Bell's theorem

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 unit 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 ${\displaystyle \lambda }$:

$${\displaystyle P({\vec {a}},{\vec {b}})=\int d\lambda \,\rho (\lambda )A({\vec {a}},\lambda )B({\vec {b}},\lambda )\,,}$$

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]

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.[16][17] In this thought experiment, Victor generates a set of three spin-1/2 particles described by the quantum state

$${\displaystyle |\psi \rangle ={\frac {1}{\sqrt {2}}}(|000\rangle -|111\rangle )\,,}$$

where as above, ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$ 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, ${\displaystyle x\oplus y\oplus z=0}$, 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 ${\displaystyle a\oplus b\oplus c=x\lor y\lor z}$. 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.

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}

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.

This type of argument gained attention when an instance of it was advanced by John Conway and Simon Kochen under the name of the free will theorem.[29][30][31] The Conway–Kochen theorem uses a pair of entangled qutrits and a Kochen–Specker configuration discovered by Asher Peres.[32]

As Bell pointed out, some predictions of quantum mechanics can be replicated in local hidden variable models, including special cases of correlations produced from entanglement. This topic has been studied systematically in the years since Bell's theorem. In 1989, Reinhard Werner introduced what are now called Werner states, joint quantum states for a pair of systems that yield EPR-type correlations but also admit a hidden-variable model.[33] Werner states are bipartite quantum states that are invariant under unitaries of symmetric tensor-product form:

$${\displaystyle \rho _{AB}=(U\otimes U)\rho _{AB}(U^{\dagger }\otimes U^{\dagger })\,.}$$

In 2004, Robert Spekkens introduced a toy model that starts with the premise of local, discretized degrees of freedom and then imposes a "knowledge balance principle" that restricts how much an observer can know about those degrees of freedom, thereby making them into hidden variables. The allowed states of knowledge ("epistemic states") about the underlying variables ("ontic states") mimic some features of quantum states. Correlations in the toy model can emulate some aspects of entanglement, like monogamy, but by construction, the toy model can never violate a Bell inequality.[34][35]

The question of whether quantum mechanics can be "completed" by hidden variables dates to the early years of quantum theory. In his 1932 textbook on quantum mechanics, the Hungarian-born polymath John von Neumann presented what he claimed to be a proof that there could be no "hidden parameters". The validity and definitiveness of von Neumann's proof were questioned by Hans Reichenbach, in more detail by Grete Hermann, and possibly in conversation though not in print by Albert Einstein.[note 3] (Simon Kochen and Ernst Specker rejected von Neumann's key assumption as early as 1961, but did not publish a criticism of it until 1967.[41])

Einstein argued persistently that quantum mechanics could not be a complete theory. His preferred argument relied on a principle of locality:

Consider a mechanical system constituted of two partial systems A and B which have interaction with each other only during limited time. Let the ψ function before their interaction be given. Then the Schrödinger equation will furnish the ψ function after their interaction has taken place. Let us now determine the physical condition of the partial system A as completely as possible by measurements. Then the quantum mechanics allows us to determine the ψ function of the partial system B from the measurements made, and from the ψ function of the total system. This determination, however, gives a result which depends upon which of the determining magnitudes specifying the condition of A has been measured (for instance coordinates or momenta). Since there can be only one physical condition of B after the interaction and which can reasonably not be considered as dependent on the particular measurement we perform on the system A separated from B it may be concluded that the ψ function is not unambiguously coordinated with the physical condition. This coordination of several ψ functions with the same physical condition of system B shows again that the ψ function cannot be interpreted as a (complete) description of a physical condition of a unit system.[42]

The EPR thought experiment is similar, also considering two separated systems A and B described by a joint wave function. However, the EPR paper adds the idea later known as the EPR criterion of reality, according to which the ability to predict with probability 1 the outcome of a measurement upon B implies the existence of an "element of reality" within B.[43]

In 1951, David Bohm proposed a variant of the EPR thought experiment in which the measurements have discrete ranges of possible outcomes, unlike the position and momentum measurements considered by EPR.[44] The year before, Chien-Shiung Wu and Irving Shaknov had successfully measured polarizations of photons produced in entangled pairs, thereby making the Bohm version of the EPR thought experiment practically feasible.[45]

By the late 1940s, the mathematician George Mackey had grown interested in the foundations of quantum physics, and in 1957 he drew up a list of postulates that he took to be a precise definition of quantum mechanics.[46] Mackey conjectured that one of the postulates was redundant, and shortly thereafter, Andrew M. Gleason proved that it was indeed deducible from the other postulates.[47][48] Gleason's theorem provided an argument that a broad class of hidden-variable theories are incompatible with quantum mechanics.[note 4] More specifically, Gleason's theorem rules out hidden-variable models that are "noncontextual". Any hidden-variable model for quantum mechanics must, in order to avoid the implications of Gleason's theorem, involve hidden variables that are not properties belonging to the measured system alone but also dependent upon the external context in which the measurement is made. This type of dependence is often seen as contrived or undesirable; in some settings, it is inconsistent with special relativity.[5][50] The Kochen–Specker theorem refines this statement by constructing a specific finite subset of rays on which no such probability measure can be defined.[5][51]

Tsung-Dao Lee came close to deriving Bell's theorem in 1960. He considered events where two kaons were produced traveling in opposite directions, and came to the conclusion that hidden variables could not explain the correlations that could be obtained in such situations. However, complications arose due to the fact that kaons decay, and he did not go so far as to deduce a Bell-type inequality.[note 5]

Bell chose to publish his theorem in a comparatively obscure journal because it did not require page charges, and at the time it in fact paid the authors who published there. Because the journal did not provide free reprints of articles for the authors to distribute, however, Bell had to spend the money he received to buy copies that he could send to other physicists.[52] While the articles printed in the journal themselves listed the publication's name simply as Physics, the covers carried the trilingual version Physics Physique Физика to reflect that it would print articles in English, French and Russian.[40]^{: 92–100, 289}

Prior to proving his 1964 result, Bell also proved a result equivalent to the Kochen–Specker theorem (hence why the latter is sometimes also known as the Bell–Kochen–Specker or Bell–KS theorem). However, publication of this theorem was inadvertently delayed until 1966.[5][53] In that paper, Bell argued that because an explanation of quantum phenomena in terms of hidden variables would require nonlocality, the EPR paradox "is resolved in the way which Einstein would have liked least".[53]

The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as features of it date to the development of quantum mechanics during 1925–1927, and it remains one of the most commonly taught.[70] There is no definitive historical statement of what is the Copenhagen interpretation. In particular, there were fundamental disagreements between the views of Bohr and Heisenberg.[71][72][73] Some basic principles generally accepted as part of the Copenhagen collection include the idea that quantum mechanics is intrinsically indeterministic, with probabilities calculated using the Born rule,[74] and the complementarity principle: certain properties cannot be jointly defined for the same system at the same time. In order to talk about a specific property of a system, that system must be considered within the context of a specific laboratory arrangement. Observable quantities corresponding to mutually exclusive laboratory arrangements cannot be predicted together, but considering multiple such mutually exclusive experiments is necessary to characterize a system.[71] Bohr himself used complementarity to argue that the EPR "paradox" was fallacious. Because measurements of position and of momentum are complementary, making the choice to measure one excludes the possibility of measuring the other. Consequently, he argued, a fact deduced regarding one arrangement of laboratory apparatus could not be combined with a fact deduced by means of the other, and so, the inference of predetermined position and momentum values for the second particle was not valid.[37]^{: 194–197} Bohr concluded that EPR's "arguments do not justify their conclusion that the quantum description turns out to be essentially incomplete."[75]

Copenhagen-type interpretations generally take the violation of Bell inequalities as grounds to reject the assumption often called counterfactual definiteness or "realism", which is not necessarily the same as abandoning realism in a broader philosophical sense.[76][77] For example, Roland Omnès argues for the rejection of hidden variables and concludes that "quantum mechanics is probably as realistic as any theory of its scope and maturity ever will be."[78]^: 531 This is also the route taken by interpretations that descend from the Copenhagen tradition, such as consistent histories (often advertised as "Copenhagen done right"),[79] as well as QBism.[80]

The Many-worlds interpretation, also known as the Everett interpretation, is local and deterministic, as it consists of the unitary part of quantum mechanics without collapse. It can generate correlations that violate a Bell inequality because it violates an implicit assumption by Bell that measurements have a single outcome. In fact, Bell's theorem can be proven in the Many-Worlds framework from the assumption that a measurement has a single outcome. Therefore a violation of a Bell inequality can be interpreted as a demonstration that measurements have multiple outcomes.[81]

The explanation it provides for the Bell correlations is that when Alice and Bob make their measurements, they split into local branches. From the point of view of each copy of Alice, there are multiple copies of Bob experiencing different results, so Bob cannot have a definite result, and the same is true from the point of view of each copy of Bob. They will obtain a mutually well-defined result only when their future light cones overlap. At this point we can say that the Bell correlation starts existing, but it was produced by a purely local mechanism. Therefore the violation of a Bell inequality cannot be interpreted as a proof of non-locality.[82]

Most advocates of the hidden-variables idea believe that experiments have ruled out local hidden variables.[note 6] They are ready to give up locality, explaining the violation of Bell's inequality by means of a non-local hidden variable theory, in which the particles exchange information about their states. This is the basis of the Bohm interpretation of quantum mechanics, which requires that all particles in the universe be able to instantaneously exchange information with all others. One challenge for non-local hidden variable theories is to explain why this instantaneous communication can exist at the level of the hidden variables, but it cannot be used to send signals.[85] A 2007 experiment ruled out a large class of non-Bohmian non-local hidden variable theories, though not Bohmian mechanics itself.[86]

The transactional interpretation, which postulates waves traveling both backwards and forwards in time, is likewise non-local.[87]

A necessary assumption to derive Bell's theorem is that the hidden variables are not correlated with the measurement settings. This assumption has been justified on the grounds that the experimenter has "free will" to choose the settings, and that it is necessary to do science in the first place. A (hypothetical) theory where the choice of measurement is necessarily correlated with the system being measured is known as superdeterministic.[60]

A few advocates of deterministic models have not given up on local hidden variables. For example, Gerard 't Hooft has argued that superdeterminism cannot be dismissed.[88]