scholarly journals Quantum physics on a general Hilbert space

Author(s):  
Klaas Landsman
2018 ◽  
Author(s):  
Rajendra K. Bera

It now appears that quantum computers are poised to enter the world of computing and establish its dominance, especially, in the cloud. Turing machines (classical computers) tied to the laws of classical physics will not vanish from our lives but begin to play a subordinate role to quantum computers tied to the enigmatic laws of quantum physics that deal with such non-intuitive phenomena as superposition, entanglement, collapse of the wave function, and teleportation, all occurring in Hilbert space. The aim of this 3-part paper is to introduce the readers to a core set of quantum algorithms based on the postulates of quantum mechanics, and reveal the amazing power of quantum computing.


Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 581
Author(s):  
Stefan Heusler ◽  
Paul Schlummer ◽  
Malte S. Ubben

What is the origin of quantum randomness? Why does the deterministic, unitary time development in Hilbert space (the ‘4π-realm’) lead to a probabilistic behaviour of observables in space-time (the ‘2π-realm’)? We propose a simple topological model for quantum randomness. Following Kauffmann, we elaborate the mathematical structures that follow from a distinction(A,B) using group theory and topology. Crucially, the 2:1-mapping from SL(2,C) to the Lorentz group SO(3,1) turns out to be responsible for the stochastic nature of observables in quantum physics, as this 2:1-mapping breaks down during interactions. Entanglement leads to a change of topology, such that a distinction between A and B becomes impossible. In this sense, entanglement is the counterpart of a distinction (A,B). While the mathematical formalism involved in our argument based on virtual Dehn twists and torus splitting is non-trivial, the resulting haptic model is so simple that we think it might be suitable for undergraduate courses and maybe even for High school classes.


2021 ◽  
Vol 3 (4) ◽  
pp. 643-655
Author(s):  
Louis Narens

In 1933, Kolmogorov synthesized the basic concepts of probability that were in general use at the time into concepts and deductions from a simple set of axioms that said probability was a σ-additive function from a boolean algebra of events into [0, 1]. In 1932, von Neumann realized that the use of probability in quantum mechanics required a different concept that he formulated as a σ-additive function from the closed subspaces of a Hilbert space onto [0,1]. In 1935, Birkhoff & von Neumann replaced Hilbert space with an algebraic generalization. Today, a slight modification of the Birkhoff-von Neumann generalization is called “quantum logic”. A central problem in the philosophy of probability is the justification of the definition of probability used in a given application. This is usually done by arguing for the rationality of that approach to the situation under consideration. A version of the Dutch book argument given by de Finetti in 1972 is often used to justify the Kolmogorov theory, especially in scientific applications. As von Neumann in 1955 noted, and his criticisms still hold, there is no acceptable foundation for quantum logic. While it is not argued here that a rational approach has been carried out for quantum physics, it is argued that (1) for many important situations found in behavioral science that quantum probability theory is a reasonable choice, and (2) that it has an arguably rational foundation to certain areas of behavioral science, for example, the behavioral paradigm of Between Subjects experiments.


Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca

In this section we introduce the framework of quantum mechanics as it pertains to the types of systems we will consider for quantum computing. Here we also introduce the notion of a quantum bit or ‘qubit’, which is a fundamental concept for quantum computing. At the beginning of the twentieth century, it was believed by most that the laws of Newton and Maxwell were the correct laws of physics. By the 1930s, however, it had become apparent that these classical theories faced serious problems in trying to account for the observed results of certain experiments. As a result, a new mathematical framework for physics called quantum mechanics was formulated, and new theories of physics called quantum physics were developed in this framework. Quantum physics includes the physical theories of quantum electrodynamics and quantum field theory, but we do not need to know these physical theories in order to learn about quantum information. Quantum information is the result of reformulating information theory in this quantum framework. We saw in Section 1.6 an example of a two-state quantum system: a photon that is constrained to follow one of two distinguishable paths. We identified the two distinguishable paths with the 2-dimensional basis vectors and then noted that a general ‘path state’ of the photon can be described by a complex vector with |α0|2 +|α1|2 = 1. This simple example captures the essence of the first postulate, which tells us how physical states are represented in quantum mechanics. Depending on the degree of freedom (i.e. the type of state) of the system being considered, H may be infinite-dimensional. For example, if the state refers to the position of a particle that is free to occupy any point in some region of space, the associated Hilbert space is usually taken to be a continuous (and thus infinite-dimensional) space. It is worth noting that in practice, with finite resources, we cannot distinguish a continuous state space from one with a discrete state space having a sufficiently small minimum spacing between adjacent locations. For describing realistic models of quantum computation, we will typically only be interested in degrees of freedom for which the state is described by a vector in a finite-dimensional (complex) Hilbert space.


1984 ◽  
Vol 39 (3) ◽  
pp. 205-217
Author(s):  
Fritz Bopp

AbstractAs previously shown, quantum physics for single pairs of creation and annihilation processes may be derived from first principles. Quantum physics at all can be therefore considered as an interplay of such elementary processes. This is easily possible if the number of pairs of processes is finite. Difficulties arise only for infinite numbers.The difficulties are similar to those occurring in the derivation of the equation for an oscillating string from that for an oscillator chain. It is true that the spectra of both systems are not continuously connected. However, a weaker theorem is more important: The chain eigenvalue of each order converges to the string one of the same order for an infinitely growing number of oscillators of a certain kind. Therefore both systems are continuously connected in the sense of semiconvergency.Exhausting the space continuum with a sequence of lattices equably becomming infinitely large and fine, the infinitely dimensional Hilbertspace is steadily connected with the finitely dimensional one in the sense of semiconvergency. It will be shown that the Hilbert spaces in the sequence of lattices yield the suitable tool for quantum physics as an interplay in the mentioned sense. This kind of Hilbert space, the so-called rational one, must be preferred in physics rather than the real one introduced by Hilbert, since all theories in physics are based on a finite number of data.In particular, we formulate Dirac's equation in the rational Hilbert space. It is shown that, even in quantum physics, a theorem of classical physics remains true, according to which relativity results from certain principles formulating most obvious experiences. We obtain the Lorentz invariant Dirac equation mainly from a modification of Newtons definition II according to which p = Hυ/c2 (instead of p = m υ).


Universe ◽  
2021 ◽  
Vol 7 (8) ◽  
pp. 285
Author(s):  
Julio Marny Hoff da Silva ◽  
Gabriel Marcondes Caires da Rocha

We revisit the fundamental notion of continuity in representation theory, with special attention to the study of quantum physics. After studying the main theorem in the context of representation theory, we draw attention to the significant aspect of continuity in the analytic foundations of Wigner’s work. We conclude the paper by reviewing the connection between continuity, the possibility of defining certain local groups, and their relation to projective representations.


10.53733/99 ◽  
2021 ◽  
Vol 51 ◽  
pp. 65-78
Author(s):  
Berthold-Georg Englert ◽  
Heng Huat Chan

We introduce two families of multiplicative functions, which generalize the somewhat unusual function that was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum.


2019 ◽  
Vol 2 (4) ◽  

Quantum physics is non-causal, and randomness is so-called “intrinsic”. We propose no less than an 18th interpretation of it through non-Archimedean geometry to bring back causality, respect of the Kolmogorov axioms and the existence of hidden variables. For these latter ones, we show that they cannot be in any Hilbert space and hence could not be detected in any traditional experiment. We end through proposing two experiments which would prove the non-Archimedean nature of our universe. The first one consists in a new disruptive type of quantum radar. The second one explains how viscosity naturally occurs in fluid mechanics whereas Boltzmann’s approach only considers elastic shocks at the molecular scale.


Author(s):  
Muharrem Tuncay Gençoğlu ◽  
Praveen Agarwal

AbstractEmerging as a new field, quantum computation has reinvented the fundamentals of Computer Science and knowledge theory in a manner consistent with quantum physics. The fact that quantum computation has superior features and new events than classical computation provides benefits in proving mathematical theories. With advances in technology, the nonlinear partial differential equations are used in almost every area, and many difficulties have been overcome by the solutions of these equations. In particular, the complex solutions of KdV and Burgers equations have been shown to be used in modeling a simple turbulence flow. In this study, Burger-like equation with complex solutions is defined in Hilbert space and solved with an example. In addition, these solutions were analyzed. Thanks to the Quantum Burgers-Like equation, the nonlinear differential equation is solved by linearizing. The pattern changes of time made the result linear. This means that the Quantum Burgers-Like equation can be used to smoothen the sonic processing.


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