A history of the EPR paradox and its resolution
Any popular article about quantum mechanics or quantum computing is bound to contain the, now catchphrase, “spooky action at a distance”. It amazes the reader to learn that entanglement is so counterintuitive that Einstein himself expressed his disbelief. Then you read about the EPR paradox, Bell’s inequality, “hidden variables” and, before you know it, you are convinced the universe is not “locally real”.
Whenever I hear about subjects that seem too complicated to grasp, it always helps me to understand their history. What is the series of events that led people to think in such terms? Here I’ll tell the story of the EPR paradox, how it was confirmed in practice, and one possible, but weirdly uncited, resolution.
All I require from the reader is a vague understanding of Heisenberg’s uncertainty principle. Not even that, because I will explain it here. In quantum mechanics, we have to deal with pairs of so-called conjugate quantities: the more you know about one the less you know about the other. One example of such a pair is the position and momentum of a quantum particle. Once you know the position of a particle you cannot know anything about its momentum. Once you learn the momentum of the particle you lose any information about its position. The two quantities cannot be known at the same time.
Our story starts in 1935 when Einstein together with Podolsky and Rosen (EPR) want to express their discontent with a theory based on the principle outlined above. Together they wrote a paper called “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”¹. In this paper, they express the fact that, in order to be successful, a theory of physics must satisfy two conditions: 1. it must be correct, and 2. it must be complete.
A theory is correct if its predictions are in agreement with what is observed in experiments. Even during that time, it was clear that the predictions of quantum mechanics were in agreement with experiments. The theory was correct. But was it complete?
As the three authors explain, a theory is complete if every element of physical reality has a correspondent in the physical theory.
Okay, but what is physical reality? The authors explain again: “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.” In other words, if you can predict with certainty a physical quantity, then that quantity is real; it exists.
Remember that in quantum mechanics we have pairs of conjugate variables. If we know one then we can predict it with certainty, but we do not know anything about the other. This means that, according to quantum mechanics, at any given moment only one of the quantities in a conjugate pair is real in the sense of the EPR paper. In the case of position and momentum, when we know the position of a particle then the position is real, but it makes no sense to talk about the momentum. We cannot predict it with certainty. Even more: if we measure the momentum we disturb the system. The momentum is not defined, it is not real. Likewise, if we decide to measure the momentum, then, after the measurement, we know the momentum of the particle with certainty. We can predict that a second measurement of the momentum will yield the same result. Now the momentum is real. But now we cannot predict the position. The position is no longer defined. It is no longer real.
Einstein, Podolsky and Rosen then conclude: If they can show that both position and momentum are real at the same time, then it means quantum mechanics is an incomplete theory. Why? Because one of the real quantities is not explained by the theory.
To show that both quantities can be real at the same time they imagine a system of two particles. In this system, the two particles interact for a given amount of time, after which they no longer interact. The authors of the paper show with an example that it is theoretically possible that, after the interaction, the system of two particles is left in such a state that:
- if you measure the momentum of the first particle, the second particle is left in a state with a definite momentum and unknown position.
- if you measure the position of the first particle, then the second particle is left in a state with a definite position, but indefinite momentum.
We would now call this state an entangled state, but they did not call it that in the paper.
For the three authors, the second particle has both definite momentum and definite position, because it should not matter to it what quantity we measure on the first particle. Measurements on the first particle should not define the reality of the second particle since the two particles no longer interact. We would now say that we separate the particles in space so far apart that interaction between the two would require faster-than-light communication. In the words of the paper, denying the simultaneous reality of both position and momentum of the second particle would make them “depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this.”
Having proved that two conjugate quantities can exist as part of reality at the same time, Einstein, Podolski and Rosen conclude that quantum mechanics is incomplete, as it only describe one of these quantities at any given moment.
But the story doesn’t end here. This is only the beginning.
Almost 30 years later, in 1964, John Bell, in his paper “On the Einstein Podolsky Rosen Paradox”², showed that quantum mechanics cannot be made complete as Einstein and his co-authors wanted. There is no way to simply add some new variables, things that we don’t know about the system, and make quantum mechanics complete. Adding these so-called “hidden variables” and thinking that conjugate quantities have definite values, would yield results that are incompatible with the existing predictions of the quantum theory.
Bell used the spin of the electron in his paper instead of position and momentum. Measuring the spin of an electron along orthogonal axes obeys the uncertainty principle just like a pair of conjugate variables. If you measure the spin along the Z axis you lose the information about its component along the X axis.
In the following years and decades, several experiments were conducted to see what the observed results would be. Would the results of the measurements of such conjugate quantities on entangled systems be as predicted by quantum mechanics, or would they be compatible with “hidden variable” theories?
A series of experiments were carried out and it turned out that they agreed with quantum mechanics.
People now say that the world we live in is either non-local: different systems that are far apart can influence each other instantaneously; or non-real: properties do not have a value unless measured.
But there’s more. I mentioned at the start an obscure resolution to this paradox.
In June 1999 David Deutsch and Patrick Hayden published a remarkable paper called “Information Flow in Entangled Quantum Systems”³ in which they challenge the idea of non-locality. In the paper they used a theoretical abstraction of a quantity that exhibits quantum properties, the basic unit of quantum information: the qubit.
Some of you may be familiar with the state vector used in quantum computing that is acted upon by unitary quantum gates. We call this the Schrodinger model of quantum information. In their paper, Deutsch and Hayden used a different model: the Heisenberg model. For them, each qubit has associated an infinite set of observables. The information is encoded in one of these observables. If you know which observable holds the information you can measure it and retrieve the information with certainty.
In this model of computation, a qubit also has observables associated with every other qubit with which it is entangled. Information stored in these observables is not accessible through measurement on this qubit alone, but it is accessible through measurement of both entangled qubits. This is called “locally inaccessible information”. Deutsch and Hayden show that in a pair of entangled qubits, information can be stored locally, but only be accessible when the observer looks at correlations between the entangled parts.
It’s difficult to explain this view in terms of the original EPR paper where position and momentum were discussed since we are now talking about qubits, but I’ll try an analogy anyway. Talking about the two entangled particles in the EPR paper, Deutsch and Hayden would say that the measurement of the first particle does not affect the second particle in any way. Measuring momentum on the first particle does not leave the second particle in a state of definite momentum; likewise, measuring the first particle’s position does not leave the second particle in a state with a definite position. Since they are too far apart to interact, all the observers of each particle can do is perform different measurements on their respective particles. Even though the particles are entangled, the information looks random to the separate observers. It is only when they meet and compare results that they notice a correlation in their measurements. This means that the correlation between measurements of the two particles is only revealed when the information of the separate systems is brought back together by the respective observers who now meet to compare results. Fun stuff!
This is as far as the story goes for now, but let’s hope there will be a next chapter written soon. Keep all these things in mind for the next time you hear about local realism.
[1] A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev. 47, pages 777–780
[2] J. S. Bell On the Einstein Podolsky Rosen paradox, Physics Physique Fizika vol. 1, pages 195–200
[3] David Deutsch and Patrick Hayden Information flow in entangled quantum systems
