scholarly journals The power of quantum complexity

2021 ◽  
Vol 64 (3) ◽  
pp. 15-17
Author(s):  
Don Monroe
Keyword(s):  
Quantum Mechanics ◽  
Quantum Complexity ◽  
Mathematics And Physics

A theorem about computations that exploit quantum mechanics challenges longstanding ideas in mathematics and physics.

QUANTUM COMPLEXITY THEORY GOALS AND CHALLENGES

2005 ◽  
Vol 03 (01) ◽  
pp. 31-39 ◽  
Author(s):  
JOZEF GRUSKA
Keyword(s):  
Quantum Mechanics ◽  
Information Processing ◽  
Quantum Information ◽  
Complexity Theory ◽  
Quantum Information Processing ◽  
Short Review ◽  
Quantum Complexity ◽  
Quantum Complexity Theory

Quantum complexity theory is a powerful tool that provides deep insights into Quantum Information Processing (QIP) and aims to do that also for Quantum Mechanics (QM), in general. This paper is a short review of the main and new motivations, goals, tools, results and challenges of quantum complexity, oriented mainly for pedestrians.

scholarly journals “The Heisenberg Method”: Geometry, Algebra, and Probability in Quantum Theory

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 656 ◽  
Author(s):  
Arkady Plotnitsky
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Algebraic Method ◽  
Specific Character ◽  
Matrix Mechanics ◽  
Quantum Behavior ◽  
Quantum Phenomena ◽  
Physical Phenomena ◽  
Algebraic Scheme ◽  
Mathematics And Physics

The article reconsiders quantum theory in terms of the following principle, which can be symbolically represented as QUANTUMNESS → PROBABILITY → ALGEBRA and will be referred to as the QPA principle. The principle states that the quantumness of physical phenomena, that is, the specific character of physical phenomena known as quantum, implies that our predictions concerning them are irreducibly probabilistic, even in dealing with quantum phenomena resulting from the elementary individual quantum behavior (such as that of elementary particles), which in turn implies that our theories concerning these phenomena are fundamentally algebraic, in contrast to more geometrical classical or relativistic theories, although these theories, too, have an algebraic component to them. It follows that one needs to find an algebraic scheme able make these predictions in a given quantum regime. Heisenberg was first to accomplish this in the case of quantum mechanics, as matrix mechanics, whose matrix character testified to his algebraic method, as Einstein characterized it. The article explores the implications of the Heisenberg method and of the QPA principle for quantum theory, and for the relationships between mathematics and physics there, from a nonrealist or, in terms of this article, “reality-without-realism” or RWR perspective, defining the RWR principle, thus joined to the QPA principle.

Weyl, Hermann (1885–1955)

2018 ◽  
Author(s):  
T.A. Ryckman
Keyword(s):  
Twentieth Century ◽  
Quantum Mechanics ◽  
Philosophy Of Science ◽  
Related Species ◽  
Philosophy Of Mathematics ◽  
Theoretical Physics ◽  
Theory Of Relativity ◽  
Symbolic Construction ◽  
Mathematics And Science ◽  
Mathematics And Physics

A leading mathematician of the twentieth century, Weyl made fundamental contributions to theoretical physics, to philosophy of mathematics, and to philosophy of science. Weyl wrote authoritative works on the theory of relativity and quantum mechanics, as well as a classic philosophical examination of mathematics and science. He was briefly a follower of Brouwer’s intuitionism in philosophy of mathematics. Upon moving closer to Hilbert’s finitism, he articulated a conception of mathematics and physics as related species of ‘symbolic construction’.

scholarly journals On Some Consequences of the Functional Generalization of the Parallelogram Identity

2014 ◽  
Vol 47 (2) ◽  
Author(s):  
Maciej J. Maczyński
Keyword(s):  
Quantum Mechanics ◽  
Functional Form ◽  
Schwarz Inequality ◽  
Vector Spaces ◽  
Complex Vector ◽  
Standard Structure ◽  
Heisenberg's Uncertainty Principle ◽  
Normed Vector ◽  
Mathematics And Physics

AbstractThe aim of this paper is to unify the partial results, which up to now, have been dispersed in various publications in order to show the importance of the functional form of parallelogram identity in mathematics and physics. We study vector spaces admitting a real non-negative functional which satisfies an identity analogous to the parallelogram identity in normed vector spaces. We show that this generalized parallelogram identity also implies an equality analogous to the Cauchy-Schwarz inequality. We study the consequences of this identity in real and complex vector spaces, in generalized Riesz spaces and in abelian groups. We give a physical interpretation to these results. For vector spaces of observables and states, we show that the parallelogram identity implies an inequality analogous to Heisenberg’s uncertainty principle (HUP), and we show that we can obtain the standard structure of quantum mechanics from the parallelogram identity, without assuming from the beginning the HUP. The role of complex numbers in quantum mechanics is discussed.

scholarly journals The Montevideo Interpretation: How the Inclusion of a Quantum Gravitational Notion of Time Solves the Measurement Problem

Universe ◽  
2020 ◽  
Vol 6 (12) ◽  
pp. 236
Author(s):  
Rodolfo Gambini ◽  
Jorge Pullin
Keyword(s):  
Quantum Mechanics ◽  
Measurement Problem ◽  
Operational Definition ◽  
Covariant Theory ◽  
Problem Of Time ◽  
New Formalism ◽  
Definition Of ◽  
Additional Support ◽  
Quantum Complexity ◽  
Generally Covariant

We review the Montevideo Interpretation of quantum mechanics, which is based on the use of real clocks to describe physics, using the framework that was recently introduced by Höhn, Smith, and Lock to treat the problem of time in generally covariant systems. These new methods, which solve several problems in the introduction of a notion of time in such systems, do not change the main results of the Montevideo Interpretation. The use of the new formalism makes the construction more general and valid for any system in a quantum generally covariant theory. We find that, as in the original formulation, a fundamental mechanism of decoherence emerges that allows for supplementing ordinary environmental decoherence and avoiding its criticisms. The recent results on quantum complexity provide additional support to the type of global protocols that are used to prove that within ordinary—unitary—quantum mechanics, no definite event—an outcome to which a probability can be associated—occurs. In lieu of this, states that start in a coherent superposition of possible outcomes always remain as a superposition. We show that, if one takes into account fundamental inescapable uncertainties in measuring length and time intervals due to general relativity and quantum mechanics, the previously mentioned global protocols no longer allow for distinguishing whether the state is in a superposition or not. One is left with a formulation of quantum mechanics purely defined in quantum mechanical terms without any reference to the classical world and with an intrinsic operational definition of quantum events that does not need external observers.

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