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 [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." 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 (or "Bell-type") inequalities. The first laboratory experiment to observe a Bell inequality violation was conducted by John Clauser and Stuart Freedman, who published their results in 1970. 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 (perhaps by flipping a coin to decide which). Denote these measurements by A 0 {\displaystyle A_{0}} and A 1 {\displaystyle A_{1}} . Both A 0 {\displaystyle A_{0}} and A 1 {\displaystyle A_{1}} are binary measurements: the result of A 0 {\displaystyle A_{0}} is either + 1 {\displaystyle +1} or − 1 {\displaystyle -1} , and likewise for A 1 {\displaystyle A_{1}} . When Bob receives his particle, he chooses one of two measurements, B 0 {\displaystyle B_{0}} and B 1 {\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 A 0 {\displaystyle A_{0}} and obtains the result + 1 {\displaystyle +1} , then the particle she received carried a value of + 1 {\displaystyle +1} for a property a 0 {\displaystyle a_{0}} . Consider the combination 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 . {\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 a 0 {\displaystyle a_{0}} and a 1 {\displaystyle a_{1}} take the values ± 1 {\displaystyle \pm 1} , then either a 0 = a 1 {\displaystyle a_{0}=a_{1}} or a 0 = − a 1 {\displaystyle a_{0}=-a_{1}} . In the former case, the quantity ( a 0 − a 1 ) b 1 {\displaystyle (a_{0}-a_{1})b_{1}} must equal 0, while in the latter case, ( a 0 + a 1 ) b 0 = 0 {\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 ± 2 {\displaystyle \pm 2} . 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 a 0 b 0 + a 0 b 1 + a 1 b 0 − a 1 b 1 {\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 | ⟨ A 0 B 0 ⟩ + ⟨ A 0 B 1 ⟩ + ⟨ A 1 B 0 ⟩ − ⟨ A 1 B 1 ⟩ | ≤ 2 . {\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. Its derivation here depends upon two assumptions: first, that the underlying physical properties a 0 , a 1 , b 0 , {\displaystyle a_{0},a_{1},b_{0},} and b 1 {\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). Quantum mechanics can violate the CHSH inequality, as follows. Victor prepares a pair of qubits which he describes by the Bell state | ψ ⟩ = | 0 ⟩ ⊗ | 1 ⟩ − | 1 ⟩ ⊗ | 0 ⟩ 2 , {\displaystyle |\psi \rangle ={\frac {|0\rangle \otimes |1\rangle -|1\rangle \otimes |0\rangle }{\sqrt {2}}},}
where | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } are the eigenstates of one of the Pauli matrices, σ z = ( 1 0 0 − 1 ) . {\displaystyle \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}
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 σ z {\displaystyle \sigma _{z}} and σ x {\displaystyle \sigma _{x}} : A 0 = σ z , A 1 = σ x = ( 0 1 1 0 ) ; {\displaystyle A_{0}=\sigma _{z},\ A_{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}};}
and Bob measures either of the two observables B 0 = − σ x + σ z 2 , B 1 = σ x − σ z 2 . {\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: ⟨ A 0 ⊗ B 0 ⟩ = 1 2 , ⟨ A 0 ⊗ B 1 ⟩ = 1 2 , ⟨ A 1 ⊗ B 0 ⟩ = 1 2 , ⟨ A 1 ⊗ B 1 ⟩ = − 1 2 . {\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 ⟨ A 0 ⊗ B 0 ⟩ + ⟨ A 0 ⊗ B 1 ⟩ + ⟨ A 1 ⊗ B 0 ⟩ − ⟨ A 1 ⊗ B 1 ⟩ = 2 2 {\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. The value 2 2 {\displaystyle 2{\sqrt {2}}} 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, x {\displaystyle x} and y {\displaystyle y} , independently and at random. He sends bit x {\displaystyle x} to Alice and bit y {\displaystyle y} to Bob. Alice and Bob win if they return answer bits a {\displaystyle a} and b {\displaystyle b} to Victor, satisfying
x y = a + b mod 2 . {\displaystyle xy=a+b\mod 2\,.}
Or, equivalently, Alice and Bob win if the logical AND of x {\displaystyle x} and y {\displaystyle y} is the logical XOR of a {\displaystyle a} and b {\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 3 / 4 {\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 2 + 2 4 ≈ 0.85 . {\displaystyle {\frac {2+{\sqrt {2}}}{4}}\approx 0.85\,.}
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 (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 + 1 {\displaystyle +1} and − 1 {\displaystyle -1} . 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 a → {\displaystyle {\vec {a}}} and b → {\displaystyle {\vec {b}}} is
P ( a → , b → ) = − a → ⋅ b → . {\displaystyle P({\vec {a}},{\vec {b}})=-{\vec {a}}\cdot {\vec {b}}.}
In particular, if the orientation of the two detectors is the same ( a → = b → {\displaystyle {\vec {a}}={\vec {b}}} ), then the outcome of one measurement is certain to be the negative of the outcome of the other, giving P ( a → , a → ) = − 1 {\displaystyle P({\vec {a}},{\vec {a}})=-1} . And if the orientations of the two detectors are orthogonal ( a → ⋅ b → = 0 {\displaystyle {\vec {a}}\cdot {\vec {b}}=0} ), then the outcomes are uncorrelated, and P ( a → , b → ) = 0 {\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 λ {\displaystyle \lambda } : P ( a → , b → ) = ∫ d λ ρ ( λ ) A ( a → , λ ) B ( b → , λ ) , {\displaystyle P({\vec {a}},{\vec {b}})=\int d\lambda \,\rho (\lambda )A({\vec {a}},\lambda )B({\vec {b}},\lambda ),}
where ρ ( λ ) {\displaystyle \rho (\lambda )} is a probability density function. The two functions A ( a → , λ ) {\displaystyle A({\vec {a}},\lambda )} and B ( b → , λ ) {\displaystyle B({\vec {b}},\lambda )} provide the responses of the two detectors given the orientation vectors and the hidden variable: A ( a → , λ ) = ± 1 , B ( b → , λ ) = ± 1. {\displaystyle A({\vec {a}},\lambda )=\pm 1,\,B({\vec {b}},\lambda )=\pm 1.}
Crucially, the outcome of detector A {\displaystyle A} does not depend upon b → {\displaystyle {\vec {b}}} , and likewise the outcome of B {\displaystyle B} does not depend upon a → {\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 b → {\displaystyle {\vec {b}}} or to c → {\displaystyle {\vec {c}}} . From the assumption that A ( a → , λ ) = − B ( a → , λ ) {\displaystyle A({\vec {a}},\lambda )=-B({\vec {a}},\lambda )} , that is, perfect anti-correlations are observed for the same setting, Bell proves that | P ( a → , b → ) − P ( a → , c → ) | ≤ 1 + P ( b → , c → ) . {\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 while exhibiting perfect anti-correlations. For example, let the vectors a → {\displaystyle {\vec {a}}} and b → {\displaystyle {\vec {b}}} be orthogonal, and let c → {\displaystyle {\vec {c}}} lie in their plane at a 45° angle from both of them. Then P ( a → , b → ) = 0 , {\displaystyle P({\vec {a}},{\vec {b}})=0,}
while
P ( a → , c → ) = P ( b → , c → ) = − 2 2 , {\displaystyle P({\vec {a}},{\vec {c}})=P({\vec {b}},{\vec {c}})=-{\frac {\sqrt {2}}{2}},}
but
2 2 ≰ 1 − 2 2 . {\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 a → {\displaystyle {\vec {a}}} , b → {\displaystyle {\vec {b}}} , and c → . {\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. 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 (i.e., with probability equal to unity) 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 ( a → = b → {\displaystyle {\vec {a}}={\vec {b}}} ), 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.
GHZ–Mermin (1990)
Daniel Greenberger, Michael A. Horne, and Anton Zeilinger presented a four-particle thought experiment in 1990, which David Mermin then simplified to use only three particles. In this thought experiment, Victor generates a set of three spin-1/2 particles described by the quantum state | ψ ⟩ = 1 2 ( | 000 ⟩ − | 111 ⟩ ) , {\displaystyle |\psi \rangle ={\frac {1}{\sqrt {2}}}(|000\rangle -|111\rangle )\,,}
where as above, | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } are the eigenvectors of the Pauli matrix σ z {\displaystyle \sigma _{z}} . Victor then sends a particle each to Alice, Bob, and Charlie, who wait at widely separated locations. Alice measures either σ x {\displaystyle \sigma _{x}} or σ y {\displaystyle \sigma _{y}} on her particle, and so do Bob and Charlie. The result of each measurement is either + 1 {\displaystyle +1} or − 1 {\displaystyle -1} . Applying the Born rule to the three-qubit state | ψ ⟩ {\displaystyle |\psi \rangle } , Victor predicts that whenever the three measurements include one σ x {\displaystyle \sigma _{x}} and two σ y {\displaystyle \sigma _{y}} 's, the product of the outcomes will always be + 1 {\displaystyle +1} . This follows because | ψ ⟩ {\displaystyle |\psi \rangle } is an eigenvector of σ x ⊗ σ y ⊗ σ y {\displaystyle \sigma _{x}\otimes \sigma _{y}\otimes \sigma _{y}} with eigenvalue + 1 {\displaystyle +1} , and likewise for σ y ⊗ σ x ⊗ σ y {\displaystyle \sigma _{y}\otimes \sigma _{x}\otimes \sigma _{y}} and σ y ⊗ σ y ⊗ σ x {\displaystyle \sigma _{y}\otimes \sigma _{y}\otimes \sigma _{x}} . Therefore, knowing Alice's result for a σ x {\displaystyle \sigma _{x}} measurement and Bob's result for a σ y {\displaystyle \sigma _{y}} measurement, Victor can predict with probability 1 what result Charlie will return for a σ y {\displaystyle \sigma _{y}} measurement. According to the EPR criterion of reality, there would be an "element of reality" corresponding to the outcome of a σ y {\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 σ x {\displaystyle \sigma _{x}} or σ y {\displaystyle \sigma _{y}} measurement upon it. The set of all three particles would then be described by the instruction set ( a x , a y , b x , b y , c x , c y ) , {\displaystyle (a_{x},a_{y},b_{x},b_{y},c_{x},c_{y})\,,}
with each entry being either − 1 {\displaystyle -1} or + 1 {\displaystyle +1} , and each σ x {\displaystyle \sigma _{x}} or σ y {\displaystyle \sigma _{y}} measurement simply returning the appropriate value. If Alice, Bob, and Charlie all perform the σ x {\displaystyle \sigma _{x}} measurement, then the product of their results would be a x b x c x {\displaystyle a_{x}b_{x}c_{x}} . This value can be deduced from ( a