Bell's theorem
Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measurement. The first such result was introduced by John Stewart Bell in 1964, building upon the Einstein–Podolsky–Rosen paradox, which had called attention to the phenomenon of quantum entanglement.
In the context of Bell's theorem, "local" refers to the principle of locality, the idea that a particle can only be influenced by its immediate surroundings, and that interactions mediated by physical fields cannot propagate faster than the speed of light. "Hidden variables" are supposed properties of quantum particles that are not included in quantum theory but nevertheless affect the outcome of experiments. In the words of Bell, "If is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."
In his original paper, Bell analyzed independent measurements on two spatially separated particles of an entangled pair. He assumed that each outcome is determined by local hidden variables. Under this assumption, the correlations between the outcomes must obey a specific mathematical constraint. Such a constraint would later be named a Bell inequality. Bell then showed that quantum physics predicts correlations that violate this inequality. Multiple variations on Bell's theorem were put forward in the years following his original paper, using different assumptions and obtaining different Bell inequalities.
The first rudimentary experiment designed to test Bell's theorem was performed in 1972 by John Clauser and Stuart Freedman. More advanced experiments, known collectively as Bell tests, have been performed many times since. Often, these experiments have had the goal of "closing loopholes", that is, ameliorating problems of experimental design or set-up that could in principle affect the validity of the findings of earlier Bell tests. Bell tests have consistently found that physical systems obey quantum mechanics and violate Bell inequalities; which is to say that the results of these experiments are incompatible with local hidden-variable theories.
The exact nature of the assumptions required to prove a Bell-type constraint on correlations has been debated by physicists and by philosophers. While the significance of Bell's theorem is not in doubt, different interpretations of quantum mechanics disagree about what exactly it implies.
Theorem
There are many variations on the basic idea, some employing stronger mathematical assumptions than others. Significantly, Bell-type theorems do not refer to any particular theory of local hidden variables, but instead show that quantum physics violates general assumptions behind classical pictures of nature. The original theorem proved by Bell in 1964 is not the most amenable to experiment, and it is convenient to introduce the genre of Bell-type inequalities with a later example.Hypothetical characters Alice and Bob stand in widely separated locations. Their colleague Victor prepares a pair of particles and sends one to Alice and the other to Bob. When Alice receives her particle, she chooses to perform one of two possible measurements. Denote these measurements by and. Both and are binary measurements: the result of is either or, and likewise for. When Bob receives his particle, he chooses one of two measurements, and, which are also both binary.
Suppose that each measurement reveals a property that the particle already possessed. For instance, if Alice chooses to measure and obtains the result, then the particle she received carried a value of for a property. Consider the combinationBecause both and take the values, then either or. In the former case, the quantity must equal 0, while in the latter case,. So, one of the terms on the right-hand side of the above expression will vanish, and the other will equal. Consequently, if the experiment is repeated over many trials, with Victor preparing new pairs of particles, the absolute value of the average of the combination across all the trials will be less than or equal to 2. No single trial can measure this quantity, because Alice and Bob can only choose one measurement each, but on the assumption that the underlying properties exist, the average value of the sum is just the sum of the averages for each term. Using angle brackets to denote averages
This is a Bell inequality, specifically, the CHSH inequality. Its derivation here depends upon two assumptions: first, that the underlying physical properties and exist independently of being observed or measured ; and second, that Alice's choice of action cannot influence Bob's result or vice versa.
Quantum mechanics can violate the CHSH inequality, as follows. Victor prepares a pair of qubits which he describes by the Bell state
where and are the eigenstates of one of the Pauli matrices,
Victor then passes the first qubit to Alice and the second to Bob. Alice and Bob's choices of possible measurements are also defined in terms of the Pauli matrices. Alice measures either of the two observables and :
and Bob measures either of the two observables
Victor can calculate the quantum expectation values for pairs of these observables using the Born rule:
While only one of these four measurements can be made in a single trial of the experiment, the sum
gives the sum of the average values that Victor expects to find across multiple trials. This value exceeds the classical upper bound of 2 that was deduced from the hypothesis of local hidden variables. The value is in fact the largest that quantum physics permits for this combination of expectation values, making it a Tsirelson bound.
The CHSH inequality can also be thought of as a game in which Alice and Bob try to coordinate their actions. Victor prepares two bits, and, independently and at random. He sends bit to Alice and bit to Bob. Alice and Bob win if they return answer bits and to Victor, satisfying
Or, equivalently, Alice and Bob win if the logical AND of and is the logical XOR of and. Alice and Bob can agree upon any strategy they desire before the game, but they cannot communicate once the game begins. In any theory based on local hidden variables, Alice and Bob's probability of winning is no greater than, regardless of what strategy they agree upon beforehand. However, if they share an entangled quantum state, their probability of winning can be as large as
Variations and related results
Bell (1964)
Bell's 1964 paper shows that a very simple local hidden-variable model can in restricted circumstances reproduce the predictions of quantum mechanics, but then he demonstrates that, in general, such models give different predictions. Bell considers a refinement by David Bohm of the Einstein–Podolsky–Rosen thought experiment. In this scenario, a pair of particles are formed together in such a way that they are described by a spin singlet state. The particles then move apart in opposite directions. Each particle is measured by a Stern–Gerlach device, a measuring instrument that can be oriented in different directions and that reports one of two possible outcomes, representable by and. The configuration of each measuring instrument is represented by a unit vector, and the quantum-mechanical prediction for the correlation between two detectors with settings and isIn particular, if the orientation of the two detectors is the same, then the outcome of one measurement is certain to be the negative of the outcome of the other, giving. And if the orientations of the two detectors are orthogonal, then the outcomes are uncorrelated, and. Bell proves by example that these special cases can be explained in terms of hidden variables, then proceeds to show that the full range of possibilities involving intermediate angles cannot.
Bell posited that a local hidden-variable model for these correlations would explain them in terms of an integral over the possible values of some hidden parameter :
where is a probability density function. The two functions and provide the responses of the two detectors given the orientation vectors and the hidden variable:
Crucially, the outcome of detector does not depend upon, and likewise the outcome of does not depend upon, because the two detectors are physically separated. Now we suppose that the experimenter has a choice of settings for the second detector: it can be set either to or to. From the assumption that , that is, perfect anti-correlations are observed for the same setting, Bell proves that
However, it is easy to find situations where quantum mechanics violates the Bell inequality while exhibiting perfect anti-correlations. For example, let the vectors and be orthogonal, and let lie in their plane at a 45° angle from both of them. Then
while
but
Therefore, there is no local hidden-variable model that can reproduce the predictions of quantum mechanics for all choices of,, and Experimental results contradict the classical curves and match the curve predicted by quantum mechanics as long as experimental shortcomings are accounted for.
Bell's 1964 theorem requires the possibility of perfect anti-correlations: the ability to make a completely certain prediction about the result from the second detector, knowing the result from the first. The theorem builds upon the "EPR criterion of reality", a concept introduced in the 1935 paper by Einstein, Podolsky, and Rosen. This paper posits: "If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of reality corresponding to that quantity." Bell noted that this applies when the two detectors are oriented in the same direction, and so the EPR criterion would imply that some element of reality must predetermine the measurement result. Because the quantum description of a particle does not include any such element, the quantum description would have to be incomplete. In other words, Bell's 1964 paper shows that, assuming locality, the EPR criterion implies hidden variables and then he demonstrates that local hidden variables are incompatible with quantum mechanics. Because experiments cannot achieve perfect correlations or anti-correlations in practice, Bell-type inequalities based on derivations that relax this assumption are tested instead.