volume II: Synopsis
section IV: Divine Dynamics
page 29: Quantum information theory
Our current standard model of the world is expressed as quantum field theory. It postulates fields throughout space-time which govern the rate of creation and annihilation of each species of particle at each point in space-time. Quantum field theory arises at the intersection of quantum mechanics and the special theory of relativity.
In the context of a quantum theory of fields in four dimensional space-time, quantum mechanics may be understood as a one dimensional theory independent of space, concerned only with action and energy. This suggests that quantum mechanics antecedes space-time, that is space-time is an emergent phenomenon founded in some way on quantum mechanics. This suggestion is supported by the discovery that quantum mechanics appears to be 'non-local'. Standard model - Wikipedia, Anthony Zee: Quantum Field Theory in a Nutshell,, page 19, Principle of locality - Wikipedia
The world of quantum mechanics comprises a set of state vectors and operators on these vectors, in a complex Hilbert space. Each vector is set of complex numbers whose cardinal is equal to the dimension of the Hilbert space it inhabits. State vectors are normalized, meaning that the inner product of a vector with itself has the absolute value 1. Each vector is a representation of the circle group. The information carried by a vector is thus encoded by its angle with respect to another vector, not by its length. Quantum mechanics - Wikipedia, Circle group - Wikipedia
State vectors are not observable. The observable output of quantum mechanics is derived from the 'Born Rule', which tells us that the probability of a transition between two states is equal to the absolute square of the inner product of the states. The probability of a state being itself is 1, because of the normalization mentioned above. The probability of transition between orthogonal states is 0, since states whose inner product is 0 are by definition orthogonal. Intermediate probabilities are associated with states that are neither parallel nor orthogonal. Born rule - Wikipedia
The dynamics of quantum mechanic is described by operators, which are differential equations which describe rotating state vectors in Hilbert space. As state vectors rotate relative to one another, their inner products change in a periodic manner, so that the probabilities of observing particular events also change periodically, a feature which gives rise to the notion of wave-particle duality.
An experiment with particles like photons or electrons will thus often show an 'interference pattern' similar to that observed when real water waves interact. The rate at which a state rotates is given by its energy, f = E/h, where f is the rate of rotation in cycles per unit time and h is Planck's quantum of action.
Richard Feynman, who had a very strong intuitive grasp of quantum theory, realized that the mathematical mechanism of quantum mechanics could be used to simulate computation. In particular, one can match quantum mechanical operators with the elementary logical functions not, and, or and so on. Since Feynman's time quantum information theory has yielded many interesting results and led some to believe quantum computation be able to solve problems that are intractable with classical computers. Feynman: Feynman Lectures on Computation, Nielsen & Chuang: Quantum Computation and Quantum Information
The principal difficulty with quantum computation lies in the Born rule. The functional analysis upon which quantum mechanics is built is a deterministic mathematical theory yielding definite probabilities which evolve deterministically. Nevertheless, actual events remain random, so it is difficult to exploit the potential of a quantum computer.
The promise of quantum computation is too great to be put off by this difficulty since maybe there are a ways around it. Perhaps the greatest hope is that one quantum operation might perform an infinity of computations in parallel, thus yielding a speed increase over the sequential operations or classical computers. Lov K Grover: Quantum mechanics helps in searching for a needle in a haystack
This hope is based on the fact that a differential equation may have an infinity of solutions (eigenfunctions) which form a superposition. A quantum operator is considered to operate on all the eigenfunctions at once. Further, because the mathematics of quantum mechanics is continuous, some feel that it may encode an infinite amount of data. Nevertheless an observation on such a system yields only the eigenvalue of one eigenfunction. This suggests that we ask whether the superposition is real or simply a mathematical artefact. Does the observation contain all the information that was in the quantum state, or is something lost? Quantum superposition - Wikipedia
The primitive nature of quantum mechanics suggests that in itself it is not very powerful at all. In particular, quantum mechanics does not presuppose the existence of memory, so one might wonder where all the elements of superposition reside. The answer suggested here is that space is memory and that the emergence of space from quantum mechanics can be explained by the existence of fixed points in dynamic systems that map to themselves. Brouwer fixed point theorem - Wikipedia
A mathematical proof, executed by a Turing machine, is a deterministic process. It comprises a series of steps, each of which is bound to follow from the steps before, establishing a logical chain from hypothesis to conclusion. If we assume, for example, Euclid's axioms of geometry, the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides. Mathematical proof - Wikipedia
The steps in a proof are represented by states of a physical system. In the classical pencil and paper approach to mathematics, these steps are represented by marks on paper. In an electronic computer theorem proving system, the states are represented by the presence or absence of electrons in different parts of the physical computer. In the Universe at large, these steps may be represented by particles, that is stationary points or messages in space.
The simplest particle known to us is probably the photon, which is never at rest and has no rest mass. From a massive observer's point of view the Lorentz transformation tells us that a a photon has no extension and no time. The space-time interval between the point of creation of a photon and its point of annihilation is zero. A photon has spin 1, that is one quantum of action, qualifying it as a boson or messenger particle. Finally, photons may have a countable infinity of different states, corresponding to different frequencies or energies. Given these properties, we consider the photon the archetype of memory and the foundation for the emergence of space-time from quantum mechanics. We explore this emergence using the network model. Photon - Wikipedia, Lorentz transformation - Wikipedia
Quantum information theory leads us to understand the universe as an information processing system of enormous power. The problem, from a technological point of view, is that much of this power is expressed at the level of probability amplitudes, represented by complex numbers, which are to a large degree inaccessible from the 'classical' word in which we live. From a computing point of view a complex number is a two dimensional object whose behaviour can be represented by two synchronised logical machines operating in parallel. The beauty of complex numbers is that they are perfect for representing periodic functions. Thus they are good for describing the periodic operation of the 'hardware' in both classical and quantum computers and serve as an invisible foundation for the design and understanding of such machinery, as they do for our understanding of the design and construction of the universe itself.
We note here that the complex numbers were the final extension of the number system. They complete traditional algebra, providing us with means of solving all traditional algebraic problems. It is not surprising therefore, they are an essential foundation quantum mechanics, the theory that describes the computational hardware of the universe. Number - Wikipedia
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Further readingBooks
Davis, Martin, Computability and Unsolvability, Dover 1982 Preface: 'This book is an introduction to the theory of computability and non-computability ususally referred to as the theory of recursive functions. The subject is concerned with the existence of purely mechanical procedures for solving problems. . . . The existence of absolutely unsolvable problems and the Goedel incompleteness theorem are among the results in the theory of computability that have philosophical significance.'
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Feynman, Richard, Feynman Lectures on Computation, Perseus Publishing 2007 Amazon Editorial Reviews
Book Description
'The famous physicist's timeless lectures on the promise and limitations of computers
When, in 1984-86, Richard P. Feynman gave his famous course on computation at the California Institute of Technology, he asked Tony Hey to adapt his lecture notes into a book. Although led by Feynman, the course also featured, as occasional guest speakers, some of the most brilliant men in science at that time, including Marvin Minsky, Charles Bennett, and John Hopfield. Although the lectures are now thirteen years old, most of the material is timeless and presents a "Feynmanesque" overview of many standard and some not-so-standard topics in computer science such as reversible logic gates and quantum computers.'
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Lo, Hoi-Kwong, and Tim Spiller, Sandra Popescu, Introduction to Quantum Computation and Information, World Scientific 1998 Jacket: 'This book provides a pedagogical introduction to the subjects of quantum information and computation. Topics include non-locality of quantum mechanics, quantum computation, quantum cryptography, quantum error correction, fault tolerant quantum computation, as well as some experimental aspects of quantum computation and quantum cryptography. A knowledge of basic quantum mechanics is assumed.'
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Milburn, Gerard J, The Feynman Processor : Quantum Entanglement and the Computing Revolution , Perseus Jacket: 'Starting with a clear and concise description of the basic principles of quantum physics, Milburn goes on to introduce some of its most amazing, newly discovered (sic) phenomena, including quantum entanglement, the strangest property of what is already the strangest field of science. Quantum entanglement - which Einstein called "spooky action at a distance" - underlies the interdimensional connections that join seemingly unrelated events and objects. He shows how conventional computers cannot go on getting smaller and faster forever and how the unpredictability of matter at this level has enabled scientists to rethink the way that we could design, build and use the new "quantum computers". Finally Milburn takes us into the near future, when physicists and computer scientists will build new and incredible devices that will deliver a world of lightning-fast computers, unbreakable codes, and even the beginning of Star-trek like matter teleportation.'
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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.
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Zee, Anthony, Quantum Field Theory in a Nutshell, Princeton University Press 2003 Amazon book description: 'An esteemed researcher and acclaimed popular author takes up the challenge of providing a clear, relatively brief, and fully up-to-date introduction to one of the most vital but notoriously difficult subjects in theoretical physics. A quantum field theory text for the twenty-first century, this book makes the essential tool of modern theoretical physics available to any student who has completed a course on quantum mechanics and is eager to go on.
Quantum field theory was invented to deal simultaneously with special relativity and quantum mechanics, the two greatest discoveries of early twentieth-century physics, but it has become increasingly important to many areas of physics. These days, physicists turn to quantum field theory to describe a multitude of phenomena.
Stressing critical ideas and insights, Zee uses numerous examples to lead students to a true conceptual understanding of quantum field theory--what it means and what it can do. He covers an unusually diverse range of topics, including various contemporary developments,while guiding readers through thoughtfully designed problems. In contrast to previous texts, Zee incorporates gravity from the outset and discusses the innovative use of quantum field theory in modern condensed matter theory.
Without a solid understanding of quantum field theory, no student can claim to have mastered contemporary theoretical physics. Offering a remarkably accessible conceptual introduction, this text will be widely welcomed and used.
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Links
Born rule - Wikipedia, Born rule - Wikipedia, the free encyclopedia, 'The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of the Copenhagen interpretation of quantum mechanics. There have been many attempts to derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results. . . . The Born rule states that if an observable corresponding to a Hermitian operator A with discrete spectrum is measured in a system with normalized wave function (see bra-ket notation), then
the measured result will be one of the eigenvalues λ of A, and
the probability of measuring a given eigenvalue λi will equal <ψ|Pi|ψ> where Pi is the projection onto the eigenspace of A corresponding to λi'. back |
Brouwer fixed point theorem - Wikipedia, Brouwer fixed point theorem - Wikipedia, the free encyclopedia, 'Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f(x0) = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to itself. A more general form is for continuous functions from a convex compact subset K of Euclidean space to itself. back |
Circle group - Wikipedia, Circle group - Wikipedia, the free encyclopedia, ' In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane or simply the unit complex numbers.' back |
Lorentz transformation - Wikipedia, Lorentz transformation - Wikipedia, the free encyclopedia, 'In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik Lorentz. It reflects the surprising fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events.' back |
Lov K Grover, Quantum mechanics helps in searching for a needle in a haystack, 'Quantum mechanics can speed up a range of search applications over unsorted data. For example imagine a phone directory containing N names arranged in completely random order. To find someone's phone number with a probability of 50%, any classical algorithm (whether deterministic or probabilistic) will need to access the database a minimum of O(N) times. Quantum mechanical systems can be in a superposition of states and simultaneously examine multiple names. By properly adjusting the phases of various operations, successful computations reinforce each other while others interfere randomly. As a result, the desired phone number can be obtained in only O(sqrt(N)) accesses to the database.' back |
Mathematical proof - Wikipedia, Mathematical proof - Wikipedia, the free encyclopedia, 'In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference.' back |
Number - Wikipedia, Number - Wikipedia, the free encyclopedia, ' A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. For being manipulated, individual numbers need to be represented by symbols, called numerals; for example, "5" is a numeral that represents the number five. As only a small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows representing any number by a combination of ten basic numerals called digits.' back |
Photon - Wikipedia, Photon - Wikipedia, the free encyclopedia, 'A photon is an elementary particle, the quantum of all forms of electromagnetic radiation including light. It is the force carrier for electromagnetic force, even when static via virtual photons. The photon has zero rest mass and as a result, the interactions of this force with matter at long distance are observable at the microscopic and macroscopic levels.' back |
Principle of locality - Wikipedia, Principle of locality - Wikipedia, the free encyclopedia, ' In physics, the principle of locality states that an object is directly influenced only by its immediate surroundings. A theory which includes the principle of locality is said to be a "local theory". This is an alternative to the older concept of instantaneous "action at a distance". Locality evolved out of the field theories of classical physics. The concept is that for an action at one point to have an influence at another point, something in the space between those points such as a field must mediate the action. To exert an influence, something, such as a wave or particle, must travel through the space between the two points, carrying the influence. ' back |
Quantum mechanics - Wikipedia, Quantum mechanics - Wikipedia, the free encyclopedia, 'Quantum mechanics (QM; also known as quantum physics or quantum theory), including quantum field theory, is a fundamental branch of physics concerned with processes involving, for example, atoms and photons. In such processes, said to be quantized, the action has been observed to be only in integer multiples of the Planck constant. This is utterly inexplicable in classical physics.'' back |
Quantum superposition - Wikipedia, Quantum superposition - Wikipedia, the free encyclopedia, 'Quantum superposition is the application of the superposition principle to quantum mechanics. The superposition principle is the addition of the amplitudes of waves from interference. In quantum mechanics it is the sum of wavefunction amplitudes, or state vectors. It occurs when an object simultaneously "possesses" two or more possible values for an observable quantity (e.g. the position or energy of a particle)' back |
Standard model - Wikipedia, Standard model - Wikipedia, the free encyclopedia, 'The Standard Model of particle physics is a theory that describes three of the four known fundamental interactions between the elementary particles that make up all matter. It is a quantum field theory developed between 1970 and 1973 which is consistent with both quantum mechanics and special relativity. To date, almost all experimental tests of the three forces described by the Standard Model have agreed with its predictions. However, the Standard Model falls short of being a complete theory of fundamental interactions, primarily because of its lack of inclusion of gravity, the fourth known fundamental interaction, but also because of the large number of numerical parameters (such as masses and coupling constants) that must be put "by hand" into the theory (rather than being derived from first principles) . . . ' back |
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