vol III Development:

Chapter 4: Physics

page 9: Entanglement

The mathematical machinery of quantum mechanics is remarkably simple and compact yet it carries many surprises, not least of which is entanglement. Here we suggest that entanglement reflects the underlying unity of the Universe as it grows from the initial singularity. Quantum entanglement - Wikipedia

We assume that the initial singularity can be modelled as a system embedded in a zero dimensional complex Hilbert space. Quantum mechanics holds trivially in this system, and there is really nothing to say about it, any more than we can say anything about the classical Christian God, which is considered to be pure actuality (actus purus) and totally simple (omnino simplex). We may think of the zero dimensional Hilbert space as a space of one point. Space and 'point in space' are identical. There is no room to move. This is consistent with the ancient notion that God is the 'fullness of being' or pleroma. Aquinas 20, Pleroma - Wikipedia

The next step up in complexity takes us to a quantum system in one dimensional Hilbert space. This system is equivalent to a complex 'line' which is (by convention) represented on a two dimensional plane, the complex plane. Complex plane - Wikipedia.

This space represents a physical system whose energy is proportional to the rate of change of the phase of a unit vector represented on this plane. This energy is represented by the Einstein-Planck relation, E = hf. If we represent the phase of the vector by φ we can write E = ℏ dφ/dt. We imagine that this system has a countable infinity of states corresponding to different frequencies which can be placed into correspondence with the natural numbers. Planck-Einstein relation - Wikipedia

The next step in complexity is to a two state system whose behaviour is modelled by unit vectors in a two dimensional complex Hilbert space. The total energy of the system is the sum of the energies associated with each of these two vectors, often named qubits, |0> and |1>. The state of this system is represented by the expression ψ = a|0>+b|1>, where + is vector addition. To maintain the normalization of ψ we require that the complex amplitudes a and b obey the equation a² + b² = 1. Qubit - Wikipedia

This process of complexification can continue through Hilbert spaces with countable and transfinite dimensions. The fundamental axioms of quantum mechanics are indifferent to the complexity of the system. We see each of these new dimensions as a new fixed point in the divine dynamics.

Einstein, Podolsky and Rosen

Although he won a Nobel Prize for postulating that electromagnetic radiation propagates as the particles we call photons, Einstein retained a life-long suspicion of quantum mechanics. He felt that it was incomplete, in that the quantum mechanical formalism could specify the nature of quantum events and their probability but it was unable to predict exactly when or where something would happen. Nobelprize.org

Einstein, Podolsky and Rosen (EPR) wrote a paper intending to point out this incompleteness. In it they identified two remarkable properties of quantum mechanics, entanglement and non-locality, both of which suggest that the quantum mechanical structure of the Universe antecedes the familiar structure of space and time. Einstein, Podolsky and Rosen

EPR equate reality with predictability: If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.

They begin with the uncertainty principle, which may be written ΔpΔx ≈ ΔEΔt ≈ ℏ. This implies that if we know the momentum p of a system exactly we can have no knowledge of the position, x. Energy, E and time t have a similar relationship.

Quantum measurements are made with measurement operators, M. These operators, applied to a quantum system, yield the eigenvalues corresponding to the orthogonal eigenvectors which form the basis (fixed points) of the operator. Two operators with the same basis are said to commute. Different quantities, measured by different commuting operators, exist and can all be measured at the same time.

The uncertainty principle enters when we wish to make measurements with non-commuting operators, like the energy and time operators or the momentum and position operators. EPR conclude from this that 'either (1) the quantum mechanical description of reality given by the wave functions is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality.'

They then go on to show that the assumption that all the information about a quantum system is contained in the wave function leads to a contradiction. To do this they use quantum theory to model the situation where two systems whose states are known are allowed to interact for a while and are then separated. The quantum theory enables us to calculate the joint state of the two systems both when they are together and when they are separated, but the theory provides no way to calculate their individual states after separation. Further observations are needed to obtain this information.

Further, the formalism shows that a measurement on one system enables us to predict with certainty the outcome of a measurement on the other system even though they are separated. This phenomenon is known as entanglement. EPR concluded that 'no reasonable definition of reality coud be expected to permit this.' It turns out, however, that the quantum mechanical description has become established as the new reasonable definition of reality.

Entanglement

Quantum mechanical entanglement establishes a correlation between states which has no equivalent in classical physics. Entanglement introduces new ways of counting in statistics, so that we must extend classical statistical mechanics to include quantum statistical behaviour.

Classical statistical mechanics assumes that all states of an isolated system are equiprobable, that is they are uncorrelated with one another, like successive throws of a fair coin. This is the approach used in classical probability theory and classical statistical mechanics. Cercignani

Quantum mechanics introduces two new statistical schemes. Fermi-Dirac statistics apply to fermions, whose distinguishing feature is that two fermions in the same state cannot occupy the same region of space. This is known as the Pauli exclusion principle, and is responsible for the extended spatial structure of complex particles. Bose-Einstein statistics apply to bosons which have the opposite feature, a tendency to flock together into the same state. Fermi-Dirac statistics - Wikipedia, Bose-Einstein statistics - Wikipedia

Here we exemplify entanglement and its consequences using qubits, abstract two state quantum systems formed by analogy with the binary digits (bits) of classical computing. A qubit is represented by a vector in a two dimensional Hilbert space with orthonormal basis vectors |0> and |1>. Orthonormal basis - Wikipedia

We begin with two fermions (electrons) in a singlet state, that is a state with zero spin. Electrons have two spin states which are often called up and down. In the singlet state, one electron has spin up, the other down, so that the total spin is zero. Singlet - Wikipedia

Quantum theory shows that if two particles that have been in communication, such as electrons that were once in a singlet state, are spatially separated, they behave, when observed, as though they are still in contact. In the case of electrons, this means that if one electron is observed to be spin up, the other will be observed to be spin down no matter how far apart they are. Einstein dubbed this phenomenon 'spooky action at a distance'.

We write |q > = a |0> + b |1> where a and b are complex numbers constrained by the normalization condition |a |² + |b |² = 1. Physically a qubit may be realized by any two state quantum system like the spin of an electron.

We assume that two qubits a and b having once been entangled in a singlet state have been carried far apart by their owners, Alice and Bob. Alice and Bob observe their qubits in a way that allows no communication between their observations, that is they are separated in a spacelike way.

Later, they compare their results. Quantum mechanics predicts a perfect correlation. Given the singlet state, if Alice sees |0>, Bob sees (|1>) and vice versa. Bell showed in 1964 that there was no reasonable classical explanation of this correlation, although it flows naturally from the quantum formalism. Bell, d'Espagnat

Bell's result has been verified by experiment, and it has been found that this action at a distance operates at many times the velocity of light. Pan, Salart, Juan Yin et al: Bounding the speed of 'spooky action at a distance'

Quantum mechanics is prior to space

Field theories are attempts to avoid the assumption of action at a distance where distance is understood in the ordinary spatial sense or in the relativistic sense of ‘spacelike distance’. The more powerful notion of logical continuity is not specifically related to geometric continuity.

Being networked creatures we intuitively apply logical continuity in our conversations. The spatial positioning of people conversing is generally irrelevant. On the other hand to make sense, the elements of the conversation must be time ordered from the point of view of each person, which implies that there can be no conversation across a spacelike separation. Spacetime - Wikipedia

The experiments of Pan, Salart and Yin referred to above have demonstrated that entangled particles could act upon one another at a distance even though their separation was spacelike, requiring something greater than the velocity of light to account for their correlation (if it is due to communication). This is called quantum non-locality. Quantum nonlocality - Wikipedia

Let us assume that the velocity of light is finite rather than infinite because of the delay in the transmission of a photon from point to point due to error preventative encoding. Conversely, we might speculate that communications that cannot go wrong, that is communications that effectively carry no information, might require no encoding and therefore travel at infinite velocity. The observed correlations between entangled photons have been found to propagate at many times c. and the measurements are compatible with infinite velocity, ie instantaneous transmission, over many kilometres.

In practical communication networks a message originating with user A is passed down through the software layers in A’s computer to the physical layer which carries it to B’s machine. It is then passed up through B’s software layers until reaches a form that B can read. By analogy, communication between one system in the Universe must pass down to the ultimate physical layer (which we might identify with the structureless initial singularity) then up again to the peer system receiving the message.

It may be that the simplicity (high symmetry) in the lower layers of this network make encoding for error prevention unnecessary, so that instantaneous communication is possible.

Quantum entanglement and the network

We assume that systems can only interact when the space-time interval between them is zero. We can safely say that everything is to some extent entangled. This entanglement reflects the common descent of all quantum mechanical states from one primordial state. Let us guess that this universal symmetry lies deeper in the universal network than space-time and gravitation.

We take the Universe to be pure action. Symmetry, on the other hand, is absence of action, so that we understand symmetries to be the boundaries of the Universe, the borderlines between action within the Universe and inaction (ie nothing) outside it. Let us say that the founding symmetry of quantum mechanics is conservation of action.

In network terms, we understand a quantum of action as the execution of a computation, that is an act of encoding or decoding. In subsequent pages we will explore the notion that entanglement is a consequence of the conservation of action. In the case cited above of a singlet state, which has zero action, the electron spins must always be opposite to maintain this condition.

Because we are working at a layer in the Universal network beneath space (momentum) the conservation of action takes place without reference to any spatial distance that may exist between the elements of the singlet in a higher layer of the network.

(revised 21 May 2016)

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Further reading

Books

Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)

Bell, John S, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987 Jacket: JB ... is particularly famous for his discovery of a crucial difference between the predictions of conventional quantum mechanics and the implications of local causality ... This work has played a major role in the development of our current understanding of the profound nature of quantum concepts and of the fundamental limitations they impose on the applicability of classical ideas of space, time and locality.  Amazon   back

Cercignani, Carlo, Ludwig Boltzmann: The Man Who Trusted Atoms, Oxford University Press, USA 2006 'Cercignani provides a stimulating biography of a great scientist. Boltzmann's greatness is difficult to state, but the fact that the author is still actively engaged in research into some of the finer, as yet unresolved issues provoked by Boltzmann's work is a measure of just how far ahead of his time Boltzmann was. It is also tragic to read of Boltzmann's persecution by his contemporaries, the energeticists, who regarded atoms as a convenient hypothesis, but not as having a definite existence. Boltzmann felt that atoms were real and this motivated much of his research. How Boltzmann would have laughed if he could have seen present-day scanning tunnelling microscopy images, which resolve the atomic structure at surfaces! If only all scientists would learn from Boltzmann's life story that it is bad for science to persecute someone whose views you do not share but cannot disprove. One surprising fact I learned from this book was how research into thermodynamics and statistical mechanics led to the beginnings of quantum theory (such as Planck's distribution law, and Einstein's theory of specific heat). Lecture notes by Boltzmann also seem to have influenced Einstein's construction of special relativity. Cercignani's familiarity with Boltzmann's work at the research level will probably set this above other biographies of Boltzmann for a very long time to come.' Dr David J Bottomley   Amazon   back

Nielsen, Michael A, and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2000 Review: A rigorous, comprehensive text on quantum information is timely. The study of quantum information and computation represents a particularly direct route to understanding quantum mechanics. Unlike the traditional route to quantum mechanics via Schroedinger's equation and the hydrogen atom, the study of quantum information requires no calculus, merely a knowledge of complex numbers and matrix multiplication. In addition, quantum information processing gives direct access to the traditionally advanced topics of measurement of quantum systems and decoherence.' Seth Lloyd, Department of Quantum Mechanical Engineering, MIT, Nature 6876: vol 416 page 19, 7 March 2002.  Amazon   back

Papers

d'Espagnat, Bernard, "Quantum theory and reality", Scientific American, 241, 5, November 1979, page 128-140. 'Most particles or aggregates of particles that are ordinarily regarded as separate objects have interacted at some time in the past with other objects. The violation of separability seems to imply that in some sense all these objects constitute an indivisible whole. Perhaps in such a world the concept of an independently existing reality can reatain some meaning, but it will be an altered meaning and one remove from everyday experience.' (page 140). back

Pan, Jian-Wei, et al, "Experimental test of quantum nonlocality in three-photon Greenberger_horne-Zeilinger entanglement", Nature, 403, 6769, 3 February 2000, page 515-519. 'The results of three specific experiments, involving measurements of polarisation correlations between three photons lead to predictions for a fourth experiment; quantum physical predictions are mutually contradictory with expectations based on local realism. We find the results of the fourth experiment to be in agreement with the quantum prediction and in striking conflict with local realism'. back

Salart, Daniel, et al, "Testing the speed of 'spooky action at a distance'", Nature, 454, , 14 August 2008, page 861-864. 'Correlations are generally described by one of two mechanisms: either a first event influences a second one by sending information encoded in bosons or other physical carriers, or the correlated events have some common causes in their shared history. Quantum physics predicts an entirely different kind of cause for some correlations, named entanglement. This reveals itself in correlations that violate Bell inequalities (implying that they cannot be described by common causes) between space-like separated events (implying that they cannot be described by classical communication). Many Bell tests have been performed, and loopholes related to locality and detection have been closed in several independent experiments. It is still possible that a first event could influence a second, but the speed of this hypothetical influence (Einstein's 'spooky action at a distance') would need to be defined in some universal privileged reference frame and be greater than the speed of light. Here we put stringent experimental bounds on the speed of all such hypothetical influences. We performed a Bell test over more than 24 hours between two villages separated by 18 km and approximately east–west oriented, with the source located precisely in the middle. We continuously observed two-photon interferences well above the Bell inequality threshold. Taking advantage of the Earth's rotation, the configuration of our experiment allowed us to determine, for any hypothetically privileged frame, a lower bound for the speed of the influence. For example, if such a privileged reference frame exists and is such that the Earth's speed in this frame is less than 10-3 times that of the speed of light, then the speed of the influence would have to exceed that of light by at least four orders of magnitude.. back

Vedral, Vlatko, "Quantifying entanglement in macroscopic systems", Nature, 453, 7198, 19 June 2008, page 1004 - 1007. 'Traditionally, entangement was considered to be a quirk of microscopic objects that defied a common-sense explanation. Now, however, entanglement is recognized to be ubiquitous and robust. With the realization that entanglement can occur in macroscopic systems -- and with the development of experiments aimed at exploiting this fact -- new tools are required to define and quantify entanglement beyond the original microscopic framework. '. back

Links

Action at a distance - Wikipedia, Action at a distance - Wikipedia, the free encyclopedia, 'In physics, action at a distance is the concept that an object can be moved, changed, or otherwise affected without being physically touched (as in mechanical contact) by another object. That is, it is the nonlocal interaction of objects that are separated in space. 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 objects. For example, Coulomb's law and the law of universal gravitation are such early theories. More generally "action at a distance" describes the failure of early atomistic and mechanistic theories which sought to reduce all physical interaction to collision. The exploration and resolution of this problematic phenomenon led to significant developments in physics, from the concept of a field, to descriptions of quantum entanglement and the mediator particles of the Standard Model.' back

Alan Turing, On Computable Numbers, with an application to the Entscheidungsproblem, 'The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by some finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable of a real or computable variable, computable predicates and so forth. . . . ' back

Aquinas 20, Summa I, 3, 7: Whether God is altogether simple? , 'I answer that, The absolute simplicity of God may be shown in many ways. First, from the previous articles of this question. For there is neither composition of quantitative parts in God, since He is not a body; nor composition of matter and form; nor does His nature differ from His "suppositum"; nor His essence from His existence; neither is there in Him composition of genus and difference, nor of subject and accident. Therefore, it is clear that God is nowise composite, but is altogether simple. . . . ' back

Bell state - Wikipedia, Bell state - Wikipedia, the free encyclopedia, '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. . . . A Bell state is defined as a maximally entangled quantum state of two qubits. The qubits are usually thought to be spatially separated (held by Alice and Bob, respectively, to use quantum cryptography terms). Nevertheless they exhibit perfect correlations which cannot be explained without quantum mechanics.' back

Bose-Einstein statistics - Wikipedia, Bose-Einstein statistics - Wikipedia, the free encyclopedia, 'In statistical mechanics, Bose–Einstein statistics (or more colloquially B–E statistics) determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.' back

Cartesian product - Wikipedia, Cartesian product - Wikipedia, the free encyclopedia, 'In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept. Specifically, the Cartesian product of two sets X (for example the points on an x-axis) and Y (for example the points on a y-axis), denoted X ? Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y (e.g. the whole of the x-y plane):. . . A Cartesian product of two finite sets can be represented by a table, with one set as the rows and the other as the columns, and forming the ordered pairs, the cells of the table, by choosing the element of the set from the row and the column. back

Complex plane - Wikipedia, Complex plane - Wikipedia, the free encuclopedia, 'In mathematics, the complex plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.' back

Counterfactual definiteness - Wikipedia, Counterfactual definiteness - Wikipedia, the free enyclopedia, 'In some interpretations of quantum mechanics, counterfactual definiteness (CFD) is the ability to speak with meaning of the definiteness of the results of measurements that have not been performed (i.e. the ability to assume the existence of objects, and properties of objects, even when they have not been measured). A macroscopic example of CFD would be the assumption -without measurement- that a ball, thrown into the air, will return to the Earth due to gravity. CFD says that if a phenomenon (the return of an airborne ball to the Earth) has been reproducibly measured in the past, one can safely assume its presence in the future without having to refer to additional measurement events for proof of its existence. More rigorously, an interpretation of quantum mechanics satisfies CFD if it includes in the statistical population of measurement results, those measurements which are counter-factual by virtue of their being excluded by the quantum mechanical prohibition on simultaneous measurement of certain pairs of properties.' back

Einstein, Podolsky and Rosen, Can the Quantum Mechanical Description of Physical Reality be Considered Complete?, A PDF of the classic paper. 'In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false, One is thus led to conclude that the description of reality given by the wave function is not complete.' back

EPR Paradox - Wikipedia, EPR Paradox - Wikipedia, the free encyclopedia, '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.' back

Fermi-Dirac statistics - Wikipedia, Fermi-Dirac statistics - Wikipedia, the fre encyclopedia, 'In statistical mechanics, Fermi-Dirac statistics is a particular case of particle statistics developed by Enrico Fermi and Paul Dirac that determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. In other words, it is the distribution of the probabilities that each possible energy levels is occupied by a fermion. back

Gödel's incompleteness theorems - Wikipedia, Gödel's incompleteness theorems - Wikipedia, 'Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, a corollary of the first, shows that such a system cannot demonstrate its own consistency.' back

Juan Yin et al, Bounding the speed of 'spooky action at a distance', 'In the well-known EPR paper, Einstein et al. called the nonlocal correlation in quantum entanglement as `spooky action at a distance'. If the spooky action does exist, what is its speed? All previous experiments along this direction have locality loopholes and thus can be explained without having to invoke any `spooky action' at all. Here, we strictly closed the locality loopholes by observing a 12-hour continuous violation of Bell inequality and concluded that the lower bound speed of `spooky action' was four orders of magnitude of the speed of light if the Earth's speed in any inertial reference frame was less than 10^(-3) times of the speed of light.' back

Nobelprize.org, The Nobel prize in Physics 1921: Albert Einstein, 'The Nobel Prize in Physics 1921 was awarded to Albert Einstein "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect".' back

Orthonormal basis - Wikipedia, Orthonormal basis - Wikipedia, the free encyclopedia, 'In mathematics, an orthonormal basis of an inner product space V (i.e., a vector space with an inner product), or in particular of a Hilbert space H, is a set of elements whose span is dense in the space, in which the elements are mutually orthogonal and of magnitude 1. Elements in an orthogonal basis do not have to have a magnitude of 1 but must be mutually perpendicular. It is easy to change the vectors in an orthogonal basis by scalar multiples to get an orthonormal basis, and indeed this is a typical way that an orthonormal basis is constructed.' back

Outer product - Wikipedia, Outer product - Wikipedia, the free encyclopedia, 'Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. The cardinality of these operations is that of cartesian products.' back

Planck-Einstein relation - Wikipedia, Planck-Einstein relation - Wikipedia, the free encyclopedia, 'The Planck–Einstein relation. . . refers to a formula integral to quantum mechanics, which states that the energy of a photon (E) is proportional to its frequency (ν). E = hν. The constant of proportionality, h, is known as the Planck constant.' back

Pleroma - Wikipedia, Pleroma - Wikipedia, the free encyclopedia, 'Pleroma (Greek πλήρωμα) generally refers to the totality of divine powers. The word means fullness from πληρόω ("I fill") comparable to πλήρης which means "full", and is used in Christian theological contexts: both in Gnosticism generally, and by St. Paul the Apostle in Colossians 2:9 (the word is used 17 times in the NT). Pleroma is also used in the general Greek language and is used by the Greek Orthodox church in this general form since the word appears in the book of Colossians.' back

Principle of locality - Wikipedia, Principle of locality - Wikipedia, the free encyclopedia, 'In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. Experiments have shown that quantum mechanically entangled particles must violate either the principle of locality or the form of philosophical realism known as counterfactual definiteness. . . . ' back

Quantum entanglement - Wikipedia, Quantum entanglement - Wikipedia, the free encyclopedia, 'Quantum entanglement, also called the quantum non-local connection, is a possible property of a quantum mechanical state of a system of two or more objects in which the quantum states of the constituting objects are linked together so that one object can no longer be adequately described without full mention of its counterpart&mdasheven if the individual objects are spatially separated in a spacelike manner. This interconnection leads to non-classical correlations between observable physical properties of remote systems, often referred to as nonlocal correlations.' back

Quantum mechanics - Wikipedia, Quantum mechanics - Wikipedia, the free encyclopedia, 'Quantum mechanics, also known as quantum physics or quantum theory, is a theory of physics providing a mathematical description of the interaction of matter and energy.' back

Quantum nonlocality - Wikipedia, Quantum nonlocality - Wikipedia, the free encyclopedia, 'In theoretical physics, quantum nonlocality is the phenomenon by which the measurements made at a microscopic level necessarily refute one or more notions (often referred to as local realism) that are regarded as intuitively true in classical mechanics. Rigorously, quantum nonlocality refers to quantum mechanical predictions of many-system measurement correlations that cannot be simulated by any local hidden variable theory. Many entangled quantum states produce such correlations when measured, as demonstrated by Bell's theorem.' back

Qubit - Wikipedia, Qubit - Wikipedia, the free encyclopedia, 'A quantum bit, or qubit . . . is a unit of quantum information. That information is described by a state vector in a two-level quantum mechanical system which is formally equivalent to a two-dimensional vector space over the complex numbers. Benjamin Schumacher discovered a way of interpreting quantum states as information. He came up with a way of compressing the information in a state, and storing the information on a smaller number of states. This is now known as Schumacher compression. In the acknowledgments of his paper (Phys. Rev. A 51, 2738), Schumacher states that the term qubit was invented in jest, during his conversations with Bill Wootters.' back

Singlet - Wikipedia, Singlet - Wikipedia, the free encyclopedia, 'In theoretical physics, a singlet usually refers to a one-dimensional representation (e.g. a particle with vanishing spin). It may also refer to two or more particles prepared in a correlated state, such that the total angular momentum of the state is zero. Singlets frequently occur in atomic physics as one of the two ways in which the spin of two electrons can be combined; the other being a triplet. A single electron has spin 1/2, and transforms as a doublet, that is, as the fundamental representation of the rotation group SU(2). The product of two doublet representations can be decomposed into the sum of the adjoint representation (the triplet) and the trivial representation, the singlet. More prosaically, a pair of electron spins can be combined to form a state of total spin 1 and a state of spin 0. The singlet state formed from a pair of electrons has many peculiar properties, and plays a fundamental role in the EPR paradox and quantum entanglement' back

Spacetime - Wikipedia, Spacetime - Wikipedia, the free encyclopedia, 'In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of the fourth dimension. According to Euclidean space perception, the universe has three dimensions of space, and one dimension of time. By combining space and time into a single manifold, physicists have significantly simplified a large amount of physical theory, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels.' back

Tensor product - Wikipedia, Tensor product - Wikipedia, the free encyclopedia, 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, among many other structures or objects. In each case the significance of the symbol is the same: the most general bilinear operation. In some contexts, this product is also referred to as outer product.' back