Introducing the divinity of the Universe
to pave the way for scientifically credible theology
Chapter 20: Measurement: the interface between Hilbert and Minkowski spaces
Synopsis
The two slit experiment suggests that every particle is associated with waves that interfere with one another and control how the particle moves. Here we understand measurement as conversations between particles. We know that all action in the world occurs at the quantum level and there are quantum operators associated with every particle. These operators have the same dimension as the Hilbert space associated with the particle and therefore the same number of eigenvalues. We observe an unknown particle by arranging for it to interact with a known particle and looking at the result, which may be any one of the eigenstates of the known particle. The eigenvalues of this result are precise but particular values appear at random distributed by their distance apart computed by the Born Rule. This uncertainty is called the measurement problem.
Contents
20.1: The measurement problem20.2: Measurement: two particles communication is tensor product space?
20.3: Zurek: the mathematics of observation
20.4: Observers in classical physics do not affect what they look at
20.5: But quantum observations are mutual interactions
20.6: Measurement and creation
20.1: The measurement problem
The relationship between the invisible processes in Hilbert space and the visible process in Minkowski space have been a perennial issue in quantum theory often known as the measurement problem.
The Hilbert space representation of a quantum state is a vector which may be the sum of a number of orthonormal basis states corresponding to the dimensions of a Hilbert space. The principal difference between Hilbert spaces is their number of dimensions. In physical applications such spaces commonly have a countable infinity of dimensions but the simple two dimensional Hilbert space called the qubit tells us most of what we need to know. Qubit - Wikipedia
Quantum mechanics, like computer networks, is symmetrical with respect to complexity. Both are built from an atom: in quantum mechanics the atomic operator is the quantum of action. In the theory of computation we have the not-and or Sheffer stroke operator. Multiple instances of these operators, properly connected to a suitable memory, can perform all the functions needed to execute both quantum and classical computations. Sheffer stroke - Wikipedia
Our conjectures about this hidden quantum mechanical structure are based on observing particles. In the laboratory we prepare a particle in a particular state, to some extent known, representing the observer. We arrange for it to interact with the unknown particle. The information we want is carried by the particle(s) which emerge from this interaction. What we see are eigenvalues, real numbers, the frequencies of photons or the momenta and directions of massive particles.
The theory predicts that there are as many possible eigenvalues as the dimensions of the measurement operator. The terms collapse or reduction of the wave function refer to the fact that individual observations only ever reveal just one of the possible states of an unknown system. In this respect, a quantum measurement is equivalent to the emission of one symbol at a time from a communication source. The spectrum of a measurement operator corresponds to the alphabet of a source.
A radical problem facing our understanding of quantum mechanics and the development of quantum computation is illustrated by the difference between a classical bit (binary digit) and its quantum analogue, the qubit, which is a space. A classical bit has just two states, usually represented 0 and 1. These states are orthogonal, one is not the other. A qubit on the other hand is a vector formed in a two dimensional Hilbert space by adding the orthogonal basis vectors |0⟩ and |1⟩.
We write
| qubit ⟩ = a | 0 ⟩ + b |1⟩,
where a and b are complex numbers such that |a|2 + |b|2 = 1.
We may imagine | 0 ⟩ and |1 ⟩ to be the axes of a Cartesian plane. The qubit is then the set of vectors from the origin to some point on the unit circle around the origin of the plane.
When we observe a qubit, however, all we ever see is | 0 ⟩ or |1 ⟩ with relative frequency P1 = |a|2, P2 |b|2. The infinite amount of information which we suppose to have been represented by the continuous qubit turns out to be at best just one classical bit. Designers of quantum computations must try to devise some way to take advantage of this (allegedly) hidden information.
Understanding the hidden quantum information is a question that we grapple with for much of this book and which lies at the heart of what makes quantum mechanics a powerful tool for information processing. Nielsen & Chuang (2016); Quantum Computation and Quantum Information
The essence of the historical measurement problem, sometimes called the reduction or the collapse of the wave function, is why do actual measurements only yield one definite result and ignore the rest of the wide spectrum of states presumed to be represented by vectors in tensor product of the Hilbert spaces of the interacting particles?
20.2: Measurement: two particles communication is tensor product space?
We are very familiar with everyday Newtonian space. Special and general relativity extend our understanding to situations where speeds approach the velocity of light and general relativity reveals the large scale structure of the Universe.
Behind all this, explaining its behaviour, are the natural quantum computations in Hilbert space, invisible but quite well known after more than a century of intense study. Engineered applications of quantum theory like computer chips and LEDs are now components of almost every item of modern technology. Light-emitting diode - Wikipedia
Measurement in a general sense is everywhere. It may be simple, like using a ruler to discover that I am 1580 mm tall, or very complex, like the judgement made by the referee in a football match whether a contact between two players is fair or foul.
Quantum measurement is ubiquitous. Everything we see in Minkowski space is in effect a measurement of the underlying quantum process. It is the foundation of the world occurring continually at the simplest ontological level, the next step after the initial singularity. Football players are particles controlled by quantum mechanics. Every communication between discrete particles at all scales is a measurement in which they exchange information. In football the state vectors involved are huge because each player comprises trillions of trillions of trillions of elementary particles. Physicists prefer to study much simpler elementary particles, but the same rules of physics control everything.
20.3: Zurek: the mathematics of observation
Zurek's suggests that the alleged collapse of the wave function is a necessary consequence of the transmission of information between two quantum systems. The distinction between observer and observed is fictitious, in the sense that a quantum process is simply the communication channel in Hilbert space between two sources in Minkowski space. The mathematical theory of communication treats the space of all possible classical communications between two sources but its results apply to each particular communication. The mathematical expression of quantum mechanics works in the same way. Wojciech Hubert Zurek (2008): Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer and the transition from quantum to classical
We often think of a measurement as an interaction between a classical and a quantum system, but in reality it is the interaction of the quantum systems associated with the particles. One classical system is the source of a state which we call the measurement operator. The operator interacts with an unknown state attached to another classical system, yielding a classically observable result, the particle(s) created by this interaction.
A measurement interrupts an isolated system by injecting another process, represented by a measurement operator, into the isolated system. This is analogous to one person interrupting another by starting a conversation.
Zurek begins with a concise definition of standard quantum mechanics in six propositions. The first three describe its mathematical mechanism:
(1) the quantum state of a system is represented by a vector in its Hilbert space;(2) a complex system is represented by a vector in the tensor product of the Hilbert spaces of the constituent systems;
(3) the evolution of isolated quantum systems is unitary, governed by the Schrödinger equation:
iℏ ∂|ψ⟩ / ∂t = H |ψ⟩
where H is the energy (or Hamiltonian) operator. Schrödinger equation - Wikipedia
The other three show how the mathematical formalism in Hilbert space couples to the observed world:
(4) immediate repetition of a measurement yields the same outcome;
(5) measurement outcomes are restricted to an orthonormal set { | sk ⟩ } of k eigenstates of the measured observable [ie the measurement operator associated with the classical measuring system];
(6) the probability of finding a given outcome is given by the Born Rule
pk = |⟨ sk | ψ⟩ |2,
where |ψ⟩ is the preexisting state of the [measured] system. Born rule - Wikipedia,
Let us ignore Zurek’s detailed calculation and jump to his information theoretical conclusion :
He writes:
The aim of this paper is to point out that already the (symmetric and uncontroversial) postulates (1) - (3) necessarily imply selection of some preferred set of orthogonal states – that they impose the broken symmetry that is at the heart of the collapse postulate (4).
Selection of an orthonormal basis induced by information transfer – the need for spontaneous symmetry breaking that arises from the unitary axioms of quantum mechanics (1, 3) is a general and intriguing result.
This means that if information is to be transferred in an interaction all the basis states of the interacting particles cannot talk at once. One basis must be selected by breaking the symmetry of the unitary transformation believed to be occurring in the undisturbed evolution of the quantum minds of the particles.
Nature, it seems, works like the chair at a well organized meeting. She permits just one person to speak at time.
20.4: Observers in classical physics do not affect what they look at
Einstein radically revised classical physics with his theories of special and general relativity. His work struck deeper however, into the methodology of classical physics, summed up in the principle of covariance. The core idea is that the classical Universe is indifferent to observers.
When everything is moving inertially the Lorentz transformations enables each observer to transform what they see on a distant moving system to what it would look like in their own system and vice versa.
To get an arithmetic grip on the geometry of nature Einstein used Gaussian coordinates to associate real numbers to geometric points. Unlike Cartesian coordinates Gaussian coordinates describe flexible topological spaces which may be bent and stretched as long as they are not torn.
The Gauss co-ordinate system takes the place of a body of reference. Einstein writes:
The following statement corresponds to the fundamental idea of the general principle of relativity: All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature. Einstein (1916, 2005): Relativity: The Special and General Theory
The key to getting a deterministic mathematical theory out of this somewhat arbitrary topological coordinate system is that the only fixed points in nature are events and the space-time intervals between them. Whatever coordinate systems we choose must be constrained to give a one to one correspondence between identical spacetime intervals and identical differences in Gaussian coordinates.
He established a relationship between energy and spacetime distance to represent the large scale structure of the Universe. He expressed this as a field equation which connects every point in the Universe to its neighbours by contact, without an action at a distance. He exploited the topological freedom of the Gaussian coordinate system to honour the principle of general covariance: observation has no effect on classical reality.
20.5: But quantum observations are a mutual interactions
In human terms Einstein's general covariance is like dictation. Nature dictates and you see what you see. Your actual presence is irrelevant, the world goes its own way. The quantum world is much more natural. It involves conversation. Every communication is a meeting. There are always two actors and the meeting changes them both.
20.6: Measurement and creation
Von Neumann shows that quantum mechanical measurement creates entropy. This may seem counterintuitive: the alleged annihilation of quantum states implicit in measurement process would seem to decrease the entropy of the system. This may be true at the kinematic level but at the dynamic level observation leads to the selection of a real state, the outcome of the measurement. John von Neumann (2014): Mathematical Foundations of Quantum Mechanics, Chapter V, §3 Reversibility and Equilibrium Problems
Everywhere, as entities communicate the Universe is measuring and creating itself. At the most basic level this conversation in the invisible world of Hilbert space has visible effects Minkowski space. The spacetime in which we live acts as our interface with the quantum world. Every move we make sends signals to this invisible world for processing and the answer comes back to us as the result of our actions. The situation looks rather like body and mind.
Copyright:
You may copy this material freely provided only that you quote fairly and provide a link (or reference) to your source.
Notes and references
Further reading
Books
Einstein (1916, 2005), Albert, and Robert W Lawson (translator) Roger Penrose (Introduction), Robert Geroch (Commentary), David C Cassidy (Historical Essay), Relativity: The Special and General Theory, Pi Press 1916, 2005 Preface: 'The present book is intended, as far as possible, to give an exact insight into the theory of relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. ... The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated.' page 3
|
Nielsen (2016), Michael A., and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2016 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.
|
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 |
John von Neumann (2014), Mathematical Foundations of Quantum Mechanics, ' Mathematical Foundations of Quantum Mechanics by John von Neumann translated from the German by Robert T. Beyer (New Edition) edited by Nicholas A. Wheeler. Princeton UP Princeton & Oxford. Preface: ' This book is the realization of my long-held intention to someday use the resources of TEX to produce a more easily read version of Robert T. Beyer’s authorized English translation (Princeton University Press, 1955) of John von Neumann’s classic Mathematische Grundlagen der Quantenmechanik (Springer, 1932).'This content downloaded from 129.127.145.240 on Sat, 30 May 2020 22:38:31 UTC back |
Light-emitting diode - Wikipedia, Light-emitting diode - Wikipedia, ' A light-emitting diode (LED) is a semiconductor device that emits light when current flows through it. Electrons in the semiconductor recombine with electron holes, releasing energy in the form of photons. The color of the light (corresponding to the energy of the photons) is determined by the energy required for electrons to cross the band gap of the semiconductor. White light is obtained by using multiple semiconductors or a layer of light-emitting phosphor on the semiconductor device. 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 |
Schrödinger equation - Wikipedia, Schrödinger equation - Wikipedia, the free encyclopedia, ' In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. . . . In classical mechanics Newton's second law, (F = ma), is used to mathematically predict what a given system will do at any time after a known initial condition. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function").' back |
Sheffer stroke - Wikipedia, Sheffer stroke - Wikipedia, the free encyclopedia, 'In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written "|" . . . denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called nand ("not and") or the alternative denial, since it says in effect that at least one of its operands is false.' back |
Wojciech Hubert Zurek (2008), Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer and the transition from quantum to classical, 'Submitted on 17 Mar 2007 (v1), last revised 18 Mar 2008 (this version, v3)) Measurements transfer information about a system to the apparatus, and then further on – to observers and (often inadvertently) to the environment. I show that even imperfect copying essential in such situations restricts possible unperturbed outcomes to an orthogonal subset of all possible states of the system, thus breaking the unitary symmetry of its Hilbert space implied by the quantum superposition principle. Preferred outcome states emerge as a result. They provide framework for the “wavepacket collapse”, designating terminal points of quantum jumps, and defining the measured observable by specifying its eigenstates.' back |