vol 3: Development
chapter 3: Physics
page 7: Entanglement
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Entanglement
The mathematical machinery of quantum mechanics is remarkably simple and compact yet it carries many surprises, not least of which entanglement. Here we suggest that entanglement maintains the 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.
The next step takes us to a quantum system in one dimensional 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 From this point on, we see the universe growing by communication.
From an axiomatic point of view, quantum mechanics splits neatly into two parts. The first three axioms deal with isolated systems; the second three deal with communicating systems which are seen as observing or measuring one another. Quantum mechanics
Communicating systems obey postulates (iv) and (v). During their interaction, the two isolated systems become one, and states of this new system are represented by a vectors in the tensor product space of the interacting systems (postulate (i')). Tensor product - Wikipedia
Tensor product of Hilbert spaces
The simplest product of two or more sets is the Cartesian product, named for Renee Descartes. Descartes saw that all the points on a plane could be generated as a product of two perpendicular lines, and so on. Cartesian product - Wikipedia
The effect of such a product (like simple multiplication) is to replace each element of one set with all the elements of another. So the point x on the x axis of the Cartesian plane represents all the points on the line passing through x parallel to the y axis. We may say that by creating a Cartesian product the meaning of x has been expanded from a point to a line.
The product of Hilbert spaces works in a similar way. Two one dimensional spaces may be combined to create a two dimensional space. Two one dimensional operators can be changed into two dimensional operators and so on.
The two dimensional space contains points which are not accessible in either of the component one dimensional spaces. By analogy with the Cartesian plane, these are all the points except the x and y axes. We might say that multiplication is an exponentially more powerful method of creating space than addition.
Entanglement
From a classical point of view, one would expect each point in the new space to be equiprobable, and use this as a basis to work out the probabilities of events. This is the approach used in classical probability theory and classical statistical mechanics. Cercignani
The most important difference between quantum and classical mechanics is the method of computing probabilities. Quantum theory shows that if two particles have been in communication, such as electrons that were once in a singlet state, they retain a memory of this encounter when later observed. Singlet - Wikipedia
Attention was first directed to this 'spooky action at a distance' by Einstein, Podolsky and Rosen, Their paper formed part of the early debate about the interpretation and completeness of quantum mechanics. Einstein, Podolsky and Rosen
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 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.
To describe the interaction of n qubits, we require a space with 2n dimensions to which we can assign basis vectors corresponding to ordinary n place binary numbers. The vectors |00>, |01>, |10>, and |11> thus provide a basis for a two qubit space, and we may write a two qubit state as |qq > = a |00> + b |01> + c |10> + d |11>, subject to the normalization requirement that |a |2 + |b |2 + |c |2 + |d |2 = 1.
We represent one of four possible Bell states in this space by
This state cannot be broken down into a product of independent states. It is said to be entangled; in fact, maximally entangled.
We assume that the qubits a and b having once been entangled 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 state above, if Alice sees |0> (|1>), so does Bill. 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. Pan
The unity of the universe
Although the concept of an isolated state is useful theoretically, 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 manifests as space-time and gravitation, the force that controls the large scale structure of spacetime.
This guess is founded on the observation that gravitation is the one force applies equally to all states in the universe It sees only energy. Energy, time and proof
(revised 1 February, 2008)
Further reading
Books
| 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 |
| Newton, Isaac, and Julia Budenz, I. Bernard Cohen, Anne Whitman (Translators), The Principia : Mathematical Principles of Natural Philosophy, University of California Press 1999 This completely new translation, the first in 270 years, is based on the third (1726) edition, the final revised version approved by Newton; it includes extracts from the earlier editions, corrects errors found in earlier versions, and replaces archaic English with contemporary prose and up-to-date mathematical forms. ... The illuminating Guide to the Principia by I. Bernard Cohen, along with his and Anne Whitman's translation, will make this preeminent work truly accessible for today's scientists, scholars, and students. 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 tht 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 expereince.' (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 |
Links
| 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 |
| 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 |
| Quantum entanglement - Wikipedia Quantum entanglement - Wikipedia, the free encyclopedia 'Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. This leads to correlations between observable physical properties of the systems.' back |
| Quantum mechanics - Wikipedia Quantum mechanics - Wikipedia, the free encyclopedia 'Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics. The effects of quantum mechanics are typically not observable on macroscopic scales, but become evident at the atomic and subatomic level. Quantum theory generalizes all classical theories, including mechanics and electromagnetism, and provides accurate descriptions for many previously unexplained phenomena such as black body radiation and stable electron orbits.' 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 |
| Tensor product - Wikipedia Tensor product - Wikipedia, the free encyclopedia 'In mathematics, the tensor product, denoted by x, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. 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. The term "tensor product" is also used in relation to monoidal categories.' back |
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