1.5 The celebrated EPR paper
New York Times 4 May 1935
Einstein supported de Broglie’s championing of hidden variables (but not the implicit non-locality) and in 1935 he co-authored what is maybe the most famous paper ever written on quantum mechanics, the so-called EPR paper (from the authors, A. Einstein, B. Podolsky and N. Rosen: Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review 47, 777 (1935)).
Strictly speaking, the text of the EPR paper was written by Boris Podolsky. The ideas in the paper had developed over a series of meetings between Einstein and his two assistants, Podolsky and Nathan Rosen. Einstein invited Podolsky to write the paper because he reckoned Podolsky’s written English would be better than his own. Rosen was chosen because of his expertise in the particular quantum-mechanical calculations that the ideas called for. The paper turned out to be a disappointment for Einstein, who apparently had not checked it before Podolsky submitted it for publication. The problem, according to Einstein, was that the central idea was obscured by the formalism adopted by Podolsky: given that the paper is still one of the most frequently cited in the scientific literature, and having read it several times, I can understand Einstein’s disappointment!
EPR begin their argument (in which they intend to give the answer “No” to the question posed in the title) by saying that, if a theory is complete, then it must, at least, describe “every element of the physical reality”. They give an instance of what they mean: if it is possible, without disturbing a system in any way, to predict the exact value of a physical quantity within this system with certainty, “then there exists an element of physical reality corresponding to this physical quantity”. The idea of the paper is to show that it is, indeed, possible to predict such a quantity with certainty, whereas quantum mechanics fails to do so. This, they argue, would mean that quantum mechanics fails the completeness test: it would be incomplete, and, by implication, it might be saved in the form of a local theory with what later became known as hidden variables.
The central idea is stunningly simple. If two particles come sufficiently close together, then their wave functions overlap so that they are instead described by a single quantum-mechanical wave function, which still holds even after they have separated once more. When he had read the EPR paper, Schrödinger called this entanglement. The background to his choice of this term is that particles (or systems) that have never been in contact, and are therefore not entangled, can be described by their individual wave functions, and so a wave function that would describe them both would then simply be the product of their two separate wave functions. However, a wave function for particles that have been in close contact cannot be separated mathematically into two simple products – it cannot be disentangled. As a crude analogy, you can separate the expression into the product , and so you would say that x and y are separable. However, you cannot do the same with . Using the analogy, we would say that x and y are entangled here: they are non-separable.
Just to be clear, normally everything is more or less entangled with everything else, at least in the neighbourhood, and so, when we talk of two particles being entangled, what we really mean is that we have been able to isolate the two particles from being entangled with everything else for a brief moment.
Paraphrasing the EPR paper, imagine two identical particles, A and B, which are initially stationary and in such close contact that they are entangled. Now suppose they explode apart from “ground zero” in opposite directions. Being identical, they have the same mass, and so they will be travelling with equal but opposite velocities, away from each other. This means that the momentum of each particle is equal and opposite to the other, which, in turn, means that the total momentum of the two particles is always zero (because they are both initially stationary with consequently zero momentum). Suppose that Alice and Bob are two experimenters, and that Alice measures the properties of her particle A, and Bob measures the properties of his particle, B. (Alice and Bob crop up everywhere in thought experiments of all shapes and sizes in physics and philosophical literature.)
Now, although Heisenberg’s Uncertainty Principle tells us that Alice’s particle does not simultaneously have a precise position and momentum, and the same goes for Bob’s particle, nevertheless, surprisingly, you can use the Uncertainty Principle to show that the separation distance and the total momentum of these two particles do have precise values simultaneously.
So, suppose that, after the particles have separated by, say, a light-year, Alice decides at 3:00 pm to measure the precise position of her particle. She can do this, of course, at the expense of giving up any knowledge of the momentum of her particle. However, since the separation of the two particles has a precise value at all times, Bob’s particle must now also be in the state where it has a precise position at 3:00 pm.
But suppose, instead, at 3:00 pm, Alice decides to measure the precise momentum of her particle rather than its position. Of course, in doing so, she gives up any knowledge of the position of her particle, but the important point is that now Bob’s particle must also, at 3:00 pm, be in the state where it has a precise momentum.
Now here is where Einstein felt let down by Rosen’s writing. Einstein wanted the thrust of the EPR paper to be that Alice, through her decision on whether to measure position or momentum, was instantaneously controlling the state of Bob’s particle, no matter how far away it was. To Einstein, this clearly demonstrated the nonlocality of quantum mechanics, which, in his eyes, meant that quantum mechanics could not be a complete theory – there had to be an explanation for the apparent nonlocality. Podolsky, on the other hand, had taken the argument further, pointing out that Alice could choose to measure either position or momentum at 3:00 pm, and that either choice would mean that Bob’s particle also had a precise position or momentum. There would not be time for “news” of which property Alice had chosen to measure, and so Bob’s particle must already have precise values for both properties at 3:00 pm. Since that contradicted Heisenberg’s Uncertainty principle, then, argued Podolsky, quantum mechanics could not be complete, implying that it needed hidden variables to complete it. Regardless of this difference of emphasis, though, EPR remains a landmark paper, highlighting the essential nonlocality of quantum mechanics.
