanics and its completeness. To this end, EPR propose the first premise of their argument:

(P1) If a theory is complete, then every element of physical reality must have a counterpart in the physical theory.



EPR here write that the “elements of physical reality cannot be determined by a priori philosophical considerations but must be found by an appeal to the results of experiments and measurements.” This premise and the preceding addendum are seemingly innocuous, reflecting again a sort of unreflective empiricism, but there are nevertheless reasons why even the staunchest empiricist would have reason be suspicious of (P1). For one thing, the statement seems to presuppose that experiments and measurements take place in the absence of or at a level more fundamental than that of physical theory. While it is true that any theory, physical or otherwise, employs language and concepts, to describe or reflect reality, there is nevertheless a metaphysical puzzle about whether we can access such reality independent of conceptualizing. This puzzle vexed Kant in the Critique of Pure Reason where he concludes, in accord with his Transcendental Idealism, that the true nature of the Thing-in-itself cannot be known independently of the conceptual apparatus which we impose upon our sensory experience. The same Kantian puzzle here applies, mutatis mutandis: we would only ever be in a position to assess the completeness of a physical theory if we were able to access reality directly in the absence of theorizing. But measurements and experiments are conceptual activities embedded within frameworks for understanding the physical world.

EPR do not dwell on the Kantian puzzle posed by the premise but move on to a description of their second premise, which I call the Sufficiency Criterion:

(P2) If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity within a theory, then (according to the theory) there exists an element of physical reality corresponding to this physical quantity.



Except possibly in the cases of fiction or other imaginative circumstances, every non-probabilistic statement we normally make comes with an assumption (or presupposition, as Strawson (1950) would put it) of the existence of the referents of the terms (i.e. the referring expressions) in our conversation. There is hence presumably a concomitant assumption of the reality of those referents. So perhaps the specification that a theory takes certain things to be real might strike you as superfluous: aren’t all the things a theory speaks about presumed to have reality, in some sense or another?

The problem here once again concerns when a physical theory employs probabilities. When I make an uncertain assessment of an outcome (for example, at the Blackjack table), I probably take it that one of several potential events will eventually come about—that is, one of them will “have reality”. There is perhaps a temptation to say here that even prior to having witnessed the next card I take some disjunction of events to have reality. However, this would be a confused way of talking since disjunctions are properly applied to elements of logical language (i.e. to propositions) and not, at least at first glance, to spatiotemporal events.

But if a physical theory employs probabilities, what can we say about which events are real? Or, perhaps to use an antiquated expression, which events have being? Perhaps we can say the event which is assigned nonextremal probability has some kind of partial being (or reality). Indeed, maybe we could say that the degree of reality is commensurate with the measure. Call this preceding position Partialism about reality. But these considerations of Partialism are metaphysically troublesome: should we treat claims of reality (or being) as ones capable of degree just like we can say we are some degree (i.e. some distance) above sea level? Or is being a categorical property: you either have it or you don’t?

The Sufficiency Criterion is evidently a premise which attempts to sidestep these metaphysical troubles. It says that when a theory makes a prediction with certainty, then at the very least those claims are about events or objects which it takes to be real. Hence it is a sufficient condition for reality. EPR’s aim here, \texit{it appears}, is to avoid giving a complete characterization of physical reality by giving both necessary and sufficient conditions. All P2 asserts is that if a theory can make a prediction with certainty, then at the very least it must treat the contents of the prediction as real. The Sufficiency Criterion is weak enough that a Partialist should be able to accept it: the Partialist believes that when a theory makes a prediction of certainty, then the theory regards that event as being fully real.

The third and perhaps least controversial premise is a mathematical fact of quantum physics. As I mentioned above, observables within the formalism are represented using operators in Hilbert space. Einstein, Podolsky, and Rosen note that when any two operators, and , fail to commute (i.e. when ), then a prediction of certainty about a value in one observable results in an uncertain prediction about the value of the other observable. EPR write that of one such case of noncommuting operators in quantum physics is the position operator and momentum operator. In particular, if the position of an element of a quantum system has unitary probability then its momentum will have nonextremal probability. So they write that the “usual conclusion from this in quantum mechanics is that when the momentum of a particle is known, its coordinate has no physical reality.” Likewise, returning to our example of the positron and electron, if we divide space along three axes , , and , then spin along any such axis will be an observable with an associated Hermitian operator. For example, for the observable of spin on the electron I will write . Any two spin operators along orthogonal axes for a single particle will fail to commute (e.g. ).

In more contemporary language we would then say, of a single particle—say, the electron–that we have have made a measurement with certainty that it is spin one-half in the direction. Then it is in superposition of spin one-half and negative one-half in the direction. Mysteriously, some sometimes say that that the idea of superposition is something like being both one thing and another, partially or to some degree. Einstein, Podolsky, and Rosen do not employ this concept of superposition but instead configure their argument in terms of certainty, uncertainty, and predictability. So, I schematize their third premise as follows:

(P3) For any pair of observables formalized with non-commuting operators, a prediction of certainty of a value for one corresponding observable will yield an uncertain prediction for the other.



From these three premises, EPR conclude the following principle which is taken to be an intermediary conclusion. They say:

(C1) From (P1)-(P3): either (a) the quantum mechanical description of reality is not complete or (b) when the operators corresponding to two physical quantities do not commute, the two quantities cannot have simultaneous reality.



Recall that as a matter of classical logic, you may establish a disjunction “a or b” by showing that the negation of and the negation of together yield a contradiction. So Einstein, Podolsky and Rosen’s argument appears to be as follows: suppose both that the two observable values of noncommuting operators did have simultaneous reality and that quantum mechanics was complete. From the condition of completeness in P1, we would have that there is some corresponding description or conceptual counterpart of this fact in the theory. Here is now the crucial—and mistaken, I shall argue—appeal to P2. They write:

If then the wave function provided such a complete description of reality, it would contain these values; these would then be predictable.



The above statement is misleading since of course there is some general sense in which quantum mechanics “contains” all the observable values. In fact, by the Born Rule, there are the probabilities of all the eigenvalues that a Hermitian operator may take. So, in some sense they are all predictable according to the formalism of quantum mechanics. What EPR must mean here is that if quantum mechanics did contain the descriptions of both observables, then they would make those predictions certain. In other words, the Born Rule would yield unitary probability for both the value of position and for momentum (or for spin value in the -axis of the electron as well as the -axis). If quantum mechanics considered those observable values simultaneously real, then both would have certain probabilities. But this is not an appeal to the Sufficiency Criterion at all but to an alternate premise which says that certainty of a prediction is a necessary condition of what a theory takes to be real. So, the argument relies on a confusion of necessary condition with a sufficient one. I shall return to this problem in the next section in more detail.

EPR nevertheless note that if quantum mechanics yielded all predictions of noncommuting observables with certainty, then that would contradict the mathematical fact established by P3. So they believe to have shown that the negation of (a) and the negation of (b) above yield the desired contradiction. So they conclude either (a) or (b).

The next steps in the EPR argument famously propose the possibility of quantum entanglement. They consider systems like the electron and positron which I described in the introduction of this paper. Suppose that the two travel sufficiently far apart from each other so that the measurement on one particle cannot affect a measurement on the other (perhaps they would have to be light years apart). If, for example, I were to make a measurement of for the spin along the -axis for the electron, then I would know with certainty the spin in the -axis of positron would be (since the particles began with a total spin of 0 and that total quantity must be preserved). Likewise, I could repeat this argument for the values of spin along the -axis: if I were to make a measurement of for the spin the -axis, I would be in a position to make a prediction of -axis spin of the positron with certainty. Or more generally, for the purposes of the argument, the premise is:

(P4) Quantum mechanics makes predictions of certainty about the values of the observables of non-commuting operators when particles are entangled.



Since we can make these predictions with certainty, EPR argue that by an application of the Sufficiency Criterion (here applied properly), we may conclude the negation of the disjunct (b). That is:

(C2) From P4 and P2: two quantities corresponding to two noncommuting operators have simultaneous reality.



And hence they conclude finally, by an application of modus tollendo pollens with (C1):

(C3) From C1 and C2: the quantum mechanical description of reality is not complete.





The Sufficiency Criterion



In the preceding section, I argued that Einstein, Podolsky, and Rosen present their arguments as depending crucially on a premise which I have called the Sufficiency Criterion. In particular, the criterion claims that so long as we can make a prediction of certainty within a theory without disturbing that system, then the theory must regard the predicted event as being real. I argued, however, that the move from P1-P3 to C1 misapplies the Sufficiency Criterion. What the argument requires is a premise that any description of an element of reality by a theory is concomitant with the theory predicting it with certainty. That is, what they require is the following Necessity Criterion:

(X2) Any claim of the existence of an element of physical reality by a theory requires that the claim of its existence be made via a prediction of certainty.



Since P2 was required for concluding C2, EPR’s argument requires both P2 and X2. That is, predictions of certainty by a theory are both necessary and sufficient for its claims of what is real by a theory. In this section, I argue that both principles are misapplied to quantum mechanics because the probabilities yielded by the Born rule cannot possibly be understood as predictions. Instead, superpositional claims are claims about what really and presently is.

To begin, let me say that if we were to mistake the probabilities of quantum mechanics for measures about uncertainty concerning the future, then in some respects that mistake would be perfectly understandable. As Hacking (2006) (and many others) have pointed out, the idea of a probability measure was developed in the early 17th century principally as a guide for assessing the prudence of strategies in games of chance. In those contexts, probabilities do rightly concern subjective uncertainty about the future and are correctly understood (perhaps indirectly) as predictions. For example, having a high subjective probability at the Blackjack table that the next card will be less than a 5 is a prediction that the next card will be one of four kinds of cards. But of course, not all subjective uncertainty is about the future: I can presumably take a gamble about what happened in Ancient Rome (perhaps who the true murderer of Julius Caesar was). We don’t typically gamble about the past not because we don’t have uncertainty about the past but simply because settling such bets would prove to be troublesome.

To further complicate matters, our best mathematical characterization of probability is given by the Kolmogorov axioms. These axioms define a probability measure to be a real valued function on an algebra of subsets of some other underlying set . Such an algebra is a set closed under (countable) intersections and complementation. The function is a probability measure when it satisfies three axioms: (i) the maximal element of the algebra, , should have measure 1; (2) the measures of the union of any two sets in the algebra should be equal to the sum of their measures individually; and (3) all sets in the algebra receive nonnegative measure.

Though the Kolmogorov axioms serve to clarify the definition of a probability function, they do not serve to tell us anything about how or which probabilities should be applied to subjective uncertainty. Nor do they serve to give us any clarification on the metaphysical nature of chance. Indeed, many other structures of measure completely unrelated to chance and uncertainty can rightly satisfy the Kolmogorov axioms.

To illustrate this, consider the following example: I am presently the owner of a 2008 Toyota Prius, which is parked in my brother Luke’s garage in California. Suppose as a prank, while I am away in England, he decides to take apart my car piece by piece. Let us suppose further that Luke has access to a scale for automobiles, and so for some reason he decides to measure the weight of all the pieces of my Prius. Let the set of all car parts be . First, he weighs the entire scrap heap of parts which turns out to be kilograms. In turn, he measures each part individually and also all the combinations of parts—he has a lot of time on his hands. If any combination weighs kilograms, then we say the prank measure (call it Pr for short) of the combination is . Now a question presents itself: does the Pr satisfy the Kolmogorov axioms? If Luke has done his weighing correctly, yes. He has weighed the powerset of parts of the car including the whole heap, and so he has at hand a proportional measure.

Now if I asked you what interpretation of probability Luke was using in assessing this measure, you would say something was mistaken about the question. Luke is not being a frequentist even though he does measure many things in sequence. He is not being a subjectivist even though he personally does believe that this Pr measure of proportion is accurate. Moreover, no value of the Pr measure is itself a prediction even though—and here is an important point—the Pr measure could be used by itself to make predictions. For example, if for a given part its Pr measure was less than a trillionth, then I could predict that it would probably would not hurt if it fell on me some time in the future. Even though the Pr measure satisfies the Kolmogorov axioms, it simply gives us claims about what actually is presently—in particular what proportions obtain in the present. Luke’s measure therefore cannot rightly be considered a measure of chance or uncertainty.

Now I claim that what the EPR argument does not acknowledge is the possibility that (superpositional) proportions provided by the Born rule constitute exactly this sort of measure, unrelated to prediction, chance, or uncertainty. They satisfy the Kolmogorov axioms, but what they describe are presently occurrent facts. To assume that they are measures concerning uncertainty or chance is to make the same mistake in assuming that Luke’s measure is one concerning uncertainty or chance. What precisely is this measure? That is perhaps the continuing mystery of the quantum mechanical formalism.

But here is a further problem: suppose the probabilities within the quantum mechanical formalism did nevertheless include some intrinsically predictive element. What justification do we have for the Necessity Criterion and the Sufficiency Criterion? I said in the preceding section that the latter of these two, the Sufficiency Criterion had some plausible justification: all it says is that a theory making a prediction of certainty must be committed to the reality (or existence) of the event about which it makes the prediction. But even so, recall the exact wording of this premise (P2):

(P2) If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity within a theory, then (according to the theory) there exists an element of physical reality corresponding to this physical quantity.



But then is the theory committed to to the present existence of a future event? Does it commit itself to the existence of something qua the future? The potential existence of something? Configuring the claim of existence to the present tense yields many metaphysical troubles. At the very least, this is some sort of barbarism of grammar. A theory making a prediction of certainty about an event doesn’t say that something or other exists; rather it makes a prediction that something or other will exist. That is, this modification yields:

(P2’) If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity within a theory, then (according to the theory) there will exist an element of physical reality corresponding to this physical quantity.



This principle is plausible enough, but now consider its converse, which is the same modification made to Principle (X2):

(X2’) Any claim of the future existence of an element of physical reality by a theory requires that the claim of its existence be made via a prediction of certainty.



This principle is downright false. Consider the case of subjective prediction: presumably I can be predict that Sanders will win the 2020 election and not be committed with certainty to the outcome. But a principle having the form of (X2) or (X2′) is required for concluding the disjunction in C1. These critical reflections suggest that the soundness of the EPR argument should be re-evaluated.



Conclusion



I have attempted to argue two points. First, I have attempted to argue that the EPR argument assumes that because quantum mechanics employs functions which have the structure of Kolmogorov measures they must involve either something like subjective uncertainty or random chance. It is far from clear whether the superpositions identified by quantum mechanics are anything like predictions. Furthermore, even if we were to grant that the measures are predictions, I have argued here against the EPR argument on the grounds that it mistakes a necessary condition for a sufficient one. While I thought that the sufficient condition, which I called the Sufficiency Criterion, was plausible, its converse was unpalatable. I have very conspicuously avoided giving a positive account of what the probabilities in quantum mechanics really mean. I leave that as a topic for a lifetime of future research.