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 Quantum physics $Delta x , Delta p ge frac{hbar}{2}$ Quantum mechanics Introduction to... Mathematical formulation of... Fig. ... Quantum mechanics (QM, or quantum theory) is a physical science dealing with the behaviour of matter and energy on the scale of atoms and subatomic particles / waves. ... The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ... Fundamental concepts Decoherence · Interference Uncertainty · Exclusion Transformation theory Ehrenfest theorem · Measurement In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior - a feature of classical physics - and give the appearance of wavefunction collapse. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ... The term transformation theory refers to a procedure used by P. A. M. Dirac in his early formulation of quantum theory, from around 1927. ... The Ehrenfest theorem, named after Paul Ehrenfest, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... Experiments Double-slit experiment Davisson-Germer experiment Stern–Gerlach experiment EPR paradox · Popper's experiment Schrödinger's cat Double-slit diffraction and interference pattern The double-slit experiment consists of letting light diffract through two slits, which produces fringes or wave-like interference patterns on a screen. ... In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow moving electrons at a crystalline Nickel target. ... In quantum mechanics, the Sternâ€“Gerlach experiment, named after Otto Stern and Walther Gerlach, is a celebrated experiment in 1920 on deflection of particles, often used to illustrate basic principles of quantum mechanics. ... Poppers experiment is an experiment proposed by the 20th century philosopher of science Karl Popper, to test the standard interpretation (the Copenhagen interpretation) of Quantum mechanics. ... SchrÃ¶dingers Cat: When the nucleus (bottom left) decays, the Geiger counter (bottom centre) may sense it and trigger the release of the gas. ... Equations Schrödinger equation Pauli equation Klein-Gordon equation Dirac equation For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... The Pauli equation is a SchrÃ¶dinger equation which handles spin. ... The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the SchrÃ¶dinger equation. ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-Â½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... Advanced theories Quantum field theory Wightman axioms Quantum electrodynamics Quantum chromodynamics Quantum gravity Feynman diagram Quantum field theory (QFT) is the quantum theory of fields. ... In physics the Wightman axioms are an attempt of mathematically stringent, axiomatic formulation of quantum field theory. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... This article does not cite any references or sources. ... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ... Interpretations Copenhagen · Ensemble Hidden variables · Transactional Many-worlds · Consistent histories Quantum logic Consciousness causes collapse It has been suggested that Quantum mechanics, philosophy and controversy be merged into this article or section. ... The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. ... The Ensemble Interpretation, or Statistical Interpretation of Quantum Mechanics, is an interpretation that can be viewed as a minimalist interpretation. ... In physics, a hidden variable theory is urged by a minority of physicists who argue that the statistical nature of quantum mechanics implies that quantum mechanics is incomplete; it is really applicable only to ensembles of particles; new physical phenomena beyond quantum mechanics are needed to explain an individual event. ... The transactional interpretation of quantum mechanics (TIQM) by Professor John Cramer is an unusual interpretation of quantum mechanics that describes quantum interactions in terms of a standing wave formed by retarded (forward in time) and advanced (backward in time) waves. ... The many-worlds interpretation of quantum mechanics or MWI (also known as the relative state formulation, theory of the universal wavefunction, many-universes interpretation, Oxford interpretation or many worlds), is an interpretation of quantum mechanics that claims to resolve all the paradoxes of quantum theory by allowing every possible outcome... In 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. ... In mathematical physics and quantum mechanics, quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. ... Consciousness causes collapse is the theory that observation by a conscious observer is responsible for the wavefunction collapse in quantum mechanics. ... Scientists Planck · Schrödinger Heisenberg · Bohr · Pauli Dirac · Bohm · Born de Broglie · von Neumann Einstein · Feynman Everett · Others â€œPlanckâ€ redirects here. ... Bust of SchrÃ¶dinger, in the courtyard arcade of the main building, University of Vienna, Austria. ... Werner Karl Heisenberg (December 5, 1901 â€“ February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... Niels (Henrik David) Bohr (October 7, 1885 â€“ November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in 1922. ... This article is about Austrian-Swiss physicist Wolfgang Pauli. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... David Bohm. ... Max Born (December 11, 1882 in Breslau â€“ January 5, 1970 in GÃ¶ttingen) was a mathematician and physicist. ... Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892â€“March 19, 1987), was a French physicist and Nobel Prize laureate. ... For other persons named John Neumann, see John Neumann (disambiguation). ... â€œEinsteinâ€ redirects here. ... This article is about the physicist. ... Hugh Everett III (November 11, 1930 â€“ July 19, 1982) was an American physicist who first proposed the many-worlds interpretation(MWI) of quantum physics, which he called his relative state formulation. ... Below is a list of famous physicists. ... This box: view • talk • edit

In quantum mechanics, the EPR paradox is a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory. "EPR" stands for Einstein, Podolsky, and Rosen, who introduced the thought experiment in a 1935 paper to argue that quantum mechanics is not a complete physical theory. Fig. ... In philosophy, physics, and other fields, a thought experiment (from the German Gedankenexperiment) is an attempt to solve a problem using the power of human imagination. ... â€œEinsteinâ€ redirects here. ... In quantum mechanics, the EPR paradox is a thought experiment which demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. ... Nathan Rosen (March 22, 1909 â€“ December 18, 1995) was a physicist. ... 1935 (MCMXXXV) was a common year starting on Tuesday (link will display full calendar). ...

The EPR experiment yields a dichotomy. Either A dichotomy is a division into two non-overlapping or mutually exclusive and jointly exhaustive parts. ...

1. The result of a measurement performed on one part A of a quantum system has a non-local effect on the physical reality of another distant part B, in the sense that quantum mechanics can predict outcomes of some measurements carried out at B or
2. Quantum mechanics is incomplete in the sense that some element of physical reality corresponding to B cannot be accounted for by quantum mechanics (that is, some extra variable is needed to account for it.)

## Quantum Mechanics and its Interpretation GA_googleFillSlot("encyclopedia_square");

During the twentieth century quantum theory proved to be a successful theory, which describes the physical reality of the mesoscopic and microscopic world. Up to now, no method has been found to contradict the predictions made by quantum theory. This is remarkable, since measurement accuracy has increased, and the size of the systems under consideration has decreased at a fast pace. In physics and chemistry, the mesoscopic scale refers to the length scale at which one can reasonably discuss the properties of a material or phenomenon without having to discuss the behavior of individual atoms. ...

Quantum mechanics was developed with the aim of describing atoms and to explain the observed spectral lines in a measurement apparatus. The fact that quantum theory allows for an accurate description of reality is clear from many physical experiments and has probably never been seriously disputed. Interpretations of quantum phenomena are another story.

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. It has been suggested that Quantum mechanics, philosophy and controversy be merged into this article or section. ...

Quantum theory and quantum mechanics do not account for single measurement outcomes in a deterministic way. According to an accepted interpretation of quantum mechanics known as the Copenhagen interpretation, a measurement causes an instantaneous collapse of the wave function describing the quantum system, and the system after the collapse is random. The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ...

The most prominent opponent of the Copenhagen interpretation was Albert Einstein. Einstein did not believe in the idea of genuine randomness in nature, the main argument in the Copenhagen interpretation. In his view, quantum mechanics is incomplete and suggests that there had to be 'hidden' variables responsible for random measurement results. In physics, the hidden variable theory is espoused by a minority of physicists who argue that the statistical nature of quantum mechanics indicates that QM is incomplete. ...

The famous paper "Can quantum mechanical description of physical reality be considered complete?"[1], authored by Einstein, Podolsky and Rosen in 1935, condensed the philosophical discussion into a physical argument. They claim that given a specific experiment, in which the outcome of a measurement could be 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 local, in the sense that they belong to a certain point in spacetime. This element may only be influenced by events which are located in the backward light cone of this point in spacetime. Even though these claims sound reasonable and convincing, they are founded on assumptions about nature which constitute what is now known as local realism. See also: Other events of 1935 List of years in science . ... This article is about the principle of locality in physics. ... For other uses of this term, see Spacetime (disambiguation). ... In special relativity, a light cone is the pattern describing the temporal evolution of a flash of light in Minkowski spacetime. ...

The EPR paradox draws on a phenomenon predicted by quantum mechanics, known as quantum entanglement, to show that measurements performed on spatially separated parts of a quantum system can apparently have an instantaneous influence on one another. This effect is now known as "nonlocal behavior" (or colloquially as "quantum weirdness" or "spooky action at a distance"). In order to illustrate this, let us consider a simplified version of the EPR thought experiment put forth by David Bohm. It has been suggested that Quantum coherence be merged into this article or section. ... A physical theory is said to exhibit nonlocality if, in that theory, it is not possible to treat widely separated systems as independent. ... David Bohm. ...

### Measurements on an entangled state

We have a source that emits pairs of electrons, with one electron sent to destination A, where there is an observer named Alice, and another is sent to destination B, where there is an observer named Bob. According to quantum mechanics, we can arrange our source so that each emitted electron pair occupies a quantum state called a spin singlet. This can be viewed as a quantum superposition of two states, which we call I and II. In state I, electron A has spin pointing upward along the z-axis (+z) and electron B has spin pointing downward along the z-axis (-z). In state II, electron A has spin -z and electron B has spin +z. Therefore, it is impossible to associate either electron in the spin singlet with a state of definite spin. The electrons are thus said to be entangled. The names Alice and Bob are commonly used placeholders for archetypal characters in fields such as cryptography and physics. ... The names Alice and Bob are commonly used placeholders for archetypal characters in fields such as cryptography and physics. ... A quantum state is any possible state in which a quantum mechanical system can be. ... In quantum mechanics, spin singlet is a composite particle whose total internal angular momentum, i. ... Quantum superposition is the application of the superposition principle to quantum mechanics. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... It has been suggested that Quantum coherence be merged into this article or section. ...

The EPR thought experiment, performed with electrons. A source (center) sends electrons toward two observers, Alice (left) and Bob (right), who can perform spin measurements.

There is, of course, nothing special about our choice of the z-axis. For instance, suppose that Alice and Bob now decide to measure spin along the x-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. We'll call these states Ia and IIa. In state Ia, Alice's electron has spin +x and Bob's electron has spin -x. In state IIa, Alice's electron has spin -x and Bob's electron has spin +x. Therefore, if Alice measures +x, the system collapses into Ia, and Bob will get -x. If Alice measures -x, the system collapses into IIa, and Bob will get +x.

In quantum mechanics, the x-spin and z-spin are "incompatible observables", which means that there is a Heisenberg uncertainty principle operating between them: a quantum state cannot possess a definite value for both 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. Furthermore, it is fundamentally impossible to predict which outcome will appear until Bob actually performs the measurement. In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ...

So how does Bob's electron know, at the same time, which way to point if Alice decides (based on information unavailable to Bob) to measure x and also how to point if Alice measures z? Using the usual Copenhagen interpretation rules that say the wave function "collapses" at the time of measurement, there must be action at a distance or the electron must know more than it is supposed to. To make the mixed part quantum and part classical descriptions of this experiment local, we have to say that the notebooks (and experimenters) are entangled and have linear combinations of + and – written in them, like Schrödinger's Cat. SchrÃ¶dingers Cat: When the nucleus (bottom left) decays, the Geiger counter (bottom centre) may sense it and trigger the release of the gas. ...

Incidentally, although we have used spin as an example, many types of physical quantities — what quantum mechanics refers to as "observables" — can be used to produce quantum entanglement. The original EPR paper used momentum for the observable. Experimental realizations of the EPR scenario often use photon polarization, because polarized photons are easy to prepare and measure. In classical mechanics, momentum (pl. ... Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. ...

### Reality and completeness

We will now introduce two concepts used by Einstein, Podolsky, and Rosen, (EPR) which are crucial to their attack on quantum mechanics: (i) the elements of physical reality and (ii) the completeness of a physical theory.

The authors (EPR) did not directly address the philosophical meaning of an "element of physical reality". Instead, they made the assumption that if the value of any physical quantity of a system can be predicted with absolute certainty prior to performing a measurement or otherwise disturbing it, then that quantity corresponds to an element of physical reality. Note that the converse is not assumed to be true; there may be other ways for elements of physical reality to exist, but this will not affect the argument. The philosopher Socrates about to take poison hemlock as ordered by the court. ...

Next, EPR defined a "complete physical theory" as one in which every element of physical reality is accounted for. The aim of their paper was to show, using these two definitions, that quantum mechanics is not a complete physical theory.

Let us see how these concepts apply to the above thought experiment. Suppose Alice decides to measure the value of spin along the z-axis (we'll call this the z-spin.) After Alice performs her measurement, the z-spin of Bob's electron is definitely known, so it is an element of physical reality. Similarly, if Alice decides to measure spin along the x-axis, the x-spin of Bob's electron is an element of physical reality after her measurement.

We have seen that a quantum state cannot possess a definite value for both x-spin and z-spin. If quantum mechanics is a complete physical theory in the sense given above, x-spin and z-spin cannot be elements of reality at the same time. This means that Alice's decision — whether to perform her measurement along the x- or z-axis — has an instantaneous effect on the elements of physical reality at Bob's location. However, this violates another principle, that of locality.

### 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 relativity, which states that information can never be transmitted faster than the speed of light without violating causality. It is generally believed that any theory which violates causality would also be internally inconsistent, and thus deeply unsatisfactory. For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... The ASCII codes for the word Wikipedia represented in binary, the numeral system most commonly used for encoding computer information. ... â€œLightspeedâ€ redirects here. ... Causality describes the relationship between causes and effects, and is fundamental to all natural science, especially physics. ...

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 random; 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 theorem", 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. The word random is used to express lack of order, purpose, cause, or predictability in non-scientific parlance. ... The no cloning theorem is a result of quantum mechanics which forbids the creation of identical copies of an arbitrary unknown quantum state. ...

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 distance". The conclusion they drew was that quantum mechanics is not a complete theory. In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. ...

### 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 theory. In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ... In physics, the hidden variable theory is espoused by a minority of physicists who argue that the statistical nature of quantum mechanics indicates that QM is incomplete. ...

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, of course, there is an (uncountably) infinite number of axes along which Alice and Bob can perform their measurements, so there has 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

Main article: Bell's theorem

In 1964, John Bell showed that the predictions of quantum mechanics in the EPR thought experiment are actually slightly different from the predictions of a very broad class of hidden variable theories. Roughly speaking, quantum mechanics predicts much stronger statistical correlations between the measurement results performed on different axes than the hidden variable theories. These differences, expressed using inequality relations known as "Bell's inequalities", are in principle experimentally detectable. Bells theorem is the most famous legacy of the late Irish phyisicist John Bell. ... Also Nintendo emulator: 1964 (emulator). ... This article or section is not written in the formal tone expected of an encyclopedia article. ... Positive linear correlations between 1000 pairs of numbers. ... This article is about inequalities in mathematics. ...

After the publication of Bell's paper, a variety of experiments were devised to test Bell's inequalities. (As mentioned above, these experiments generally rely on photon polarization measurements.) All the experiments conducted to date have found behavior in line with the predictions of standard quantum mechanics. In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ...

However, Bell's theorem does not apply to all possible "realist" theories. It is possible to construct theories that escape its implications, and are therefore indistinguishable from quantum mechanics, though these theories are generally non-local — they are believed to violate both causality and the rules of special relativity. Some workers in the field have also attempted to formulate local hidden variable theories that exploit loopholes in actual experiments, such as the assumptions made in interpreting experimental data. However, no one has ever been able to formulate a local realist theory that can reproduce all the results of quantum mechanics. In quantum mechanics, a local hidden variable theory is one in which distant events are assumed to have no instantaneous effect on local ones. ... // In Bell test experiments, there may be experimental problems that affect the validity of the experimental findings. ... Contemporary philosophical realism, also referred to as metaphysical realism, is the belief in a reality that is completely ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc. ...

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

### Implications for quantum mechanics

Most physicists today believe that quantum mechanics is correct, and that the EPR paradox is only a "paradox" because classical intuitions do not correspond to physical reality. How EPR is interpreted regarding locality depends on the interpretation of quantum mechanics one uses. In the Copenhagen interpretation, 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 interpretation, 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. It has been suggested that Quantum mechanics, philosophy and controversy be merged into this article or section. ... The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. ... The many-worlds interpretation of quantum mechanics or MWI (also known as the relative state formulation, theory of the universal wavefunction, many-universes interpretation, Oxford interpretation or many worlds), is an interpretation of quantum mechanics that claims to resolve all the paradoxes of quantum theory by allowing every possible outcome...

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. The measurement problem is the key set of questions that every interpretation of quantum mechanics must answer. ...

Technologies relying on quantum entanglement are now being developed. In quantum cryptography, entangled particles are used to transmit signals that cannot be eavesdropped upon without leaving a trace. In quantum computation, entangled quantum states are used to perform computations in parallel, which may allow certain calculations to be performed much more quickly than they ever could be with classical computers. Quantum cryptography is an approach based on quantum physics for secure communications. ... To eavesdrop is to surreptitiously overhear a private conversation. ... Molecule of alanine used in NMR implementation of error correction. ... Parallel computing is the simultaneous execution of the same task (split up and specially adapted) on multiple processors in order to obtain results faster. ...

### The classical approximation

From the point of view of a direct or "many worlds" interpretation, in which classical physics and ordinary language are only approximations to quantum mechanics, it is understandable that insisting on applying the approximation in the same ways all the time leads to strange results. For example, one might expect to be able to use geometric optics to describe the optical properties of a telescope, because it is large with respect to the wavelength of light. However telescopes (especially if space-based) are designed to measure such small angles that wave effects are nonetheless significant. Similarly, when EPR designed their experiment to be sensitive to subtleties of quantum mechanics, they made it sensitive to just how the classical approximation is applied. Classical Newtonian physics has, formally, been replaced by quantum mechanics on the small scale and relativity on the large scale. ... The many-worlds interpretation of quantum mechanics or MWI (also known as the relative state formulation, theory of the universal wavefunction, many-universes interpretation, Oxford interpretation or many worlds), is an interpretation of quantum mechanics that claims to resolve all the paradoxes of quantum theory by allowing every possible outcome... See also list of optical topics. ...

### Retrocausality

The idea suggests that photons, and some other elementary particles, have the ability to communicate backwards in time. Using this form of communication, it is made sure that the action of quantum observation at one place is transferred instantaneously to the entangled particle which is at a distance. Richard Feynman and other prominent physicists once supported this idea as worth considering, but it still remains untested. Currently John Cramer, a physicist at the University of Washington, is researching on it, and work is going on to experimentally verify this idea.[2]

## Mathematical formulation

The above discussion can be expressed mathematically using the quantum mechanical formulation of spin. The spin degree of freedom for an electron is associated with a two-dimensional Hilbert space 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 Sx, Sy, and Sz respectively, can be represented using the Pauli matrices: In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... The Pauli matrices are a set of 2 Ã— 2 complex Hermitian and unitary matrices. ...

$S_x = frac{hbar}{2} begin{bmatrix} 0&11&0end{bmatrix}, quad S_y = frac{hbar}{2} begin{bmatrix} 0&-ii&0end{bmatrix}, quad S_z = frac{hbar}{2} begin{bmatrix} 1&00&-1end{bmatrix}$

where $hbar$ stands for Planck's constant divided by . A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...

The eigenstates of Sz are represented as In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

$left|+zrightrang leftrightarrow begin{bmatrix}10end{bmatrix}, quad left|-zrightrang leftrightarrow begin{bmatrix}01end{bmatrix}$

and the eigenstates of Sx are represented as

$left|+xrightrang leftrightarrow frac{1}{sqrt{2}} begin{bmatrix}11end{bmatrix}, quad left|-xrightrang leftrightarrow frac{1}{sqrt{2}} begin{bmatrix}1-1end{bmatrix}$

The Hilbert space of the electron pair is $H otimes H$, the tensor product of the two electrons' Hilbert spaces. The spin singlet state is In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...

$left|psirightrang = frac{1}{sqrt{2}} bigg(left|+zrightrang otimes left|-zrightrang - left|-zrightrang otimes left|+zrightrang bigg)$

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

$left|psirightrang = frac{1}{sqrt{2}} bigg(left|+ -rightrang - left|- +rightrang bigg)$

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

$left|psirightrang = frac{-1}{sqrt{2}} bigg(left|+xrightrang otimes left|-xrightrang - left|-xrightrang otimes left|+xrightrang bigg)$

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 Sz (or Sx), Bob's value of Sz (or Sx) is uniquely determined, and therefore corresponds to an "element of physical reality". This follows from the principles of measurement in quantum mechanics. When Sz is measured, the system state ψ collapses into an eigenvector of Sz. 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

$left| +z rightrangle otimes left| phirightrangle quad phi in H$

For the spin singlet, the new state is

$left| +z rightrangle otimes left| -z rightrangle.$

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

$left| -z rightrangle otimes left| phirightrangle quad phi in H$

which means that the new state is

$left|-zrightrangle otimes left|+zrightrangle$

This implies that the measurement for Sz 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 Sx and Sz 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, In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g &#8722;1 h &#8722;1 gh, often denoted by [ g, h ]. It is equal to the groups identity if...

$left[ S_x, S_zright] = - ihbar S_y ne 0$

along with the Heisenberg uncertainty relation

$lang (Delta S_x)^2 rang lang (Delta S_z)^2 rang ge frac{1}{4} left|lang left[ S_x, S_zright] rang right|^2$

In quantum mechanics, Bells Theorem states that a Bell inequality must be obeyed under any local hidden variable theory but can in certain circumstances be violated under quantum mechanics (QM). ... The Bell states are a concept in quantum information science and represent the simplest possible examples of entanglement. ... Bells theorem is the most famous legacy of the late Irish phyisicist John Bell. ... Introduction The CHSH Bell test is an application of Bells theorem, intended to distinguish between quantum mechanics (QM) and local hidden variable theories. ... Counterfactual definiteness or CFD is a property of some interpretations of quantum mechanics but not others. ... In quantum mechanics, a local hidden variable theory is one in which distant events are assumed to have no instantaneous effect on local ones. ... In quantum information, quantum teleportation, or entanglement-assisted teleportation, is a technique that transfers a quantum state to an arbitrarily distant location using a distributed entangled state and the transmission of some classical information. ... Synchronicity is the experience of two or more events which occur in a meaningful manner, but which are causally inexplicable to the person or persons experiencing them. ...

## References

### Selected papers

• A. Aspect, Bell's inequality test: more ideal than ever, Nature 398 189 (1999). [3]
• J.S. Bell, On the Einstein-Poldolsky-Rosen paradox, Physics 1 195 (1964).
• J.S. Bell, Bertlmann's Socks and the Nature of Reality. Journal de Physique 42 (1981).
• N. Bohr, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 48, 696 (1935) [4]
• P.H. Eberhard, Bell's theorem without hidden variables. Nuovo Cimento 38B1 75 (1977).
• P.H. Eberhard, Bell's theorem and the different concepts of locality. Nuovo Cimento 46B 392 (1978).
• A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47 777 (1935). [5]
• A. Fine, Hidden Variables, Joint Probability, and the Bell Inequalities. Phys. Rev. Lett. 48, 291 (1982).[6]
• 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).[7]
• 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). [8]
• M. Smerlak, C. Rovelli, Relational EPR [9]

### Books

• J.S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, 1987). ISBN 0-521-36869-3
• J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994), pp. 174-187, 223-232. ISBN 0-201-53929-2
• F. Selleri, Quantum Mechanics Versus Local Realism: The Einstein-Podolsky-Rosen Paradox (Plenum Press, New York, 1988) ISBN 0-306-42739-7
• Roger Penrose, The Road to Reality (Alfred A. Knopf, 2005; Vintage Books, 2006 )

Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... The Road to Reality is a book by the British mathematical physicist Roger Penrose, published in 2004. ...

Results from FactBites:

 The Einstein-Podolsky-Rosen Argument in Quantum Theory (7636 words) EPR is about the interpretation of state vectors ("wave functions") and employs the standard state vector reduction formalism (von Neumann's "projection postulate"). Although the "cat paradox" is usually cited in connection with the problem of quantum measurement (Measurement in Quantum Theory) and treated as a paradox separate from EPR, its origin is here as a compact version of the EPR argument for incompleteness. To go back to the EPR dilemma between locality and completeness, it would appear from the Bell theorem that Einstein's strategy of maintaining locality, and thereby concluding that the quantum description is incomplete, may have fixed on the wrong horn.
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