Bell's theorem

# Bell's theorem

Discussion

Encyclopedia
In theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

, Bell's theorem (a.k.a. Bell's inequality) is a no-go theorem
No-go theorem
In theoretical physics, a no-go theorem is a theorem that states that a particular situation is not physically possible.-Examples of no-go theorems:* Bell's theorem* Coleman–Mandula theorem* Haag-Lopuszanski-Sohnius theorem* Earnshaw's theorem...

, loosely stating that:
The theorem has great importance for physics and the philosophy of science
Philosophy of science
The philosophy of science is concerned with the assumptions, foundations, methods and implications of science. It is also concerned with the use and merit of science and sometimes overlaps metaphysics and epistemology by exploring whether scientific results are actually a study of truth...

, as it implies that quantum physics must necessarily violate either the principle of locality
Principle of locality
In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. Experiments have shown that quantum mechanically entangled particles must violate either the principle of locality or the form of philosophical realism known as counterfactual...

or counterfactual definiteness
Counterfactual definiteness
In some interpretations of quantum mechanics, counterfactual definiteness is the ability to speak with meaning of the definiteness of the results of measurements that have not been performed...

. It is the most famous legacy of the physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

John Stewart Bell
John Stewart Bell
John Stewart Bell FRS was a British physicist from Northern Ireland , and the originator of Bell's theorem, a significant theorem in quantum physics regarding hidden variable theories.- Early life and work :...

.

Results of tests of Bell's theorem
Bell test experiments
The Bell test experiments serve to investigate the validity of the entanglement effect in quantum mechanics by using some kind of Bell inequality...

agree with the predictions of quantum mechanical theory, and demonstrate that some quantum effects appear to travel faster than light. Hence the class of tenable hidden variable theories are limited to the non-local
Nonlocality
In Classical physics, nonlocality is the direct influence of one object on another, distant object. In Quantum mechanics, nonlocality refers to the absence of a local, realist model in agreement with quantum mechanical predictions.Nonlocality may refer to:...

variety. However, none of the tests of the theorem performed to date has fulfilled all of the requisite conditions implicit in the theorem. Accordingly, none of the results are totally conclusive.

## Overview

Bell’s theorem implies that the concept of local realism
Local hidden variable theory
In quantum mechanics, a local hidden variable theory is one in which distant events are assumed to have no instantaneous effect on local ones....

, favoured by Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

, yields predictions that disagree with those of quantum mechanical theory. Because numerous experiments agree with the predictions of quantum mechanical theory, and show correlations that are stronger than could be explained by local hidden variables, the concept of local realism is thus refuted as an explanation of the physical phenomena under test, and superluminal effects
Nonlocality
In Classical physics, nonlocality is the direct influence of one object on another, distant object. In Quantum mechanics, nonlocality refers to the absence of a local, realist model in agreement with quantum mechanical predictions.Nonlocality may refer to:...

are evidenced.
The theorem applies to any quantum system of two entangled
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

qubits. The most common examples concern systems of particles that are entangled in spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

or polarization.

Following the argument in the Einstein–Podolsky–Rosen (EPR) paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...

paper (but using the example of spin, as in David Bohm
David Bohm
David Joseph Bohm FRS was an American-born British quantum physicist who contributed to theoretical physics, philosophy, neuropsychology, and the Manhattan Project.-Youth and college:...

's version of the EPR argument), Bell considered an experiment in which there are "a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions." Each is sent to two distant locations at which measurements of spin are performed, along axes that are independently chosen. Each measurement yields a result of either spin-up (+) or spin-down (−).

The probability of the same result being obtained at the two locations varies, depending on the relative angles at which the two spin measurements are made, and is subject to some uncertainty for all relative angles other than perfectly parallel alignments (0° or 180°). Bell's theorem thus applies only to the statistical results from many trials of the experiment. Symbolically, the correlation between results for a single pair can be represented as either "+1" for a match, or "−1" for a non-match. While measuring the spin of these entangled particles along parallel axes will always result in identical (i.e., perfectly correlated) results, measurement at perpendicular directions will have only a 50% chance of matching (i.e., will have a 50% probability of an uncorrelated result). These basic cases are illustrated in the table below.
 Same axis Correlation: ( Orthogonal axes Correlation ( Pair 1 Pair 2 Pair 3 Pair 4 … Pair n AliceAlice and BobThe names Alice and Bob are commonly used placeholder names for archetypal characters in fields such as cryptography and physics. The names are used for convenience; for example, "Alice sends a message to Bob encrypted with his public key" is easier to follow than "Party A sends a message to Party..., 0° + − − + … + BobAlice and BobThe names Alice and Bob are commonly used placeholder names for archetypal characters in fields such as cryptography and physics. The names are used for convenience; for example, "Alice sends a message to Bob encrypted with his public key" is easier to follow than "Party A sends a message to Party..., 0° + − − + … + +1 +1 +1 +1 … +1 ) / n = +1 (100% identical) Pair 1 Pair 2 Pair 3 Pair 4 … Pair n Alice, 0° + − + − … − Bob, 90° − − + + … − −1 +1 +1 −1 … −1 ) / n = 0 (50% identical)

With the measurements oriented at intermediate angles between these basic cases, the existence of local hidden variables would imply a linear variation in the correlation
Quantum correlation
In Bell test experiments the term quantum correlation has come to mean the expectation value of the product of the outcomes on the two sides. In other words, the expected change in physical characteristics as one quantum system passes through an interaction site...

. However, according to quantum mechanical theory, the correlation varies as the cosine of the angle. Experimental results match the curve predicted by quantum mechanics.

Bell achieved his breakthrough by first deriving the results that local realism would necessarily yield. Without making any assumptions about the specific form of the theory beyond requirements of basic consistency, the mathematical inequality he discovered was clearly at odds with the results (described above) predicted by quantum mechanics and, later, observed experimentally. Thus, Bell's theorem rules out local hidden variables as a viable explanation of quantum mechanics (though it still leaves the door open for non-local hidden variables). Bell concluded:
Over the years, Bell's theorem has undergone a wide variety of experimental tests. Various common deficiencies in the testing of the theorem have been identified, including the detection loophole and the communication loophole. Over the years experiments have been gradually improved to better address these loopholes, but no experiment to date has simultaneously fully addressed all of them. To date, Bell's theorem is supported by a substantial body of evidence and is treated as a fundamental principle of physics in mainstream quantum mechanics textbooks. However, no principle of physics can ever be absolutely beyond question; some theorists argue that experimental loopholes or hidden assumptions refute the theorem's validity, though most physicists accept that experiments confirm the violation of Bell inequalities.

## Importance of the theorem

Bell's theorem, derived in his seminal 1964 paper titled On the Einstein Podolsky Rosen paradox, has been called "the most profound in science". The title of the article refers to the famous paper by Einstein, Podolsky and Rosen
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...

that challenged the completeness of quantum mechanics. In his paper, Bell started from the same two assumptions as did EPR, namely (i) reality (that microscopic objects have real properties determining the outcomes of quantum mechanical measurements), and (ii) locality (that reality is not influenced by measurements performed simultaneously at a large distance). Bell was able to derive from those two assumptions an important result, namely Bell's inequality, implying that at least one of the assumptions must be false.

In two respects Bell's 1964 paper was a step forward compared to the EPR paper: firstly, it considered more hidden variables
Hidden variable theory
Historically, in physics, hidden variable theories were espoused by some physicists who argued that quantum mechanics is incomplete. These theories argue against the orthodox interpretation of quantum mechanics, which is the Copenhagen Interpretation...

than merely the element of physical reality
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...

in the EPR paper; and, more importantly, Bell's inequality was liable to be experimentally tested, thus yielding the opportunity to convert the question of local realism from philosophy to physics. Whereas Bell's paper deals only with deterministic hidden variable theories, Bell's theorem was later generalized to stochastic
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...

theories as well, and it was also realised that the theorem can even be proven without introducing hidden variables.

After the EPR paper, quantum mechanics was in an unsatisfactory position: either it was incomplete, in the sense that it failed to account for some elements of physical reality, or it violated the principle of a finite propagation speed of physical effects. In a modified version of the EPR thought experiment, two hypothetical observers
Observation
Observation is either an activity of a living being, such as a human, consisting of receiving knowledge of the outside world through the senses, or the recording of data using scientific instruments. The term may also refer to any data collected during this activity...

, now commonly referred to as Alice and Bob
Alice and Bob
The names Alice and Bob are commonly used placeholder names for archetypal characters in fields such as cryptography and physics. The names are used for convenience; for example, "Alice sends a message to Bob encrypted with his public key" is easier to follow than "Party A sends a message to Party...

, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state. It is the conclusion of EPR that once Alice measures spin in one direction (e.g. on the x axis), Bob's measurement in that direction is determined with certainty, as being the opposite outcome to that of Alice, whereas immediately before Alice's measurement Bob's outcome was only statistically determined (i.e., was only a probability, not a certainty); thus, either the spin in each direction is an element of physical reality, or the effects travel from Alice to Bob instantly.

In QM, predictions are formulated in terms of probabilities
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

— for example, the probability that an electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

will be detected in a particular place, or the probability that its spin is up or down. The idea persisted, however, that the electron in fact has a definite position and spin, and that QM's weakness is its inability to predict those values precisely. The possibility existed that some unknown theory, such as a hidden variables theory, might be able to predict those quantities exactly, while at the same time also being in complete agreement with the probabilities predicted by QM. If such a hidden variables theory exists, then because the hidden variables are not described by QM the latter would be an incomplete theory.

Two assumptions drove the desire to find a local realist theory:
1. Objects have a definite state that determines the values of all other measurable properties, such as position and momentum.
2. Effects of local actions, such as measurements
Measurement in quantum mechanics
The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus....

, cannot travel faster than the speed of light (in consequence of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

). Thus if observers are sufficiently far apart, a measurement made by one can have no effect on a measurement made by the other.

In the form of local realism used by Bell, the predictions of the theory result from the application of classical probability theory to an underlying parameter space. By a simple argument based on classical probability, he showed that correlations between measurements
Measurement in quantum mechanics
The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus....

are bounded in a way that is violated by QM.

Bell's theorem seemed to put an end to local realism.

According to Bell's theorem, either quantum mechanics or local realism is wrong, as they are mutually exclusive. The paper noted that "it requires little imagination to envisage the experiments involved actually being made", to determine which of them is correct, but it took many years and many improvements in technology to perform them.

The Bell test experiments
Bell test experiments
The Bell test experiments serve to investigate the validity of the entanglement effect in quantum mechanics by using some kind of Bell inequality...

have been interpreted as showing that the Bell inequalities are violated in favour of QM. The no-communication theorem
No-communication theorem
In quantum information theory, a no-communication theorem is a result which gives conditions under which instantaneous transfer of information between two observers is impossible. These results can be applied to understand the so-called paradoxes in quantum mechanics such as the EPR paradox or...

shows that the observers cannot use the effect to communicate (classical) information to each other faster than the speed of light
Faster-than-light
Faster-than-light communications and travel refer to the propagation of information or matter faster than the speed of light....

, but the ‘fair sampling’ and ‘no enhancement’ assumptions require more careful consideration (below).

John Bell's paper examines both John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

's 1932 proof of the incompatibility of hidden variables with QM and the seminal 1935 EPR paper on the subject by Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

and his colleagues.

## Bell inequalities

Bell inequalities concern measurements made by observers on pairs of particles that have interacted and then separated. According to quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

they are entangled
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

, while local realism would limit the correlation of subsequent measurements of the particles.

Different authors subsequently derived inequalities similar to Bell´s original inequality, and these are here collectively termed Bell inequalities. All Bell inequalities describe experiments in which the predicted result from quantum entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

differs from that flowing from local realism. The inequalities assume that each quantum-level object has a well-defined state that accounts for all its measurable properties and that distant objects do not exchange information faster than the speed of light. These well-defined states are typically called hidden variables
Hidden variable theory
Historically, in physics, hidden variable theories were espoused by some physicists who argued that quantum mechanics is incomplete. These theories argue against the orthodox interpretation of quantum mechanics, which is the Copenhagen Interpretation...

, the properties that Einstein posited when he stated his famous objection to quantum mechanics: "God does not play dice."

Bell showed that under quantum mechanics, the mathematics of which contains no local hidden variables, the Bell inequalities can nevertheless be violated: the properties of a particle are not clear, but may be correlated with those of another particle due to quantum entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

, allowing their state to be well defined only after a measurement is made on either particle. That restriction agrees with the Heisenberg uncertainty principle, a fundamental concept in quantum mechanics.

In Bell's words:
In probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, repeated measurements
Measurement in quantum mechanics
The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus....

of system properties can be regarded as repeated sampling of random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s. In Bell's experiment, Alice can choose a detector setting to measure either or and Bob can choose a detector setting to measure either or . Measurements of Alice and Bob may be somehow correlated with each other, but the Bell inequalities say that if the correlation stems from local random variables, there is a limit to the amount of correlation one might expect to see.

### Original Bell's inequality

The original inequality that Bell derived was:

where C is the "correlation" of the particle pairs and a, b and c settings of the apparatus. This inequality is not used in practice. For one thing, it is true only for genuinely "two-outcome" systems, not for the "three-outcome" ones (with possible outcomes of zero as well as +1 and −1) encountered in real experiments. For another, it applies only to a very restricted set of hidden variable theories, namely those for which the outcomes on both sides of the experiment are always exactly anticorrelated when the analysers are parallel, in agreement with the quantum mechanical prediction.

A simple limit of Bell's inequality has the virtue of being completely intuitive. If the result of three different statistical coin-flips A, B, and C have the property that:
1. A and B are the same (both heads or both tails) 99% of the time
2. B and C are the same 99% of the time

then A and C are the same at least 98% of the time. The number of mismatches between A and B (1/100) plus the number of mismatches between B and C (1/100) are together the maximum possible number of mismatches between A and C.

In quantum mechanics, by letting A, B, and C be the values of the spin of two entangled particles measured relative to some axis at 0 degrees, θ degrees, and 2θ degrees respectively, the overlap of the wavefunction between the different angles is proportional to . The probability that A and B give the same answer is , where is proportional to θ. This is also the probability that B and C give the same answer. But A and C are the same 1 − (2ε)2 of the time. Choosing the angle so that , A and B are 99% correlated, B and C are 99% correlated and A and C are only 96% correlated.

Imagine that two entangled particles in a spin singlet are shot out to two distant locations, and the spins of both are measured in the direction A. The spins are 100% correlated (actually, anti-correlated but for this argument that is equivalent). The same is true if both spins are measured in directions B or C. It is safe to conclude that any hidden variables that determine the A,B, and C measurements in the two particles are 100% correlated and can be used interchangeably.

If A is measured on one particle and B on the other, the correlation between them is 99%. If B is measured on one and C on the other, the correlation is 99%. This allows us to conclude that the hidden variables determining A and B are 99% correlated and B and C are 99% correlated. But if A is measured in one particle and C in the other, the results are only 96% correlated, which is a contradiction. The intuitive formulation is due to David Mermin
David Mermin
Nathaniel David Mermin is a solid-state physicist at Cornell University best known for the eponymous Mermin-Wagner theorem and his application of the term "Boojum" to superfluidity, and for the quote "Shut up and calculate!"Together with Neil W...

, while the small-angle limit is emphasized in Bell's original article.

### CHSH inequality

In addition to Bell's original inequality, the form given by John Clauser
John Clauser
John Francis Clauser is an American theoretical and experimental physicist known for contributions to the foundations of quantum mechanics, in particular the Clauser-Horne-Shimony-Holt inequality....

, Michael Horne, Abner Shimony
Abner Shimony
Abner Shimony is an American physicist and philosopher of science specializing in quantum theory.-Career:Shimony obtained his BA in Mathematics and Philosophy from Yale University in 1948, and an MA in Philosophy from the University of Chicago in 1950. He obtained his Ph.D...

and R. A. Holt, (the CHSH form
CHSH inequality
In physics, the CHSH Bell test is an application of Bell's theorem, intended to distinguish between the entanglement hypothesis of quantum mechanics and local hidden variable theories. CHSH stands for John Clauser, Michael Horne, Abner Shimony and Richard Holt, who described it in a much-cited...

) is especially important, as it gives classical limits to the expected correlation
Correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

for the above experiment conducted by Alice and Bob:

where C denotes correlation.

Correlation of observables X, Y is defined as

This is a non-normalized form of the correlation coefficient
Pearson product-moment correlation coefficient
In statistics, the Pearson product-moment correlation coefficient is a measure of the correlation between two variables X and Y, giving a value between +1 and −1 inclusive...

considered in statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

(see Quantum correlation
Quantum correlation
In Bell test experiments the term quantum correlation has come to mean the expectation value of the product of the outcomes on the two sides. In other words, the expected change in physical characteristics as one quantum system passes through an interaction site...

).

To formulate Bell's theorem, we formalize local realism as follows:
1. There is a probability space and the observed outcomes by both Alice and Bob result by random sampling of the parameter .
2. The values observed by Alice or Bob are functions of the local detector settings and the hidden parameter only. Thus
• Value observed by Alice with detector setting is
• Value observed by Bob with detector setting is

Implicit in assumption 1) above, the hidden parameter space has a probability measure and the expectation of a random variable X on with respect to is written

where for accessibility of notation we assume that the probability measure has a density.

Bell's inequality. The CHSH inequality (1) holds under the hidden variables assumptions above.

For simplicity, let us first assume the observed values are +1 or −1; we remove this assumption in Remark 1 below.

Let . Then at least one of

is 0. Thus

and therefore

Remark 1: The correlation inequality (1) still holds if the variables , are allowed to take on any real values between −1 and +1. Indeed, the relevant idea is that each summand in the above average is bounded above by 2. This is easily seen as true in the more general case:

To justify the upper bound 2 asserted in the last inequality, without loss of generality, we can assume that

In that case

Remark 2: Though the important component of the hidden parameter in Bell's original proof is associated with the source and is shared by Alice and Bob, there may be others that are associated with the separate detectors, these others being independent. This argument was used by Bell in 1971, and again by Clauser and Horne in 1974, to justify a generalisation of the theorem forced on them by the real experiments, in which detectors were never 100% efficient. The derivations were given in terms of the averages of the outcomes over the local detector variables. The formalisation of local realism was thus effectively changed, replacing A and B by averages and retaining the symbol but with a slightly different meaning. It was henceforth restricted (in most theoretical work) to mean only those components that were associated with the source.

However, with the extension proved in Remark 1, CHSH inequality still holds even if the instruments themselves contain hidden variables. In that case,
averaging over the instrument hidden variables gives new variables:

on , which still have values in the range [−1, +1] to which we can apply the previous result.

## Bell inequalities are violated by quantum mechanical predictions

In the usual quantum mechanical formalism
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...

, the observables X and Y are represented as self-adjoint operator
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...

s on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

. To compute the correlation, assume that X and Y are represented by matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

in a finite dimensional space and that X and Y commute; this special case suffices for our purposes below. The von Neumann measurement postulate states: a series of measurements
Measurement in quantum mechanics
The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus....

of an observable X on a series of identical systems in state produces a distribution of real values. By the assumption that observables are finite matrices, this distribution is discrete. The probability of observing λ is non-zero if and only if λ is an eigenvalue of the matrix X and moreover the probability is

where EX (λ) is the projector corresponding to the eigenvalue λ. The system state immediately after the measurement
Measurement in quantum mechanics
The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus....

is

From this, we can show that the correlation of commuting observables X and Y in a pure state is

We apply this fact in the context of the EPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...

. The measurements performed by Alice and Bob are spin measurements on electrons. Alice can choose between two detector settings labelled a and a′; these settings correspond to measurement of spin along the z or the x axis. Bob can choose between two detector settings labelled b and b′; these correspond to measurement of spin along the z′ or x′ axis, where the x′ – z′ coordinate system is rotated 135° relative to the xz coordinate system. The spin observables are represented by the 2 × 2 self-adjoint matrices:

These are the Pauli spin matrices normalized so that the corresponding eigenvalues are +1, −1. As is customary
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

, we denote the eigenvectors of Sx by

Let be the spin singlet state for a pair of electrons discussed in the EPR paradox. This is a specially constructed state described by the following vector in the tensor product

Now let us apply the CHSH formalism to the measurements that can be performed by Alice and Bob.

The operators , correspond to Bob's spin measurements along x′ and z′. Note that the A operators commute with the B operators, so we can apply our calculation for the correlation. In this case, we can show that the CHSH inequality fails. In fact, a straightforward calculation shows that

and

so that

Bell's Theorem: If the quantum mechanical formalism is correct, then the system consisting of a pair of entangled electrons cannot satisfy the principle of local realism. Note that is indeed the upper bound for quantum mechanics called Tsirelson's bound
Tsirelson's bound
Tsirelson's bound, also known as Tsirelson's inequality, or in another transliteration, Cirel'son's inequality, is an inequality that imposed an upper limit to quantum mechanical correlations between distant events...

. The operators giving this maximal value are always isomorphic to the Pauli matrices.

## Practical experiments testing Bell's theorem

Experimental tests can determine whether the Bell inequalities required by local realism hold up to the empirical evidence.

Bell's inequalities are tested by "coincidence counts" from a Bell test experiment such as the optical one shown in the diagram. Pairs of particles are emitted as a result of a quantum process, analysed with respect to some key property such as polarisation direction, then detected. The setting (orientations) of the analysers are selected by the experimenter.

Bell test experiments to date overwhelmingly violate Bell's inequality. Indeed, a table of Bell test experiments performed prior to 1986 is given in 4.5 of Redhead, 1987. Of the thirteen experiments listed, only two reached results contradictory to quantum mechanics; moreover, according to the same source, when the experiments were repeated, "the discrepancies with QM could not be reproduced".

Nevertheless, the issue is not conclusively settled. According to Shimony's 2004 Stanford Encyclopedia overview article:
To explore the 'detection loophole', one must distinguish the classes of homogeneous and inhomogeneous Bell inequality.

The standard assumption in Quantum Optics is that "all photons of given frequency, direction and polarization are identical" so that photodetectors treat all incident photons on an equal basis. Such a fair sampling assumption generally goes unacknowledged, yet it effectively limits the range of local theories to those that conceive of the light field as corpuscular. The assumption excludes a large family of local realist theories, in particular, Max Planck's description. We must remember the cautionary words of Albert Einstein shortly before he died: "Nowadays every Tom, Dick and Harry ('jeder Kerl' in German original) thinks he knows what a photon is, but he is mistaken".

Objective physical properties for Bell’s analysis (local realist theories) include the wave amplitude of a light signal. Those who maintain the concept of duality, or simply of light being a wave, recognize the possibility or actuality that the emitted atomic light signals have a range of amplitudes and, furthermore, that the amplitudes are modified when the signal passes through analyzing devices such as polarizers and beam splitters. It follows that not all signals have the same detection probability.

### Two classes of Bell inequalities

The fair sampling problem was faced openly in the 1970s. In early designs of their 1973 experiment, Freedman and Clauser used fair sampling in the form of the Clauser-Horne-Shimony-Holt (CHSH) hypothesis. However, shortly afterwards Clauser and Horne made the important distinction between inhomogeneous (IBI) and homogeneous (HBI) Bell inequalities. Testing an IBI requires that we compare certain coincidence rates in two separated detectors with the singles rates of the two detectors. Nobody needed to perform the experiment, because singles rates with all detectors in the 1970s were at least ten times all the coincidence rates. So, taking into account this low detector efficiency, the QM prediction actually satisfied the IBI. To arrive at an experimental design in which the QM prediction violates IBI we require detectors whose efficiency exceeds 82% for singlet states, but have very low dark rate and short dead and resolving times. This is well above the 30% achievable so Shimony’s optimism in the Stanford Encyclopedia, quoted in the preceding section, appears over-stated.

### Practical challenges

Because detectors don't detect a large fraction of all photons, Clauser and Horne recognized that testing Bell's inequality requires some extra assumptions. They introduced the No Enhancement Hypothesis (NEH):
Given this assumption, there is a Bell inequality between the coincidence rates with polarizers and coincidence rates without polarizers.

The experiment was performed by Freedman and Clauser, who found that the Bell's inequality was violated. So the no-enhancement hypothesis cannot be true in a local hidden variables model. The Freedman-Clauser experiment reveals that local hidden variables imply the new phenomenon of signal enhancement:
This is perhaps not surprising, as it is known that adding noise to data can, in the presence of a threshold, help reveal hidden signals (this property is known as stochastic resonance
Stochastic resonance
Stochastic resonance is a phenomenon that occurs in a threshold measurement system when an appropriate measure of information transfer is maximized in the presence of a non-zero level of stochastic input noise thereby lowering the response...

). One cannot conclude that this is the only local-realist alternative to Quantum Optics, but it does show that the word loophole is biased. Moreover, the analysis leads us to recognize that the Bell-inequality experiments, rather than showing a breakdown of realism or locality, are capable of revealing important new phenomena.

## Theoretical challenges

Most advocates of the hidden variables idea believe that experiments have ruled out local hidden variables. They are ready to give up locality, explaining the violation of Bell's inequality by means of a non-local hidden variable theory
Hidden variable theory
Historically, in physics, hidden variable theories were espoused by some physicists who argued that quantum mechanics is incomplete. These theories argue against the orthodox interpretation of quantum mechanics, which is the Copenhagen Interpretation...

, in which the particles exchange information about their states. This is the basis of the Bohm interpretation
Bohm interpretation
The de Broglie–Bohm theory, also called the pilot-wave theory, Bohmian mechanics, and the causal interpretation, is an interpretation of quantum theory. In addition to a wavefunction on the space of all possible configurations, it also includes an actual configuration, even in situations where...

of quantum mechanics, which requires that all particles in the universe be able to instantaneously exchange information with all others. A recent experiment ruled out a large class of non-Bohmian non-local hidden variable theories.

If the hidden variables can communicate with each other faster than light, Bell's inequality can easily be violated. Once one particle is measured, it can communicate the necessary correlations to the other particle. Since in relativity the notion of simultaneity is not absolute, this is unattractive. One idea is to replace instantaneous communication with a process that travels backwards in time along the past Light cone
Light cone
A light cone is the path that a flash of light, emanating from a single event and traveling in all directions, would take through spacetime...

. This is the idea behind a transactional interpretation
Transactional interpretation
The transactional interpretation of quantum mechanics describes quantum interactions in terms of a standing wave formed by retarded and advanced waves. It was first proposed in 1986 by John G...

of quantum mechanics, which interprets the statistical emergence of a quantum history as a gradual coming to agreement between histories that go both forward and backward in time.

A few advocates of deterministic models have not given up on local hidden variables. For example, Gerard 't Hooft has argued that the superdeterminism
Superdeterminism
In the context of quantum mechanics, superdeterminism is a term that has been used to describe a hypothetical class of theories which evade Bell's theorem by virtue of being completely deterministic. Bell's theorem depends on the assumption of counterfactual definiteness, which technically does...

loophole cannot be dismissed.

The quantum mechanical wavefunction can also provide a local realistic description, if the wavefunction values are interpreted as the fundamental quantities that describe reality. Such an approach is called a many-worlds interpretation
Many-worlds interpretation
The many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction, but denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternative histories and futures are real, each representing an...

of quantum mechanics. In this view, two distant observers both split into superpositions when measuring a spin. The Bell inequality violations are no longer counterintuitive, because it is not clear which copy of the observer B observer A will see when going to compare notes. If reality includes all the different outcomes, locality in physical space (not outcome space) places no restrictions on how the split observers can meet up.

This implies that there is a subtle assumption in the argument that realism is incompatible with quantum mechanics and locality. The assumption, in its weakest form, is called counterfactual definiteness
Counterfactual definiteness
In some interpretations of quantum mechanics, counterfactual definiteness is the ability to speak with meaning of the definiteness of the results of measurements that have not been performed...

. This states that if the results of an experiment are always observed to be definite, there is a quantity that determines what the outcome would have been even if you don't do the experiment.

Many worlds interpretations are not only counterfactually indefinite, they are factually indefinite. The results of all experiments, even ones that have been performed, are not uniquely determined.

E. T. Jaynes pointed out two hidden assumptions in Bell Inequality that could limit its generality:
1. Bell interpreted conditional probability P(X|Y) as a causal inference, i.e. Y exerted a causal inference on X in reality. However, P(X|Y) actually only means logical inference (deduction). Causes cannot travel faster than light or backward in time, but deduction can.
2. Bell's inequality does not apply to some possible hidden variable theories. It only applies to a certain class of local hidden variable theories. In fact, it might have just missed the kind of hidden variable theories that Einstein is most interested in.

## Final remarks

The violations of Bell's inequalities, due to quantum entanglement, just provide the definite demonstration of something that was already strongly suspected, that quantum physics cannot be represented by any version of the classical picture of physics. Some earlier elements that had seemed incompatible with classical pictures included apparent complementarity
Complementarity (physics)
In physics, complementarity is a basic principle of quantum theory proposed by Niels Bohr, closely identified with the Copenhagen interpretation, and refers to effects such as the wave–particle duality...

and (hypothesized) wavefunction collapse
Wavefunction collapse
In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in a superposition of several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer...

. Complementarity
Complementarity (physics)
In physics, complementarity is a basic principle of quantum theory proposed by Niels Bohr, closely identified with the Copenhagen interpretation, and refers to effects such as the wave–particle duality...

is now seen not as an independent ingredient of the quantum picture but rather as a direct consequence of the Quantum decoherence
Quantum decoherence
In quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...

expected from the quantum formalism itself. The possibility of wavefunction collapse
Wavefunction collapse
In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in a superposition of several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer...

is now seen as one possible problematic ingredient of some interpretations
Interpretation of quantum mechanics
An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations...

, rather than as an essential part of quantum mechanics. The Bell violations show that no resolution of such issues can avoid the ultimate strangeness of quantum behavior.

The EPR paper "pinpointed" the unusual properties of the entangled states, e.g. the above-mentioned singlet state, which is the foundation for present-day applications of quantum physics, such as quantum cryptography
Quantum cryptography
Quantum key distribution uses quantum mechanics to guarantee secure communication. It enables two parties to produce a shared random secret key known only to them, which can then be used to encrypt and decrypt messages...

; one application involves the measurement of quantum entanglement as a physical source of bits for Rabin's
Michael O. Rabin
Michael Oser Rabin , is an Israeli computer scientist and a recipient of the Turing Award.- Biography :Rabin was born in 1931 in Breslau, Germany, , the son of a rabbi. In 1935, he emigrated with his family to Mandate Palestine...

oblivious transfer
Oblivious transfer
In cryptography, an oblivious transfer protocol is a type of protocol in which a sender transfers one of potentially many pieces of information to a receiver, but remains oblivious as to what piece has been transferred....

protocol. This strange non-locality was originally supposed to be a Reductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

, because the standard interpretation could easily do away with action-at-a-distance by simply assigning to each particle definite spin-states. Bell's theorem showed that the "entangledness" prediction of quantum mechanics have a degree of non-locality that cannot be explained away by any local theory.

In well-defined Bell experiments (see the paragraph on "test experiments") one can now falsify either quantum mechanics or Einstein's quasi-classical assumptions: currently many experiments of this kind have been performed, and the experimental results support quantum mechanics, though some believe that detectors give a biased sample of photons, so that until nearly every photon pair generated is observed there will be loopholes.

What is powerful about Bell's theorem is that it doesn't refer to any particular physical theory. What makes Bell's theorem unique and powerful is that it shows that nature violates the most general assumptions behind classical pictures, not just details of some particular models. No combination of local deterministic and local random variables can reproduce the phenomena predicted by quantum mechanics and repeatedly observed in experiments.

• Bell test experiments
Bell test experiments
The Bell test experiments serve to investigate the validity of the entanglement effect in quantum mechanics by using some kind of Bell inequality...

• CHSH Bell test
• Counterfactual definiteness
Counterfactual definiteness
In some interpretations of quantum mechanics, counterfactual definiteness is the ability to speak with meaning of the definiteness of the results of measurements that have not been performed...

• Fundamental Fysiks Group
Fundamental Fysiks Group
The Fundamental Fysiks Group was founded in San Francisco in May 1975 by two physicists, Elizabeth Rauscher and George Weissmann, at the time both graduate students at the University of California, Berkeley. The group held informal discussions on Friday afternoons to explore the philosophical...

• GHZ experiment
GHZ experiment
GHZ experiments are a class of physics experiments that may be used to generate starkly contrasting predictions from local hidden variable theory and quantum mechanical theory, and permit immediate comparison with actual experimental results. A GHZ experiment is similar to a test of Bell's...

• Local hidden variable theory
Local hidden variable theory
In quantum mechanics, a local hidden variable theory is one in which distant events are assumed to have no instantaneous effect on local ones....

• Leggett inequality
Leggett inequality
The Leggett inequalities, named for Anthony James Leggett, who derived them, are a related pair of mathematical expressions concerning the correlations of properties of entangled particles...

• Leggett–Garg inequality
• Measurement in quantum mechanics
Measurement in quantum mechanics
The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus....

• Mott problem
Mott problem
In quantum mechanics, the Mott problem is a paradox that illustrates some of the difficulties of understanding the nature of wave function collapse and measurement in quantum mechanics...

• Quantum entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

• Quantum mechanical Bell test prediction
Quantum mechanical Bell test prediction
In physics, the quantum mechanical Bell test prediction is the prediction that quantum mechanics would give for the correlation probabilities for a set of measurements performed on a quantum entangled state...

• Renninger negative-result experiment
Renninger negative-result experiment
In quantum mechanics, the Renninger negative-result experiment is a thought experiment that illustrates some of the difficulties of understanding the nature of wave function collapse and measurement in quantum mechanics...

The following are intended for general audiences.
• Amir D. Aczel, Entanglement: The greatest mystery in physics (Four Walls Eight Windows, New York, 2001).
• A. Afriat and F. Selleri, The Einstein, Podolsky and Rosen Paradox (Plenum Press, New York and London, 1999)
• J. Baggott, The Meaning of Quantum Theory (Oxford University Press, 1992)
• N. David Mermin, "Is the moon there when nobody looks? Reality and the quantum theory", in Physics Today, April 1985, pp. 38–47.
• Louisa Gilder, The Age of Entanglement: When Quantum Physics Was Reborn (New York: Alfred A. Knopf, 2008)
• Brian Greene, The Fabric of the Cosmos (Vintage, 2004, ISBN 0-375-72720-5)
• Nick Herbert, Quantum Reality: Beyond the New Physics (Anchor, 1987, ISBN 0-385-23569-0)
• D. Wick, The infamous boundary: seven decades of controversy in quantum physics (Birkhauser, Boston 1995)
• R. Anton Wilson, Prometheus Rising (New Falcon Publications, 1997, ISBN 1-56184-056-4)
• Gary Zukav "The Dancing Wu Li Masters" (Perennial Classics, 2001, ISBN 0-06-095968-1)