This article reviews the predictions of quantum mechanics (QM) for one- and two-particle Stern–Gerlach experiments and then frames Bell's results, which rule out hidden-variable alternatives to QM, as attempts by a skeptical Eve to fool Alice and Bob with (first example) classical probability mixtures of non-entangled quantum states and (second example) a classical hidden-variable theory. With hidden variables, Eve can succeed when Alice and Bob limit themselves to two Stern–Gerlach directions, but always fails for some choices of three directions. For three random directions, hidden-variable theories are impossible (we show) exactly 2/3 of the time. All the calculations in this article are available in Mathematica and Python files in the supplementary material, allowing readers to experiment with their own variants.
This article reviews the predictions of quantum mechanics (QM) for one- and two-particle Stern–Gerlach experiments and then frames Bell's results, which rule out hidden-variable alternatives to QM, as attempts by a skeptical Eve to fool Alice and Bob with (first example) classical probability mixtures of non-entangled quantum states and (second example) a classical hidden-variable theory. With hidden variables, Eve can succeed when Alice and Bob limit themselves to two Stern–Gerlach directions, but always fails for some choices of three directions. For three random directions, hidden-variable theories are impossible (we show) exactly 2/3 of the time. All the calculations in this article are available in Mathematica and Python files in the supplementary material, allowing readers to experiment with their own variants.
Introduction The paradox of Einstein, Podolsky and Rosen was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty. There have been attempts to show that even without such a separability or locality requirement no ‘hidden variable’ interpretation of quantum mechanics is possible. These attempts have been examined elsewhere and found wanting. Moreover, a hidden variable interpretation of elementary quantum theory has been explicitly constructed. That particular interpretation has indeed a grossly non-local structure. This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions. Formulation With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern–Gerlach magnets, on selected components of the spins σ 1 and σ 2 .
The EPR-Bohm experiment is studied in a classical model with hidden variables. The main hypothesis of Bell about this experiment must be modified in a non-trivial way to allow for the fact that our model is non-linear and so non fully deterministic. With our classical model, we then arrive at the same angular behaviour than the one that is predicted by Quantum Mechanics for the correlated measurement of spin projections on the polarizer axis. If the spin-spin interaction predicted by our particle model, or more precisely in the present case, the spin-polarizer field axis interaction, is not able to align the spin axis with the polarizer axis, then our classical result differs by a factor 3 from the quantum mechanical result. It is surprising to find that the angular behaviour is easier to reproduce by a classical model than the amount of probability. Naively, one would have expected the reverse.
We discuss whether the Stern-Gerlach experiment accepts hidden-variables theories. We discuss that the existence of two spin-1/2 pure states ∣↑⟩ and ∣↓⟩ rules out the existence of probability space of specific quantum measurement. If we detect ∣↑⟩, then measurement outcome is +1. If we detect ∣↓⟩, then measurement outcome is −1. This hidden-variables theory does not accept the transition probability ∣⟨↑∣↓⟩∣2=0. Therefore we have to give up the hidden-variables theory. This implies the Stern-Gerlach experiment cannot accept the specific hidden-variables theory. And we study whether quantum phase factor accepts hidden-variables theories. We use the transition probability for two spin-1/2 pure states (∣↑⟩+∣↓⟩)/2 and (∣↑⟩+eiθ∣↓⟩)/2. It is cos2(θ/2). We discuss that the phase factor does not accept another specific hidden-variables theory. We explore the phase factor is indeed a quantum effect, not classical. Our research gives a new insight to the quantum information processing which relies on quantum phase factor, such as Deutsch's algorithm.
Now that we have a few more concepts under our belts, I would like to go back and revisit in some more detail just what John Bell proved in 1964 about entangled states. He considered a simple quantum situation like the entangled ions in Alice and Bob’s pocket watches and then set about to compare the predictions of quantum theory for such a setup to the predictions of a classical hidden variable theory that had all of Einstein’s constraints: reality, locality, and certainty. Remember-there was already the Bohm hidden variable theory, well known in 1964, which did predict the same outcome as quantum theory, but this Bohm theory doesn’t count because it violates Einstein’s locality condition. (Einstein thumbs his nose at the Bohmians.) Bell’s goal was to see if he could construct a local classical hidden variable theory that would agree with all the predictions of quantum theory and satisfy Einstein’s three conditions. If such a hidden variable theory could be found, then it would be tempting to just replace quantum theory with that hidden variable theory and all the spookiness would just disappear. But in order for this project to succeed, the classical hidden variable theory had to obey all three of Einstein’s reality, locality, and certainty conditions and reproduce all the predictions of quantum theory. at was just too tall of an order to ask of any local hidden variable theory, Bell discovered. Something, somewhere, had to give.
In the early 1950’s, Bohm simplified the 1935 EPR state from continuous coordinate-momentum space to discrete spin space by introducing a singlet state of two spin 1/2 particles. A more practical example of the EPR-Bohm state concerns the polarization states of photon pairs, such as the spin-zero state of a high energy photon pair disintegrated from the annihilation of positronium. Is it possible to have a realistic theory which provides correct predictions of the behavior of a particle similar to quantum theory and, at the same time, respects the description of physical reality by EPR as “complete”? Bohm and his followers have attempted a “hidden variable theory” to formulate the physical reality into the wavefunction of a particle or a pair of particles. The experimental testing of Bell’s inequality started from the early 1970’s. Most of the historical experiments concluded the violation of the Bell’s inequalities and thus disproved the local hidden variable theory.
Prologue A memorial conference for John Bell, open to the public, was held at Rutgers University around 20 years ago. I gave a short talk there on Bell's views about David Bohm's “hidden variables” formulation of quantum mechanics, a version of quantum mechanics often called the de Broglie–Bohm theory or Bohmian mechanics. This theory was in fact discovered by Louis de Broglie in 1927. In 1952, it was rediscovered and developed by Bohm, who was the first to appreciate its connection to the predictions of standard quantum mechanics. I did not publish the talk. I have decided that it would be appropriate to publish a very lightly edited version of it here, in this volume devoted to the work of Bell on the foundation of quantum mechanics. One reason for doing so is that the connection between Bell and Bohm continues to be somewhat underplayed, with the strength of his advocacy of Bohmian mechanics not properly appreciated. For example, about half of the papers in Bell's collected works on the foundations of quantum mechanics deal with Bohmian mechanics. But in his fine introduction to the revised edition of this great book [1], Alain Aspect mentions this theory in only a single sentence, and parenthetically at that. For several decades after Bell proved his nonlocality theorem, based in part on Bell's inequality, it was widely claimed that Bell had shown that hidden variables – and Bohmian mechanics in particular – were impossible, that they were incompatible with the predictions of quantum mechanics. For example, the great physicist Eugene Wigner, who, unlike most of his contemporaries, was profoundly concerned with the conceptual foundations of quantum mechanics and usually wrote on the subject with great clarity and insight, has written that This [hidden variables] is an interesting idea and even though few of us were ready to accept it, it must be admitted that the truly telling argument against it was produced as late as 1965, by J. S. Bell … This appears to give a convincing argument against the hidden variables theory.