The

**EPR paradox** is a topic in quantum physics and the

philosophy of scienceThe 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...

concerning the measurement and description of microscopic systems (such as individual

photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s,

electronThe 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...

s or

atomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s) by the methods of quantum physics. It refers to the

dichotomyA dichotomy is any splitting of a whole into exactly two non-overlapping parts, meaning it is a procedure in which a whole is divided into two parts...

that either the measurement of a

physical quantityA physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.-Definition of a physical quantity:Formally, the International Vocabulary of Metrology, 3rd edition defines quantity as:...

in one

systemIn physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...

must affect the measurement of a physical quantity in another, spatially separate, system or the description of reality given by a wave function must be

incompleteIncompleteness of quantum physics is the assertion that the state of a physical system, as formulated by quantum mechanics, does not give a complete description for the system, assuming the usual philosophical requirements ....

.

This challenge to the

Copenhagen interpretationThe Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

of quantum physics (that only the position or momentum of a particle, but not both, can be known with certainty) originated from the consequences of a

thought experimentA thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...

authored in 1935 by

EinsteinAlbert 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...

,

PodolskyBoris Yakovlevich Podolsky , was an American physicist of Russian Jewish descent.-Education:In 1896, Boris Podolsky was born into a poor Jewish family in Taganrog, in what was then the Russian Empire, and he moved to the United States in 1913...

and

RosenNathan Rosen was an American-Israeli physicist noted for his study on the structure of the hydrogen molecule and his work with Albert Einstein and Boris Podolsky on entangled wave functions and the EPR paradox.-Background:Nathan Rosen was born into a Jewish family in Brooklyn, New York...

. The paper they authored indicated what seemed to be a flaw in the interpretation. The experiment involved two systems that initially interact with each other and are then separated. Then the position or momentum of one of the systems is measured, and due to the known relationship between the (measured) value of the first particle and the value of the second particle, the observer is aware of that value in the second particle. A measurement of the other value is then made on the second particle, and, once again, due to the relationship between the two particles, that value is then known in the first particle. This outcome seems to violate the uncertainty principle, as both the position and momentum of a single particle would be known with certainty.

Einstein struggled to the end of his life for a theory that could better comply with

causalityCausality is the relationship between causes and effects. It is considered to be fundamental to all natural science, especially physics. Causality is also a topic studied from the perspectives of philosophy and statistics....

, protesting against the view that there exists no objective physical reality other than that which is revealed through measurement interpreted in terms of quantum mechanical formalism.

However, since Einstein's death, experiments analogous to that of the EPR paradox have been carried out, starting in 1976 by French scientists at the

Saclay Nuclear Research CentreThe Saclay Nuclear Research Centre in the Essonne department of northern France is one of 9 laboratories belonging to the CEA. It is the administrative headquarters of the CEA and also hosts a laboratory....

. These experiments appear to show that the local realism theory is false.

## Quantum mechanics and its interpretation

Since the early twentieth century, quantum theory has proved to be successful in describing accurately the physical reality of the mesoscopic and microscopic world, in multiple reproducible physics experiments.

Quantum mechanicsQuantum 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...

was developed with the aim of describing atoms and explaining the observed spectral lines in a measurement apparatus. Although disputed, it has yet to be seriously challenged. Philosophical interpretations of quantum phenomena, however, are another matter: the question of how to interpret the mathematical formulation of quantum mechanics has given rise to a variety of different answers from people of different philosophical backgrounds (see

Interpretation of quantum mechanicsAn 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...

).

Quantum theory and quantum mechanics do not provide single measurement outcomes in a deterministic way. According to the theory of quantum mechanics known as the

Copenhagen interpretationThe Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

, measurement causes an instantaneous collapse of the wave function describing the quantum system into an eigenstate of the observable state that was measured.

The most prominent opponent of the

Copenhagen interpretationThe Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

is

Albert EinsteinAlbert 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...

. In his view, quantum mechanics is incomplete. Commenting on this, other writers (such as

John von NeumannJohn 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,...

and

David BohmDavid Joseph Bohm FRS was an American-born British quantum physicist who contributed to theoretical physics, philosophy, neuropsychology, and the Manhattan Project.-Youth and college:...

) have suggested that consequently there would have to be

'hidden' variablesHistorically, 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...

responsible for random measurement results, something which was not expressly claimed in the original paper.

That paper, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?"

http://prola.aps.org/abstract/PR/v47/i10/p777_1, authored by Einstein, Podolsky and Rosen in

1935The year 1935 in science and technology involved some significant events, listed below.-Geology:* Charles Richter and Beno Gutenberg develop the Richter magnitude scale for quantifying earthquakes.-History of science:...

, condensed the philosophical discussion into a physical argument. They claim that given a specific experiment, in which the outcome of a measurement is known before the measurement takes place, there must exist something in the real world, an "element of reality", which determines the measurement outcome. They postulate that these elements of reality are

localIn 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...

, in the sense that each belongs to a certain point in

spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

. Each element may only be influenced by events which are located in the backward

light coneA light cone is the path that a flash of light, emanating from a single event and traveling in all directions, would take through spacetime...

of its point in spacetime (i.e. the past). These claims are founded on assumptions about nature which constitute what is now known as local realism.

Though the EPR paper has often been taken as an exact expression of Einstein's views, it was primarily authored by Podolsky, based on discussions at the

Institute for Advanced StudyThe Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...

with Einstein and Rosen. Einstein later expressed to

Erwin SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

that, "it did not come out as well as I had originally wanted; rather, the essential thing was, so to speak, smothered by the formalism." In 1948 Einstein presented a less formal account of

his local realist ideasIn 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...

.

## Description of the paradox

The original EPR paradox challenges the prediction of quantum mechanics that it is impossible to know both the position and the momentum of a quantum particle. This can be extended to other pairs of physical properties.

### EPR paper

The original paper describes what happens to "two systems I and II, which we permit to interact ...", and, after some time, "we suppose that there is no longer any interaction between the two parts." In the words of Kumar (2009), it has "Two particles, A and B, [which] interact briefly and then move off in opposite directions." According to Heisenberg's uncertainty principle, it is impossible to measure both the momentum and the position of particle B, say, exactly. However, it is possible to measure the exact position of particle A and the exact momentum of particle B. By calculation, therefore, with the exact position of particle A known, the exact position of particle B can be known. Also, with the exact momentum of particle B known, the exact momentum of particle A can be worked out. "EPR argued that they had proved that ... particle B can have simultaneously exact values of position and momentum."

This is a paradox in Quantum Mechanics: The theory predicts that both values cannot be known for a particle, and yet the EPR thought experiment shows that they can. "Therefore, the quantum mechanical description of physical reality, EPR conclude, is incomplete." The paper says: "We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete."

The EPR paper ends with:

While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.

### Greene version

The paradox was explained in a different way by

GreeneBrian Greene is an American theoretical physicist and string theorist. He has been a professor at Columbia University since 1996. Greene has worked on mirror symmetry, relating two different Calabi-Yau manifolds...

and others, using electron spin.

A

positronThe positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...

and an

electronThe 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...

are emitted from a source, by

pionIn particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....

decay, so that their spins are opposite; one particle’s spin about any axis is the negative of the other's. But making a measurement of a particle’s spin about one axis disturbs the particle, so that its spin about any other axis cannot be measured.

However, the measurement of the electron’s spin about the x-axis discloses the positron’s spin about the x-axis. Since this measurement has not disturbed the positron in any way, it cannot be that the positron only came to have that state when it was measured, because it was not measured.

The positron’s spin about the y-axis can also be measured, so that it is known the positron had a definite spin about two axes – much more information than the positron is capable of holding, and a "hidden variable" according to some interpretations of EPR.

### Measurements on an entangled state

We have a source that emits electron-positron pairs, with the electron sent to destination A, where there is an observer named

AliceThe 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...

, and the positron sent to destination B, where there is an observer named

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...

. According to quantum mechanics, we can arrange our source so that each emitted pair occupies a

quantum state called a spin singlet. The particles are thus said to be

entangledQuantum 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...

. This can be viewed as a

quantum superpositionQuantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...

of two states, which we call state I and state II. In state I, the electron has

spinIn 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,...

pointing upward along the z-axis (+z) and the positron has spin pointing downward along the z-axis (-z). In state II, the electron has spin -z and the positron has spin +z. Therefore, it is impossible (without measuring) to know the definite state of spin of either particle in the spin singlet.

Alice now measures the spin along the z-axis. She can obtain one of two possible outcomes: +z or -z. Suppose she gets +z. According to the

Copenhagen interpretationThe Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

of quantum mechanics, the quantum state of the system

collapsesIn 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...

into state I. The quantum state determines the probable outcomes of any measurement performed on the system. In this case, if Bob subsequently measures spin along the z-axis, there is 100% probability that he will obtain -z. Similarly, if Alice gets -z, Bob will get +z.

There is, of course, nothing special about choosing the z-axis: according to quantum mechanics the spin singlet state may equally well be expressed as a superposition of spin states pointing in the x direction. Suppose that Alice and Bob had decided to measure spin along the x-axis. We'll call these states Ia and IIa. In state Ia, Alice's electron has spin +x and Bob's positron has spin -x. In state IIa, Alice's electron has spin -x and Bob's positron has spin +x. Therefore, if Alice measures +x, the system 'collapses' into state Ia, and Bob will get -x. If Alice measures -x, the system collapses into state IIa, and Bob will get +x.

Whatever axis their spins are measured along, they are always found to be opposite. This can only be explained if the particles are linked in some way. Either they were created with a definite (opposite) spin about every axis—a "hidden variable" argument— or they are linked so that one electron "feels" which axis the other is having its spin measured along, and becomes its opposite about that one axis—an "entanglement" argument. Moreover, if the two particles have their spins measured about different axes, once the electron's spin has been measured about the x-axis (and the positron's spin about the x-axis deduced), the positron's spin about the y-axis will no longer be certain, as if (a) it knows that the measurement has taken place, or (b) it has a definite spin already, about a second axis—a hidden variable.

In quantum mechanics, the x-spin and z-spin are "incompatible observables", meaning there is a

Heisenberg uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

operating between them: a quantum state cannot possess a definite value for both of these variables. Suppose Alice measures the z-spin and obtains +z, so that the quantum state collapses into state I. Now, instead of measuring the z-spin as well, Bob measures the x-spin. According to quantum mechanics, when the system is in state I, Bob's x-spin measurement will have a 50% probability of producing +x and a 50% probability of -x. It is impossible to predict which outcome will appear until Bob actually performs the measurement.

**Here is the crux of the matter**. You might imagine that, when Bob measures the x-spin of his positron, he would get an answer with absolute certainty, since prior to this he hasn't disturbed his particle at all. But Bob's positron has a 50% probability of producing +x and a 50% probability of -x—so the outcome is not certain. Bob's positron "knows" that Alice's electron has been measured, and its z-spin detected, and hence B's z-spin calculated, so its x-spin is uncertain.

Put another way, how does Bob's positron know which way to point if Alice decides (based on information unavailable to Bob) to measure x (i.e. to be the opposite of Alice's electron's spin about the x-axis) and also how to point if Alice measures z, since it is only supposed to know one thing at a time? The Copenhagen interpretation rules that say the wave function "collapses" at the time of measurement, so there must be action at a distance (entanglement) or the positron must know more than it's supposed to (hidden variables).

**Here is the paradox summed up:**
It is one thing to say that physical measurement of the first particle's momentum affects uncertainty in its own position, but to say that measuring the first particle's momentum affects the uncertainty in the position of the other is another thing altogether. Einstein, Podolsky and Rosen asked how can the second particle "know" to have precisely defined momentum but uncertain position? Since this implies that one particle is communicating with the other instantaneously across space, i.e. faster than light, this is the "paradox".

Incidentally, Bell used spin as his example, but many types of physical quantities—what quantum mechanics refer to as "observables"—can be used. The EPR paper used

momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

for the observable. Experimental realisations of the EPR scenario often use

photon polarizationPhoton polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. Individual photons are completely polarized...

, because polarized photons are easy to prepare and measure.

### Locality in the EPR experiment

The principle of locality states that physical processes occurring at one place should have no immediate effect on the elements of reality at another location. At first sight, this appears to be a reasonable assumption to make, as it seems to be a consequence of

special relativitySpecial 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...

, which states that

informationInformation in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...

can never be transmitted faster than the

speed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

without violating

causalityCausality is the relationship between causes and effects. It is considered to be fundamental to all natural science, especially physics. Causality is also a topic studied from the perspectives of philosophy and statistics....

. It is generally believed that any theory which violates causality would also be internally inconsistent, and thus deeply unsatisfactory.

It turns out that the usual rules for combining quantum mechanical and classical descriptions violate the principle of locality without violating causality. Causality is preserved because there is no way for Alice to transmit messages (i.e. information) to Bob by manipulating her measurement axis. Whichever axis she uses, she has a 50% probability of obtaining "+" and 50% probability of obtaining "-", completely at

randomRandomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....

; according to quantum mechanics, it is fundamentally impossible for her to influence what result she gets. Furthermore, Bob is only able to perform his measurement once: there is a fundamental property of quantum mechanics, known as the "

no cloning theoremThe no-cloning theorem is a result of quantum mechanics that forbids the creation of identical copies of an arbitrary unknown quantum state. It was stated by Wootters, Zurek, and Dieks in 1982, and has profound implications in quantum computing and related fields.The state of one system can be...

", which makes it impossible for him to make a million copies of the electron he receives, perform a spin measurement on each, and look at the statistical distribution of the results. Therefore, in the one measurement he is allowed to make, there is a 50% probability of getting "+" and 50% of getting "-", regardless of whether or not his axis is aligned with Alice's.

However, the principle of locality appeals powerfully to physical intuition, and Einstein, Podolsky and Rosen were unwilling to abandon it. Einstein derided the quantum mechanical predictions as "

spooky action at a distanceIn physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. This term was used most often in the context of early theories of gravity and electromagnetism to describe how an object responds to the influence of distant...

". The conclusion they drew was that quantum mechanics is not a complete theory.

In recent years, however, doubt has been cast on EPR's conclusion due to developments in understanding

localityIn 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...

and especially

quantum decoherenceIn 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...

. The word

localityIn 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...

has several different meanings in physics. For example, in

quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

"locality" means that quantum fields at different points of space do not interact with one another. However, quantum field theories that are "local" in this sense appear to violate the

principle of localityIn 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...

as defined by EPR, but they nevertheless do not violate locality in a more general sense.

Wavefunction collapseIn 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...

can be viewed as an epiphenomenon of

quantum decoherenceIn 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...

, which in turn is nothing more than an effect of the underlying local time evolution of the wavefunction of a system and all of its environment. Since the underlying behaviour doesn't violate local causality, it follows that neither does the additional effect of wavefunction collapse, whether real or apparent. Therefore, as outlined in the example above, neither the EPR experiment nor any quantum experiment demonstrates that

faster-than-lightFaster-than-light communications and travel refer to the propagation of information or matter faster than the speed of light....

signaling is possible.

### Hidden variables

There are several ways to resolve the EPR paradox. The one suggested by EPR is that quantum mechanics, despite its success in a wide variety of experimental scenarios, is actually an incomplete theory. In other words, there is some yet undiscovered theory of nature to which quantum mechanics acts as a kind of statistical approximation (albeit an exceedingly successful one). Unlike quantum mechanics, the more complete theory contains variables corresponding to all the "elements of reality". There must be some unknown mechanism acting on these variables to give rise to the observed effects of "non-commuting quantum observables", i.e. the Heisenberg uncertainty principle. Such a theory is called a

hidden variable theoryHistorically, 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...

.

To illustrate this idea, we can formulate a very simple hidden variable theory for the above thought experiment. One supposes that the quantum spin-singlet states emitted by the source are actually approximate descriptions for "true" physical states possessing definite values for the z-spin and x-spin. In these "true" states, the electron going to Bob always has spin values opposite to the electron going to Alice, but the values are otherwise completely random. For example, the first pair emitted by the source might be "(+z, -x) to Alice and (-z, +x) to Bob", the next pair "(-z, -x) to Alice and (+z, +x) to Bob", and so forth. Therefore, if Bob's measurement axis is aligned with Alice's, he will necessarily get the opposite of whatever Alice gets; otherwise, he will get "+" and "-" with equal probability.

Assuming we restrict our measurements to the z and x axes, such a hidden variable theory is experimentally indistinguishable from quantum mechanics. In reality, there may be an infinite number of axes along which Alice and Bob can perform their measurements, so there would have to be an infinite number of independent hidden variables. However, this is not a serious problem; we have formulated a very simplistic hidden variable theory, and a more sophisticated theory might be able to patch it up. It turns out that there is a much more serious challenge to the idea of hidden variables.

#### Bell's inequality

In 1964,

John BellJohn 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 :...

showed that the predictions of quantum mechanics in the EPR thought experiment are significantly different from the predictions of a particular class of hidden variable theories (the local hidden variable theories). Roughly speaking, quantum mechanics has a much stronger statistical

correlationIn 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....

with measurement results performed on different axes than do these hidden variable theories. These differences, expressed using inequality relations known as "Bell's inequalities", are in principle experimentally detectable. Later work by Eberhard showed that the key properties of local hidden variable theories which lead to Bell's inequalities are

localityIn 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...

and counter-factual definiteness. Any theory in which these principles apply produces the inequalities.

Arthur FineArthur Fine is an American philosopher of science teaching at the University of Washington . Before moving to UW he taught for many years at Northwestern University and, before that, at Cornell University and the University of Illinois at Chicago...

subsequently showed that any theory satisfying the inequalities can be modeled by a local hidden variable theory.

After the publication of Bell's paper, a variety of experiments were devised to test Bell's inequalities (experiments which generally rely on

photon polarizationPhoton polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. Individual photons are completely polarized...

measurement). All the experiments conducted to date have found behavior in line with the predictions of standard quantum mechanics theory.

However, Bell's theorem does not apply to all possible

philosophically realistContemporary philosophical realism is the belief that our reality, or some aspect of it, is ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc....

theories. It is a common misconception that quantum mechanics is inconsistent with all notions of philosophical realism, but realist interpretations of quantum mechanics are possible, although, as discussed above, such interpretations must reject either locality or counter-factual definiteness. Mainstream physics prefers to keep locality, while striving also to maintain a notion of realism that nevertheless rejects counter-factual definiteness. Examples of such mainstream realist interpretations are the

consistent historiesIn quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology...

interpretation and the

transactional interpretationThe 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...

. Fine's work showed that, taking locality as a given, there exist scenarios in which two statistical variables are correlated in a manner inconsistent with counter-factual definiteness, and that such scenarios are no more mysterious than any other, despite the inconsistency with counter-factual definiteness seeming 'counter-intuitive'.

Violation of locality is difficult to reconcile with

special relativitySpecial 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...

, and is thought to be incompatible with the principle of causality. On the other hand the

Bohm interpretationThe 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 keeps counter-factual definiteness while introducing a conjectured non-local mechanism in form of the '

quantum potentialThe quantum potential is a central concept of the de Broglie–Bohm formulation of quantum mechanics, introduced by David Bohm in 1952.Initially presented under the name quantum-mechanical potential, subsequently quantum potential, it was later elaborated upon by Bohm and Basil Hiley in its...

', defined as one of the terms of the Schrödinger equation. Some workers in the field have also attempted to formulate hidden variable theories that exploit loopholes in actual experiments, such as the assumptions made in interpreting experimental data, although no theory has been proposed that can reproduce all the results of quantum mechanics.

There are also individual EPR-like experiments that have no local hidden variables explanation. Examples have been suggested by

David BohmDavid Joseph Bohm FRS was an American-born British quantum physicist who contributed to theoretical physics, philosophy, neuropsychology, and the Manhattan Project.-Youth and college:...

and by Lucien Hardy.

### Einstein's hope for a purely algebraic theory

The

Bohm interpretationThe 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 hypothesizes that the state of the universe evolves smoothly through time with no collapsing of quantum wavefunctions. One problem for the

Copenhagen interpretationThe Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

is to precisely define wavefunction collapse. Einstein maintained that quantum mechanics is physically incomplete and logically unsatisfactory. In "The Meaning of Relativity," Einstein wrote, "One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory and must lead to an attempt to find a purely

algebraic theoryThe Haag-Kastler axiomatic framework for quantum field theory, named after Rudolf Haag and Daniel Kastler, is an application to local quantum physics of C*-algebra theory. It is therefore also known as Algebraic Quantum Field Theory...

for the representation of reality. But nobody knows how to find the basis for such a theory."

If time, space, and energy are secondary features derived from a substrate below the

Planck scaleIn particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1019 GeV at which quantum effects of gravity become strong...

, then Einstein's hypothetical algebraic system might resolve the EPR paradox (although Bell's theorem would still be valid).

Edward FredkinEdward Fredkin is an early pioneer of digital physics. In recent work, he uses the term digital philosophy . His primary contributions include his work on reversible computing and cellular automata...

in the

Fredkin Finite Nature HypothesisIn digital physics, the Fredkin Finite Nature Hypothesis states that ultimately all quantities of physics, including space and time, are discrete and finite. All measurable physical quantities arise from some Planck scale substrate for multiverse information processing...

has suggested an informational basis for Einstein's hypothetical algebraic system. If physical reality is totally finite, then the Copenhagen interpretation might be an approximation to an information processing system below the Planck scale.

### "Acceptable theories" and the experiment

According to the present view of the situation, quantum mechanics flatly contradicts Einstein's philosophical postulate that any acceptable physical theory must fulfill "local realism".

In the EPR paper (1935) the authors realised that quantum mechanics was inconsistent with their assumptions, but Einstein nevertheless thought that quantum mechanics might simply be augmented by hidden variables (i.e. variables which were, at that point, still obscure to him), without any other change, to achieve an acceptable theory. He pursued these ideas until the end of his life, in 1955, over twenty years.

In contrast,

John BellJohn 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 :...

, in his 1964 paper, showed that quantum mechanics and the class of hidden variable theories Bell investigated would lead to different experimental results: different by a factor of for certain correlations. So the issue of "acceptability", up to that time mainly concerning theory, finally became experimentally decidable.

There are many

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

, e.g. those of

Alain AspectAlain Aspect is a French physicist noted for his experimental work on quantum entanglement....

and others. They support the predictions of quantum mechanics rather than the class of hidden variable theories Bell investigated. According to

Karl PopperSir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...

these experiments showed that the class of "hidden variables" Bell investigated is erroneous.

### Implications for quantum mechanics

Most physicists today believe that quantum mechanics is correct, and that the EPR paradox is a "paradox" only because classical intuitions do not correspond to physical reality. How EPR is interpreted regarding locality depends on the

interpretation of quantum mechanicsAn 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...

one uses. In the

Copenhagen interpretationThe Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

, it is usually understood that instantaneous wavefunction collapse does occur. However, the view that there is no causal instantaneous effect has also been proposed within the Copenhagen interpretation: in this alternate view, measurement affects our ability to define (and measure) quantities in the physical system, not the system itself. In the

many-worlds interpretationThe 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...

, a kind of locality is preserved, since the effects of irreversible operations such as measurement arise from the relativization of a global state to a subsystem such as that of an observer.

The EPR paradox has deepened our understanding of quantum mechanics by exposing the fundamentally non-classical characteristics of the measurement process. Prior to the publication of the EPR paper, a measurement was often visualized as a physical disturbance inflicted directly upon the measured system. For instance, when measuring the position of an electron, one imagines shining a light on it, thus disturbing the electron and producing the quantum mechanical uncertainties in its position. Such explanations, which are still encountered in popular expositions of quantum mechanics, are debunked by the EPR paradox, which shows that a "measurement" can be performed on a particle without disturbing it directly, by performing a measurement on a distant entangled particle. In fact,

Yakir AharonovYakir Aharonov is an Israeli physicist specializing in quantum physics. He is a Professor of Theoretical Physics and the James J. Farley Professor of Natural Philosophy at Chapman University in California. He is also a distinguished professor in Perimeter Institute.He also serves as a professor...

and his collaborators have developed a whole theory of so-called

Weak measurementWeak measurements are a type of quantum measurement, where the measured system is very weakly coupled to the measuring device. After the measurement the measuring device pointer is shifted by what is called the "weak value". So that a pointer initially pointing at zero before the measurement would...

.

Technologies relying on quantum entanglement are now being developed. In

quantum cryptographyQuantum 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...

, entangled particles are used to transmit signals that cannot be

eavesdroppedEavesdropping is the act of secretly listening to the private conversation of others without their consent, as defined by Black's Law Dictionary...

upon without leaving a trace. In quantum computation, entangled quantum states are used to perform computations in

parallelParallel computing is a form of computation in which many calculations are carried out simultaneously, operating on the principle that large problems can often be divided into smaller ones, which are then solved concurrently . There are several different forms of parallel computing: bit-level,...

, which may allow certain calculations to be performed much more quickly than they ever could be with classical computers.

## Mathematical formulation

The above discussion can be expressed mathematically using the

quantum mechanical formulation of spinIn 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,...

. The spin degree of freedom for an electron is associated with a two-dimensional complex

Hilbert spaceThe 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...

H, with each quantum state corresponding to a vector in that space. The operators corresponding to the spin along the x, y, and z direction, denoted S

_{x}, S

_{y}, and S

_{z} respectively, can be represented using the

Pauli matricesThe Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...

:

where

stands for Planck's constant divided by 2π.

The eigenstates of S

_{z} are represented as

and the eigenstates of S

_{x} are represented as

The Hilbert space of the electron pair is

, the

tensor productIn mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

of the two electrons' Hilbert spaces. The spin singlet state is

where the two terms on the right hand side are what we have referred to as state I and state II above.

From the above equations, it can be shown that the spin singlet can also be written as

where the terms on the right hand side are what we have referred to as state Ia and state IIa.

To illustrate how this leads to the violation of local realism, we need to show that after Alice's measurement of S

_{z} (or S

_{x}), Bob's value of S

_{z} (or S

_{x}) is uniquely determined, and therefore corresponds to an "element of physical reality". This follows from the principles of

measurement in quantum mechanicsThe 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....

. When S

_{z} is measured, the system state ψ collapses into an eigenvector of S

_{z}. If the measurement result is +z, this means that immediately after measurement the system state undergoes an orthogonal projection of ψ onto the

space of states of the form

For the spin singlet, the new state is

Similarly, if Alice's measurement result is -z, the system undergoes an orthogonal projection onto

which means that the new state is

This implies that the measurement for S

_{z} for Bob's electron is now determined. It will be -z in the first case or +z in the second case.

It remains only to show that S

_{x} and S

_{z} cannot simultaneously possess definite values in quantum mechanics. One may show in a straightforward manner that no possible vector can be an eigenvector of both matrices. More generally, one may use the fact that the operators do not commute,

along with the Heisenberg uncertainty relation

## See also

- 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...

- Bell state
The Bell states are a concept in quantum information science and represent the simplest possible examples of entanglement. They are named after John S. Bell, as they are the subject of his famous Bell inequality. An EPR pair is a pair of qubits which jointly are in a Bell state, that is, entangled...

- Bell's theorem
In theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...

- CHSH Bell test
- Coherence (physics)
In physics, coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave....

- Counter-factual definiteness
- Fredkin Finite Nature Hypothesis
In digital physics, the Fredkin Finite Nature Hypothesis states that ultimately all quantities of physics, including space and time, are discrete and finite. All measurable physical quantities arise from some Planck scale substrate for multiverse information processing...

- Ghirardi-Rimini-Weber theory
The Ghirardi–Rimini–Weber theory, or GRW, is a collapse theory in quantum mechanics. GRW differs from other collapse theories by proposing that wave function collapse happens spontaneously. GRW is an attempt to avoid the measurement problem in quantum mechanics...

- 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...

- 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...

- 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....

- 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...

- 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....

- Measurement problem
The measurement problem in quantum mechanics is the unresolved problem of how wavefunction collapse occurs. The inability to observe this process directly has given rise to different interpretations of quantum mechanics, and poses a key set of questions that each interpretation must answer...

- Penrose interpretation
The Penrose interpretation is a prediction of Sir Roger Penrose about the relationship between quantum mechanics and general relativity. Penrose proposes that a quantum state remains in superposition until the difference of space-time curvature attains a significant level...

- Philosophy of information
The philosophy of information is the area of research that studies conceptual issues arising at the intersection of computer science, information technology, and philosophy.It includes:...

- Philosophy of physics
In philosophy, the philosophy of physics studies the fundamental philosophical questions underlying modern physics, the study of matter and energy and how they interact. The philosophy of physics begins by reflecting on the basic metaphysical and epistemological questions posed by physics:...

- Pondicherry interpretation
Born and raised in Germany, Ulrich Mohrhoff joined the Sri Aurobindo International Centre of Education , Pondicherry , a department of the Sri Aurobindo Ashram as an undergraduate student in 1972. From 1974 to 1978 he studied physics at the University of Göttingen, Germany, and at the Indian...

- Popper's experiment
Popper's experiment is an experiment proposed by the 20th century philosopher of science Karl Popper, an advocate of an objective interpretation of quantum mechanics. He wanted to test the Copenhagen interpretation, a popular subjectivist interpretation of quantum mechanics...

- 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...

- 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 gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

- Quantum information
In quantum mechanics, quantum information is physical information that is held in the "state" of a quantum system. The most popular unit of quantum information is the qubit, a two-level quantum system...

- Quantum pseudo-telepathy
Quantum pseudo-telepathy is a phenomenon in quantum game theory resulting in anomalously high success rates in coordination games between separated players. These high success rates would require communication between the players in a purely classical world; however, the game is set up such that...

- Quantum teleportation
Quantum teleportation, or entanglement-assisted teleportation, is a process by which a qubit can be transmitted exactly from one location to another, without the qubit being transmitted through the intervening space...

- Quantum Zeno effect
The quantum Zeno effect is a name coined by George Sudarshan and Baidyanath Misra of the University of Texas in 1977 in their analysis of the situation in which an unstable particle, if observed continuously, will never decay. One can nearly "freeze" the evolution of the system by measuring it...

- Sakurai's Bell inequality
The intention of a Bell inequality is to serve as a test of local realism or local hidden variable theories as against quantum mechanics, applying Bell's theorem, which shows them to be incompatible. Not all the Bell's inequalities that appear in the literature are in fact fit for this purpose...

- Synchronicity
Synchronicity is the experience of two or more events that are apparently causally unrelated or unlikely to occur together by chance and that are observed to occur together in a meaningful manner...

- Wave function collapse
- Wheeler-Feynman absorber theory
- Zero-point field

### Selected papers

- A. Fine, Hidden Variables, Joint Probability, and the Bell Inequalities. Phys. Rev. Lett. 48, 291 (1982).http://prola.aps.org/abstract/PRL/v48/i5/p291_1
- A. Fine, Do Correlations need to be explained?, in Philosophical Consequences of Quantum Theory: Reflections on Bell's Theorem, edited by Cushing & McMullin (University of Notre Dame Press, 1986).
- L. Hardy, Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett.
**71** 1665 (1993).http://prola.aps.org/abstract/PRL/v71/i11/p1665_1
- M. Mizuki, A classical interpretation of Bell's inequality. Annales de la Fondation Louis de Broglie
**26** 683 (2001).
- P. Pluch, "Theory for Quantum Probability", PhD Thesis University of Klagenfurt (2006)
- M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe and D. J. Wineland, Experimental violation of a Bell's inequality with efficient detection, Nature
**409**, 791-794 (15 February 2001). http://www.nature.com/nature/journal/v409/n6822/full/409791a0.html
- M. Smerlak, C. Rovelli, Relational EPR http://arxiv.org/abs/quant-ph/0604064

### Books

- John S. Bell (1987) Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press. ISBN 0-521-36869-3.
- Arthur Fine
Arthur Fine is an American philosopher of science teaching at the University of Washington . Before moving to UW he taught for many years at Northwestern University and, before that, at Cornell University and the University of Illinois at Chicago...

(1996) The Shaky Game: Einstein, Realism and the Quantum Theory, 2nd ed. Univ. of Chicago Press.
- J.J. Sakurai, J. J. (1994) Modern Quantum Mechanics. Addison-Wesley: 174–187, 223-232. ISBN 0-201-53929-2.
- Selleri, F. (1988) Quantum Mechanics Versus Local Realism: The Einstein-Podolsky-Rosen Paradox. New York: Plenum Press. ISBN 0-306-42739-7
- Leon Lederman, L., Teresi, D. (1993). The God Particle: If the Universe is the Answer, What is the Question? Houghton Mifflin Company, pages 21, 187 to 189.
- John Gribbin
John R. Gribbin is a British science writer and a visiting Fellow in astronomy at the University of Sussex.- Biography :John Gribbin graduated with his bachelor's degree in physics from the University of Sussex in 1966. Gribbin then earned his master of science degree in astronomy in 1967, also...

(1984) In Search of Schroedinger's Cat. Black Swan. ISBN 9780552125550

.

## External links