# Bell's theorem

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. The "local" in this case 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 can only occur at speeds no greater than the speed of light. "Hidden variables" are hypothetical properties possessed by quantum particles, properties that are undetectable but still affect the outcome of experiments. In the words of physicist John Stewart Bell, for whom this family of results is named, "If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."[1] The term is broadly applied to a number of different derivations, the first of which was introduced by Bell in a 1964 paper titled "On the Einstein Podolsky Rosen Paradox". Bell's paper was a response to a 1935 thought experiment that Albert Einstein, Boris Podolsky and Nathan Rosen used to argue that quantum physics is an "incomplete" theory.[2][3] By 1935, it was already recognized that the predictions of quantum physics are probabilistic. Einstein, Podolsky and Rosen presented a scenario that involves preparing a pair of particles such that the quantum state of the pair is entangled, and then separating the particles to an arbitrarily large distance. The experimenter has a choice of possible measurements that can be performed on one of the particles. When they choose a measurement and obtain a result, the quantum state of the other particle apparently collapses instantaneously into a new state depending upon that result, no matter how far away the other particle is. This suggests that either the measurement of the first particle somehow also interacted with the second particle at faster than the speed of light, or that the entangled particles had some unmeasured property which pre-determined their final quantum states before they were separated. Therefore, assuming locality, quantum mechanics must be incomplete, because it cannot give a complete description of the particle's true physical characteristics. In other words, quantum particles, like electrons and photons, must carry some property or attributes not included in quantum theory, and the uncertainties in quantum theory's predictions would then be due to ignorance or unknowability of these properties, later termed "hidden variables".

Bell carried the analysis of quantum entanglement much further. He deduced that if measurements are performed independently on the two separated particles of an entangled pair, then the assumption that the outcomes depend upon hidden variables within each half implies a mathematical constraint on how the outcomes on the two measurements are correlated. This constraint would later be named the Bell inequality. Bell then showed that quantum physics predicts correlations that violate this inequality. Consequently, the only way that hidden variables could explain the predictions of quantum physics is if they are "nonlocal", which is to say that somehow the two particles were able to interact instantaneously no matter how widely the two particles are separated.[4][5] Multiple variations on Bell's theorem were put forward in the following years, introducing other closely related conditions generally known as Bell (or "Bell-type") 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. To date, 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 any local hidden variable theory.[6][7] 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, its full implications for the interpretation of quantum mechanics remain unresolved.

Contents 1 Theorem 2 Variations and related results 2.1 Bell (1964) 2.2 GHZ–Mermin 2.3 Kochen–Specker theorem 2.4 Free will theorem 2.5 Quasiclassical entanglement 3 History 3.1 Background 3.2 Bell's publications 4 Experiments 4.1 Detection loophole 4.2 Locality loophole 4.3 Coincidence loophole 4.4 Memory loophole 5 Interpretations of Bell's theorem 5.1 The Copenhagen Interpretation 5.2 Many-worlds interpretation of quantum mechanics 5.3 Non-local hidden variables 5.4 Superdeterminism 6 See also 7 Notes 8 References 9 Further reading 10 External links Theorem There are many variations on the basic idea, some employing stronger mathematical assumptions than others.[8] 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.[9] Alice and Bob stand in widely separated locations. 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 (perhaps by flipping a coin to decide which). Denote these measurements by {displaystyle A_{0}} and {displaystyle A_{1}} . Both {displaystyle A_{0}} and {displaystyle A_{1}} are binary measurements: the result of {displaystyle A_{0}} is either {displaystyle +1} or {displaystyle -1} , and likewise for {displaystyle A_{1}} . When Bob receives his particle, he chooses one of two measurements, {displaystyle B_{0}} and {displaystyle B_{1}} , which are also both binary.

Suppose that each measurement reveals a property that the particle already possessed. For instance, if Alice chooses to measure {displaystyle A_{0}} and obtains the result {displaystyle +1} , then the particle she received carried a value of {displaystyle +1} for a property {displaystyle a_{0}} .[note 1] Consider the following combination: {displaystyle a_{0}b_{0}+a_{0}b_{1}+a_{1}b_{0}-a_{1}b_{1}=(a_{0}+a_{1})b_{0}+(a_{0}-a_{1})b_{1},.} Because both {displaystyle a_{0}} and {displaystyle a_{1}} take the values {displaystyle pm 1} , then either {displaystyle a_{0}=a_{1}} or {displaystyle a_{0}=-a_{1}} . In the former case, {displaystyle (a_{0}-a_{1})b_{1}=0} , while in the latter case, {displaystyle (a_{0}+a_{1})b_{0}=0} . So, one of the terms on the right-hand side of the above expression will vanish, and the other will equal {displaystyle pm 2} . Consequently, if the experiment is repeated over many trials, with Victor preparing new pairs of particles, the average value of the combination {displaystyle a_{0}b_{0}+a_{0}b_{1}+a_{1}b_{0}-a_{1}b_{1}} 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, {displaystyle langle A_{0}B_{0}rangle +langle A_{0}B_{1}rangle +langle A_{1}B_{0}rangle -langle A_{1}B_{1}rangle leq 2,.} This is a Bell inequality, specifically, the CHSH inequality.[9]: 115  Its derivation here depends upon two assumptions: first, that the underlying physical properties {displaystyle a_{0},a_{1},b_{0},} and {displaystyle b_{1}} exist independently of being observed or measured (sometimes called the assumption of realism); and second, that Alice's choice of action cannot influence Bob's result or vice versa (often called the assumption of locality).[9]: 117  Quantum mechanics can violate the CHSH inequality, as follows. Victor prepares a pair of qubits which he describes by the Bell state {displaystyle |psi rangle ={frac {|01rangle -|10rangle }{sqrt {2}}}.} Victor then passes the first qubit to Alice and the second to Bob. Alice and Bob's choices of possible measurements are defined by the Pauli matrices. Alice measures either of the two observables {displaystyle sigma _{z}} and {displaystyle sigma _{x}} : {displaystyle A_{0}=sigma _{z}={begin{pmatrix}1&0\0&-1end{pmatrix}}, A_{1}=sigma _{x}={begin{pmatrix}0&1\1&0end{pmatrix}};} and Bob measures either of the two observables {displaystyle B_{0}=-{frac {sigma _{x}+sigma _{z}}{sqrt {2}}}, B_{1}={frac {sigma _{x}-sigma _{z}}{sqrt {2}}}.} Victor can calculate the quantum expectation values for pairs of these observables using the Born rule: {displaystyle langle A_{0}otimes B_{0}rangle ={frac {1}{sqrt {2}}},langle A_{0}otimes B_{1}rangle ={frac {1}{sqrt {2}}},langle A_{1}otimes B_{0}rangle ={frac {1}{sqrt {2}}},langle A_{1}otimes B_{1}rangle =-{frac {1}{sqrt {2}}},.} While only one of these four measurements can be made in a single trial of the experiment, the sum {displaystyle langle A_{0}otimes B_{0}rangle +langle A_{0}otimes B_{1}rangle +langle A_{1}otimes B_{0}rangle -langle A_{1}otimes B_{1}rangle =2{sqrt {2}}} 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.[9]: 116  The value {displaystyle 2{sqrt {2}}} is in fact the largest that quantum physics permits for this combination of expectation values, making it a Tsirelson bound.[12]: 140  An illustration of the CHSH game: the referee, Victor, sends a bit each to Alice and to Bob, and Alice and Bob each send a bit back to the referee.

The CHSH inequality can also be thought of as a game in which Alice and Bob try to coordinate their actions.[13][14] Victor prepares two bits, {displaystyle x} and {displaystyle y} , independently and at random. He sends bit {displaystyle x} to Alice and bit {displaystyle y} to Bob. Alice and Bob win if they return answer bits {displaystyle a} and {displaystyle b} to Victor, satisfying {displaystyle xy=a+bmod 2,.} Or, equivalently, Alice and Bob win if the logical AND of {displaystyle x} and {displaystyle y} is the logical XOR of {displaystyle a} and {displaystyle b} . 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 {displaystyle 3/4} , 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 {displaystyle {frac {2+{sqrt {2}}}{4}}approx 0.85,.} Variations and related results Bell (1964) Bell's 1964 paper points out that under restricted conditions, local hidden variable models can reproduce the predictions of quantum mechanics. He then demonstrates that this cannot hold true in general.[3] Bell considers a refinement by David Bohm of the Einstein–Podolsky–Rosen (EPR) 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 (which is an example of an entangled 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 {displaystyle +1} and {displaystyle -1} . The configuration of each measuring instrument is represented by a vector, and the quantum-mechanical prediction for the correlation between two detectors with settings {displaystyle {vec {a}}} and {displaystyle {vec {b}}} is {displaystyle P({vec {a}},{vec {b}})=-{vec {a}}cdot {vec {b}},.} In particular, if the orientation of the two detectors is the same ( {displaystyle {vec {a}}={vec {b}}} ), then the outcome of one measurement is certain to be the negative of the outcome of the other, giving {displaystyle P({vec {a}},{vec {a}})=-1} . And if the orientations of the two detectors are orthogonal ( {displaystyle {vec {a}}cdot {vec {b}}=0} ), then the outcomes are uncorrelated, and {displaystyle P({vec {a}},{vec {b}})=0} . 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 {displaystyle lambda } : {displaystyle P({vec {a}},{vec {b}})=int dlambda ,rho (lambda )A({vec {a}},lambda )B({vec {b}},lambda ),,} where {displaystyle rho (lambda )} is a probability density function. The two functions {displaystyle A({vec {a}},lambda )} and {displaystyle B({vec {b}},lambda )} provide the responses of the two detectors given the orientation vectors and the hidden variable: {displaystyle A({vec {a}},lambda )=pm 1,,B({vec {b}},lambda )=pm 1,.} Crucially, the outcome of detector {displaystyle A} does not depend upon {displaystyle {vec {b}}} , and likewise the outcome of {displaystyle B} does not depend upon {displaystyle {vec {a}}} , 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 {displaystyle {vec {b}}} or to {displaystyle {vec {c}}} . Bell proves that the difference in correlation between these two choices of detector setting must satisfy the inequality {displaystyle |P({vec {a}},{vec {b}})-P({vec {a}},{vec {c}})|leq 1+P({vec {b}},{vec {c}}),.} However, it is easy to find situations where quantum mechanics violates the Bell inequality.[15]: 425–426  For example, let the vectors {displaystyle {vec {a}}} and {displaystyle {vec {b}}} be orthogonal, and let {displaystyle {vec {c}}} lie in their plane at a 45° angle from both of them. Then {displaystyle P({vec {a}},{vec {b}})=0,,} while {displaystyle P({vec {a}},{vec {c}})=P({vec {b}},{vec {c}})=-{frac {sqrt {2}}{2}},,} but {displaystyle {frac {sqrt {2}}{2}}nleq 1-{frac {sqrt {2}}{2}},.} Therefore, there is no local hidden variable model that can reproduce the predictions of quantum mechanics for all choices of {displaystyle {vec {a}}} , {displaystyle {vec {b}}} , and {displaystyle {vec {c}}} . Experimental results contradict the classical curves and match the curve predicted by quantum mechanics as long as experimental shortcomings are accounted for.[8] Bell's 1964 theorem requires the possibility of perfect anti-correlations: the ability to make a probability-1 prediction about the result from the second detector, knowing the result from the first. This is related to 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 (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity."[2] GHZ–Mermin Main article: GHZ experiment Greenberger, Horne, and Zeilinger presented a four-particle thought experiment, which Mermin then simplified to use only three particles.[16][17] In this thought experiment, Victor generates a set of three spin-1/2 particles described by the quantum state {displaystyle |psi rangle ={frac {1}{sqrt {2}}}(|000rangle -|111rangle ),,} where as above, {displaystyle |0rangle } and {displaystyle |1rangle } are the eigenvectors of the Pauli matrix {displaystyle sigma _{z}} . Victor then sends a particle each to Alice, Bob, and Charlie, who wait at widely-separated locations. Alice measures either {displaystyle sigma _{x}} or {displaystyle sigma _{y}} on her particle, and so do Bob and Charlie. The result of each measurement is either {displaystyle +1} or {displaystyle -1} . Applying the Born rule to the three-qubit state {displaystyle |psi rangle } , Victor predicts that whenever the three measurements include one {displaystyle sigma _{x}} and two {displaystyle sigma _{y}} 's, the product of the outcomes will always be {displaystyle +1} . This follows because {displaystyle |psi rangle } is an eigenvector of {displaystyle sigma _{x}otimes sigma _{y}otimes sigma _{y}} with eigenvalue {displaystyle +1} , and likewise for {displaystyle sigma _{y}otimes sigma _{x}otimes sigma _{y}} and {displaystyle sigma _{y}otimes sigma _{y}otimes sigma _{x}} . Therefore, knowing Alice's result for a {displaystyle sigma _{x}} measurement and Bob's result for a {displaystyle sigma _{y}} measurement, Victor can predict with probability 1 what result Charlie will return for a {displaystyle sigma _{y}} measurement. According to the EPR criterion of reality, there would be an "element of reality" corresponding to the outcome of a {displaystyle sigma _{y}} measurement upon Charlie's qubit. Indeed, this same logic applies to both measurements and all three qubits. Per the EPR criterion of reality, then, each particle contains an "instruction set" that determines the outcome of a {displaystyle sigma _{x}} or {displaystyle sigma _{y}} measurement upon it. The set of all three particles would then be described by the instruction set {displaystyle (a_{x},a_{y},b_{x},b_{y},c_{x},c_{y}),,} with each entry being either {displaystyle -1} or {displaystyle +1} , and each {displaystyle sigma _{x}} or {displaystyle sigma _{y}} measurement simply returning the appropriate value.

If Alice, Bob, and Charlie all perform the {displaystyle sigma _{x}} measurement, then the product of their results would be {displaystyle a_{x}b_{x}c_{x}} . This value can be deduced from {displaystyle (a_{x}b_{y}c_{y})(a_{y}b_{x}c_{y})(a_{y}b_{y}c_{x})=a_{x}b_{x}c_{x}a_{y}^{2}b_{y}^{2}c_{y}^{2}=a_{x}b_{x}c_{x},,} because the square of either {displaystyle -1} or {displaystyle +1} is {displaystyle 1} . Each factor in parentheses equals {displaystyle +1} , so {displaystyle a_{x}b_{x}c_{x}=+1,,} and the product of Alice, Bob, and Charlie's results will be {displaystyle +1} with probability unity. But this is inconsistent with quantum physics: Victor can predict using the state {displaystyle |psi rangle } that the measurement {displaystyle sigma _{x}otimes sigma _{x}otimes sigma _{x}} will instead yield {displaystyle -1} with probability unity.

This thought experiment can also be recast as a traditional Bell inequality or, equivalently, as a nonlocal game in the same spirit as the CHSH game.[18] In it, Alice, Bob, and Charlie receive bits {displaystyle x,y,z} from Victor, promised to always have an even number of ones, that is, {displaystyle xoplus yoplus z=0} , and send him back bits {displaystyle a,b,c} . They win the game if {displaystyle a,b,c} have an odd number of ones for all inputs except {displaystyle x=y=z=0} , when they need to have an even number of ones. That is, they win the game iff {displaystyle aoplus boplus c=xlor ylor z} . With local hidden variables the highest probability of victory they can have is 3/4, whereas using the quantum strategy above they win it with certainty. This is an example of quantum pseudo-telepathy.

Kochen–Specker theorem Main article: Kochen–Specker theorem In quantum theory, orthonormal bases for a Hilbert space represent measurements than can be performed upon a system having that Hilbert space. Each vector in a basis represents a possible outcome of that measurement.[note 2] Suppose that a hidden variable {displaystyle lambda } exists, so that knowing the value of {displaystyle lambda } would imply certainty about the outcome of any measurement. Given a value of {displaystyle lambda } , each measurement outcome — that is, each vector in the Hilbert space — is either impossible or guaranteed. A Kochen–Specker configuration is a finite set of vectors made of multiple interlocking bases, with the property that a vector in it will always be impossible when considered as belonging to one basis and guaranteed when taken as belonging to another. In other words, a Kochen–Specker configuration is an "uncolorable set" that demonstrates the inconsistency of assuming a hidden variable {displaystyle lambda } can be controlling the measurement outcomes.[23]: 196–201  Free will theorem Main article: Free will theorem The Kochen–Specker type of argument, using configurations of interlocking bases, can be combined with the idea of measuring entangled pairs that underlies Bell-type inequalities. This was noted beginning in the 1970s by Kochen,[24] Heywood and Redhead,[25] Stairs,[26] and Brown and Svetlichny.[27] As EPR pointed out, obtaining a measurement outcome on one half of an entangled pair implies certainty about the outcome of a corresponding measurement on the other half. The "EPR criterion of reality" posits that because the second half of the pair was not disturbed, that certainty must be due to a physical property belonging to it.[28] In other words, by this criterion, a hidden variable {displaystyle lambda } must exist within the second, as-yet unmeasured half of the pair. No contradiction arises if only one measurement on the first half is considered. However, if the observer has a choice of multiple possible measurements, and the vectors defining those measurements form a Kochen–Specker configuration, then some outcome on the second half will be simultaneously impossible and guaranteed.

The most prevalent loopholes in real experiments are the detection and locality loopholes.[60] The detection loophole is opened when a small fraction of the particles (usually photons) are detected in the experiment, making it possible to explain the data with local hidden variables by assuming that the detected particles are an unrepresentative sample. The locality loophole is opened when the detections are not done with a spacelike separation, making it possible for the result of one measurement to influence the other without contradicting relativity. In some experiments there may be additional defects that make local-hidden-variable explanations of Bell test violations possible.[61] Although both the locality and detection loopholes had been closed in different experiments, a long-standing challenge was to close both simultaneously in the same experiment. This was finally achieved in three experiments in 2015.[62][63][64][65][66] Regarding these results, Alain Aspect writes that "... no experiment ... can be said to be totally loophole-free," but he says the experiments "remove the last doubts that we should renounce" local hidden variables, and refers to examples of remaining loopholes as being "far fetched" and "foreign to the usual way of reasoning in physics."[67] Detection loophole A common problem in optical Bell tests is that only a small fraction of the emitted photons are detected. It is then possible that the correlations of the detected photons are unrepresentative: although they show a violation of a Bell inequality, if all photons were detected the Bell inequality would actually be respected. This was first noted by Pearle in 1970,[68] who devised a local hidden variable model that faked a Bell violation by letting the photon be detected only if the measurement setting was favourable. The assumption that this does not happen, i.e., that the small sample is actually representative of the whole is called the fair sampling assumption.