EPR

In 1935, Einstein co-authored with Boris Podolsky and Nathan Rosen the most famous criticism of the idea that quantum mechanical wave functions provide com- plete descriptions of physical states. The argument of the EPR paper has the same basic structure as that of the earlier and simpler “boxes” type argument: if quantum mechanics is complete this would imply a violation of locality. Or equivalently: if we believe in the principle of relativistic local causality, we must reject the completeness doctrine.

It was discovered only rather recently that the entire text of the EPR paper – “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?”

[6] – was written by Podolsky (after conversations with Einstein and Rosen) and sent in for publication before Einstein had even seen the manuscript. Einstein wrote, in a private letter to Schrödinger, that the main point of the argument had not been made very clear: “the essential thing is, so to speak, smothered by the formalism [2].” So we should be a bit cautious about treating the EPR paper as providing an accurate presentation of Einstein’s views. But the paper is so famous and so important that we will review it rather carefully. The subsequent section then discusses some of Einstein’s own later expressions of the same basic argument.

Here is the abstract of the EPR paper, which lays out the argument to be presented:

In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality.

Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete [6].

The structure here is a bit convoluted, so let us delve in and try to understand it better.

The explanation of what it means for a theory to be “complete” seems clear and uncontroversial. In the paper, EPR elaborate what they describe as a necessary condition for calling a theory “complete”: “every element of the physical reality must have a counterpart in the physical theory”. The overall goal of the paper will thus be to establish the existence ofmoreelements of reality than have counterparts in quantum wave functions. In particular, the argument can be understood as an attempt to establish that a single particle can havebotha definite momentumanda definite position, something that is forbidden in quantum mechanics since the position and momentum operators do not commute. (This implies that there is no wave function that is simultaneously an eigenstate of both position and momentum.)

In order to try to establish the existence of these properties, EPR require a “suffi- cient condition for the reality of a physical quantity”. As they elaborate in the main text, this criterion is as follows:

4.2 EPR 97

If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity[6].

As we will discuss in Chap.6, this criterion became the focal point of Bohr’s attempt to rebut the EPR argument. But it has always seemed perfectly valid to me. In any case, it is a little hard to understand the idea, so let’s think it through with a simple concrete example.

Suppose someone hands you a shoebox with a wine glass in it and you want to determine whether the wine glass is shattered, or intact. One way of doing this might be to shake the box vigorously and listen for the tinkling sound of shattered pieces of glass hitting one another. But if the question is whether the glass in the box was shattered originally, when the box was first handed to you, this method does not really work: the act of shaking the box may very well result in an originally intact glass shattering! So hearing tinkling glass pieces inside wouldnoteffectively establish that the glass had already been broken prior to your shaking. The shaking itself may have brought that shattered state about.

By contrast, suppose (to complicate the scenario slightly) that the glass came to be in the box by the following procedure. There were two glasses on the shelf; one of them was perfectly intact, and one of them was already broken. Then your trusted friend flipped a coin and thereby randomly selected one of the glasses to seal up in the box; suppose we are certain that he was extremely careful so that, if the intact glass was the one selected for inclusion in the box, the glass was not broken during the act of putting it in the box. The second glass is then left on the shelf and the cupboard door is closed. Now in this situation, another method of determining the state of the glass in the box presents itself: simply open the cupboard and see which glass is there! If the intact glass is there in the cupboard, it must be the already-shattered glass that is in the box, and vice versa. This way of determining the contents of the box – in which we never interact directly with the box or its contents at all – ensures that the determined state of the glass in the box faithfully represents the true original state of the glass. We preclude the possibility that our act of determining the state has somehow affected and changed the state. This is the basic scheme that EPR will use to try to show that (in a certain special situation) a particle can be said to possess simultaneously definite values of both position and momentum.

EPR thus consider the following situation. Suppose two particles have interacted and gotten into an entangled state but then spatially separated so they are now far apart from one another. See Fig.4.4. Assuming the particles are well-separated in regard to theirycoordinates, we then focus our attention on the degrees of freedom x1andx2. Suppose in particular that the particles are in the following entangled state:

(x1,x2)=δ(x1−x2)=

δ(x1−x)δ(x2−x)d x (4.6) Pictured in the two-dimensional configuration space, this state is a “ridge” along the diagonal linex1=x2. One can think of it as a superposition of states, over all possible values ofx, in which both particles are definitely located at positionx. That is, the

y1

x1

y2

x2

Particle 1 Particle 2

Fig. 4.4 Two particles which have previously interacted are spatially separated but remain in an entangled state. In particular, the spatial degrees of freedomx₁andx₂are entangled. We assume that, say, the part of the quantum state associated with theycoordinates is a simple product of two well-separated wave packets, centered (say) aty₁=0 andy₂=0 for the coordinate systems shown here. (Note that we use distinct coordinate systems for the two particles such that, for example, y₁=0 andy₂=0 are perhaps a million miles apart from one another!) So the particles are entangled (in so far as their positions alongxare concerned) but they are unambiguously well-separated in space in regard to theirypositions

state does not attribute a definite position to particle 1 or to particle 2 – both particles are maximally smeared out. But they are smeared out in a perfectly correlated way:

measurement of the positionx₁of particle 1 immediately tells us the positionx₂of particle 2 because (even though neitherx₁ norx₂has a well-defined value prior to such measurements)x1andx2are definitely equal to one another.

But this means it is possible to determine the position of particle 2indirectly– without disturbing the physical state of particle 2 at all – by measuring the position x1of its distant entangled partner. And so, by the reality criterion, it follows that the distant particle must alreadyhavea definite position even when no such position is attributed to it by the pre-measurement wave function, Eq. (4.6). And note that this is already sufficient to show that that pre-measurement wave function did not provide a complete description of the state of the two particles: particle 2 has a definite position, but the wave function doesn’t tell us about this at all.

EPR, however, go farther. The two-particle wave function can be re-written in this alternative (but mathematically equivalent) form

(x₁,x₂)=δ(x₁−x₂)= 1 2π

e^{i k(x1−x2)}dk = 1 2π

e^{i kx1}e^{−i kx2}dk. (4.7) The last expression can be understood as saying that the state is a superposition – over all possible values ofk– of states in which particle 1 has momentum p1=k and particle 2 has momentum p2 = −k. So the state can also, alternatively, be understood as a state in which neither particle has any definite momentum value, but the momenta of the two particles are perfectly (anti-) correlated: p₁= −p₂.

4.2 EPR 99 But this means it is possible to determine the momentum of particle 2indirectly– without disturbing it at all – by measuring the momentum p1of its distant entangled partner. And so, by the reality criterion, it follows that the distant particle must already havea definite momentum even when no such momentum is attributed to it by the pre-measurement wave function. It is, in short, the same story again with momentum as it was before with position. So in addition to possessing a definite position (about which the pre-measurement wave function was silent), particle 2 apparently also possesses a definitemomentum(about which the pre-measurement wave function was also silent). So that wave function provides, at best, a decidedly – a doubly – incomplete description of the state of the particle. There are at least these two physical properties, position and momentum, which in reality have sharp well-defined values, for which there is no corresponding element in the theoretical description. And in a way it’s even worse than that, for by establishing the real existence of both position and momentum (for the one distant particle) EPR show not only that the particular wave function in Eq. (4.6) fails to provide a complete description, but that no wave function possibly could provide a complete description. For there is, simply, no such thing as a wave function that is simultaneously a position and momentum eigenstate.

That is the essential argument. But I have explained it here in my own words, and my version doesn’t appear to correspond perfectly to the logical structure of the EPR paper’s abstract. Let us try to understand that. First of all, what should we make of this disjunction (from the paper’s abstract), “that either (1) the quantum- mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality”? This sounds very complicated but is actually quite trivial. Suppose two operators (for example position and momen- tum) fail to commute. Clearly, either the corresponding physical properties (1)can have simultaneous reality, or (2)cannothave simultaneous reality. If theycan, then quantum mechanics is necessarily incomplete, because there is no wave function that is simultaneously an eigenstate for (i.e., there is no wave function that simultane- ously attributes definite real values to) the two properties in question. So the trivial disjunction I wrote two sentences back is equivalent to the one from the EPR text.

Now, in the paper, EPR continue the argument as follows. Having established the disjunction between (1) and (2) just discussed, they write:

Starting then with the assumption that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quantities, with noncom- muting operators, can have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2). We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete.

Where and what, exactly, is the argument described in the first sentence here? It is again somewhat obscure and confusing, but actually this is just the essential argument we reviewed before. The way it is presented in the text is along the following lines.

Thinking of the state as =

δ(x1−x)δ(x2−x)d xit is obvious that, if we measure the position of particle 1 and find it, say, atx1 =X, the state of the two-particle system collapsestoδ(x1−X)δ(x2−X)which (being a product state) implies that we can

attribute the following wave function to particle 2:

ψ2=δ(x2−X). (4.8)

On the other hand, thinking of the (2-particle pre-measurement) state as =

2π1

e^{i kx1}e^{−i kx2}dkit is obvious that, if we measure the momentum of particle 1 and find, say,p₁=P, the state of the two-particle systemcollapsesto_2π¹ e^{i P x1/}e^{−i P x2/} which (being a product state) implies that we can attribute the following wave func- tion to particle 2:

ψ2=e^i(−P/)x2. (4.9)

EPR write: “Thus,it is possible to assign two different wave functions ... to the same reality(the second system after the interaction with the first).”

But then,assuming that wave functions provide a complete description of the state of the particle, i.e., assuming the negation of statement (1) from before, we have – from Eq. (4.8) – that particle 2 has a definite position, and – from Eq. (4.9) – that particle 2 has a definite momentum. Which indeed contradicts statement (2) from before.

Thus, the way it is presented in the actual EPR paper, the argument has the following extremely convoluted structure: either (1) or (2), but denying (1) requires one to also deny (2), and so one cannot consistently deny (1); that is, one must accept (1). That is, to be sure, logically valid. But it is also needlessly convoluted. The heart of the argument is simply the idea that, for spatially-separated but appropriately- entanged pairs of particles, we can determine, with certainty, the value of some property of one of the particles without actuallymessing with itat all, but by instead messing with its entangled partner and using the correlations built into the entangled state to infer something about the undisturbed particle. It is just like the example of the wine glass in the box. No wonder Einstein thought Podolsky’s version of the argument was unnecessarily confusing!