Quantum quandary #2: A wolf in sheep’s clothing

The experiment of Greenberger, Horne, and Zeilinger (Mermin's version)

Maximilian Schlosshauer said it best:

An encounter with quantum mechanics is not unlike an encounter with a wolf in sheep’s clothing. Disguised in sleek axiomatic appearance, at first quantum mechanics looks harmless enough. But beware: a moment later, it may sneak up from behind and whack you over the head with some thoroughly mind-boggling questions. Indeed, it’s hard to imagine how we could ever cook up another physical theory that’s as simultaneously innocuous and cunning. A theory whose formalism can be written down on a napkin whilst attempts to interpret it fill entire libraries. A theory that has seen astonishing experimental confirmation yet leaves us increasingly perplexed the more we think about it.

In this letter I wish to discuss one of the most delicious ways in which quantum mechanics whacks us over the head. As a warming-up exercise, consider the following game. We have two teams, the “players” (Alice, Bob, and Carol) and the “interrogators.” These are the rules: Either all three players are asked to assign a value to X, or one player is asked to assign a value to X, while the other two players are asked to assign a value to Y. The permitted answers are  +1 or -1. In the first case, the players win if and only if the product of their answers equals -1. In the second case, the players win if and only if the product of their answers equals +1. Otherwise they lose. Once the questions are asked, the players are not allowed to communicate with each other. Before that, they may decide on a strategy. Is there a fail-safe strategy? Can the players make sure that they will win no matter what?

Suppose the players decide on a set of pre-agreed answers. Let’s call them X(A), X(B), X(C) and Y(A), Y(B), Y(C). This means that if Alice is asked to assign a value to X, she will assign the value X(A), and if she is asked to assign a value to Y, she will assign the value Y(A). Ditto for Bob and Carol. The challenge then is to find six values, each being equal to either +1 or -1, which will satisfy the following four equations:

X(A) x X(B) x X(C) = -1 

X(A) x Y(B) x Y(C) = +1 

Y(A) x X(B) x Y(C) = +1 

Y(A) x Y(B) x X(C) = +1

You may want to try filling in the six boxes with values +1 or -1 in such a way that these four equations are satisfied.

To see that this cannot be done, one only has to multiply the left-hand sides of the last  three equations. Because each of the Y values appears twice in the result, and because the square of both +1 and -1 is +1, the product of the three left-hand sides is simply the product of  X(A), X(B), and X(C). Because the product of the corresponding right-hand sides equals +1, the last three equations imply that

X(A) x X(B) x X(C) = +1. 

But this contradicts the first equation. Hence if there is a fail-safe strategy, it cannot rely on pre-agreed answers.

There does indeed exist a fail-safe strategy. Instead of relying on pre-agreed answers, it relies on the outcomes of measurements performed on three particles in a certain quantum state. A quantum state is something that can be prepared but cannot be described in the language by which we ordinarily describe states of affairs. The state of a quantum system can only be described statistically, by assigning probabilities to the possible outcomes of measurements to which the system can be subjected. Quantum mechanics makes it possible to create a system consisting of three particles in such a way that the spin of each particle can be measured, that the possible outcomes of each measurement are +1 or -1, and that they satisfy the above four equations.

Spin is a physical quantity that can be measured with respect to any axis. The number of possible outcomes for a given axis depends on the type of particle used. Here we use particles for which this number is two. Thus there are two possible outcomes: +1 for “spin up” and −1 for “spin down” (with respect to the chosen axis). The three-particle state is created in such a way that whenever their spins are measured with respect to the x axis, the product of the outcomes will be −1, and whenever one spin is measured with respect to the x axis and the two other spins are measured with respect to the y axis, the product of the outcomes will be +1.

Now suppose that each player has one of the three particles at their disposal. When asked to assign a value to X, Alice will measure the spin of her particle with respect to the x axis and assign to X the outcome of her measurement: X(A) = +1 if this is “up” and X(A) = -1 if it is “down.” Likewise, when asked to assign a value to Y, she will measure the spin of her particle with respect to the y axis, in which case Y(A) = +1 if her outcome is “up” and Y(A) = -1 if it is “down.” Proceeding in this way, the players are sure to win.

In classical (i.e., pre-quantum) physics, a measurement serves to reveal an intrinsic property or value—i.e., a property or value that exists regardless of being measured. What about the three spin measurements? Do they reveal properties that the three particles already had before the measurements were made? Would the three spins possess the values observed if they had not been measured? The answer is a definite No, for if it were a Yes, the six values X(A), X(B), X(C) and Y(A), Y(B), Y(C) would have to satisfy the above four equations, and we already know that this is impossible. What holds for pre-agreed answers also holds for pre-existent values.

In a seminal paper of 1935, Albert Einstein, writing with his postdoctoral research associates Boris Podolsky and Nathan Rosen, had argued that

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.

If Einstein, Podolsky, and Rosen were right, there would be three elements of physical reality X(A), X(B), and X(C) each having the value +1 or -1, each waiting to be revealed by the outcomes of two far-away Y measurements. There would also be three elements of reality Y(A), Y(B), and Y(C) each having the value +1 or -1, each waiting to be revealed by the outcomes of a far-away X measurement and a far-away Y measurement. Using the second equation one could conclude that X(A) equals the product of Y(B) and Y(C); using the third equation one could conclude that X(B) equals the product of Y(A) and Y(C); and using the fourth equation one could conclude that X(C) equals the product of Y(A) and Y(B). One could then conclude that the product of X(A), X(B), and X(C) equals the product of these three products. And since this only contains the squares of Y(A), Y(B), and Y(C), one could further conclude that the product of X(A), X(B), and X(C) equals +1. Yet if the three X values are measured, their product invariably turns out to be -1. Concludes Mermin:

So farewell elements of reality! And farewell in a hurry. The compelling hypothesis that they exist can be refuted by a single measurement of the three x components: The elements of reality require the product of the three outcomes invariably to be +1; but invariably the product of the three outcomes is -1.

Each of the four predictions which are based on a single equation is legitimate, inasmuch as the three measurements involved are compatible. This means, among other more technical things, that they can be made simultaneously. The prediction that is made by combining three equations, on the other hand, is illegitimate, inasmuch as it involves incompatible measurements, such as those of X(A) and Y(A). The spin of a given particle cannot be measured simultaneously with respect to two different axes.

How can we explain the possibility of making the legitimate predictions? How, for example, is it possible to predict the value of X(C) by measuring Y(A) and Y(B), regardless of the order in which the three measurements are made, and regardless of the distances between the three particles? We know of two possible explanations. The first is that the three particles are created in a state that predetermines the outcome of each of the six possible measurements. The second is that the outcomes of a pair of Y measurements somehow influence the outcome of the only compatible X measurement—instantaneously and at any distance. The first explanation is ruled out by quantum mechanics (specifically, by the absence of pre-existent values), and the second is ruled out by the theory of relativity, which is another corner stone of contemporary physics. In short: no viable explanation is known.

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The three-particle setup used in this letter was first discussed by N. David Mermin in a Reference Frame column in Physics Today (“What’s wrong with these elements of reality?,” June 1990, pp. 9-11). It is a re-formulation of a setup originally thought up by Daniel  M. Greenberger, Michael Horne, and Anton Zeilinger (“Going beyond Bell’s theorem. In M. Kafatos, ed., Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, Kluwer, 1989, pp. 69-72).