scholarly journals Effective Number Theory: Counting the Identities of a Quantum State

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1273
Author(s):  
Ivan Horváth ◽  
Robert Mendris
Keyword(s):  
Number Theory ◽  
Quantum Mechanics ◽  
Quantum Computation ◽  
Quantum State ◽  
Quantum Dynamics ◽  
Quantum Physics ◽  
Quantum Algorithm ◽  
Effective Number ◽  
Additive Theory ◽  
Participation Number

Quantum physics frequently involves a need to count the states, subspaces, measurement outcomes, and other elements of quantum dynamics. However, with quantum mechanics assigning probabilities to such objects, it is often desirable to work with the notion of a “total” that takes into account their varied relevance. For example, such an effective count of position states available to a lattice electron could characterize its localization properties. Similarly, the effective total of outcomes in the measurement step of a quantum computation relates to the efficiency of the quantum algorithm. Despite a broad need for effective counting, a well-founded prescription has not been formulated. Instead, the assignments that do not respect the measure-like nature of the concept, such as versions of the participation number or exponentiated entropies, are used in some areas. Here, we develop the additive theory of effective number functions (ENFs), namely functions assigning consistent totals to collections of objects endowed with probability weights. Our analysis reveals the existence of a minimal total, realized by the unique ENF, which leads to effective counting with absolute meaning. Touching upon the nature of the measure, our results may find applications not only in quantum physics, but also in other quantitative sciences.

scholarly journals Unification of Commutative Relativity and Noncommutative Quantum Dynamics Via Cyclic Groupoids and Spacetime Fusion Categories

2018 ◽  
Author(s):  
Arturo Tozzi ◽  
James F. Peters
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Quantum Dynamics ◽  
Quantum Physics ◽  
Binary Operation ◽  
Mathematical Framework ◽  
Mathematical Approach ◽  
Fusion Categories ◽  
Microscopic World ◽  
Noncommutative Quantum

The unexploited unification of general relativity and quantum physics is a painstaking issue that prevents physicists to properly understanding the whole of Nature. Here we propose a pure mathematical approach that introduces the problem in terms of group theory.  Indeed, we build a cyclic groupoid (a nonempty set with a binary operation defined on it) that encompasses both the theories as subsets, making it possible to join together two of their most dissimilar experimental results, i.e., the commutativity detectable in our macroscopic relativistic world and the noncommutativity detectable in the quantum, microscopic world.  This approach, combined with the Connes fusion operator, leads to a mathematical framework useful in the investigation of relativity/quantum mechanics relationships. 

scholarly journals Towards the Unification of Quantum Dynamics, Relativity and Living Organisms

2018 ◽  
Author(s):  
Arturo Tozzi ◽  
James F. Peters ◽  
John S. Torday
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Group Theory ◽  
Quantum Dynamics ◽  
Quantum Physics ◽  
Binary Operation ◽  
Relativity Theory ◽  
Mathematical Framework ◽  
Mathematical Approach ◽  
Living Organisms

The unexploited unification of quantum physics, general relativity and biology is a keystone that paves the way towards a better understanding of the whole of Nature.  Here we propose a mathematical approach that introduces the problem in terms of group theory.  We build a cyclic groupoid (a nonempty set with a binary operation defined on it) that encompasses the three frameworks as subsets, representing two of their most dissimilar experimental results, i.e., 1) the commutativity detectable both in our macroscopic relativistic world and in biology; 2) and the noncommutativity detectable both in the microscopic quantum world and in biology.  This approach leads to a mathematical framework useful in the investigation of the three apparently irreconcilable realms.  Also, we show how cyclic groupoids encompassing quantum mechanics, relativity theory and biology might be equipped with dynamics that can be described by paths on the twisted cylinder of a Möbius strip.   

Neuromorphic Adiabatic Quantum Computation

2009 ◽  
pp. 352-375
Author(s):  
Shigeo Sato ◽  
Mitsunaga Kinjo
Keyword(s):  
Neural Networks ◽  
Quantum Computation ◽  
Quantum Dynamics ◽  
Quantum Computer ◽  
Quantum Algorithm ◽  
Descent Method ◽  
Gradient Descent Method ◽  
Quantum Superposition ◽  
Adiabatic Quantum Computation ◽  
Computation Algorithms

The advantage of quantum mechanical dynamics in information processing has attracted much interest, and dedicated studies on quantum computation algorithms indicate that a quantum computer has remarkable computational power in certain tasks. Quantum properties such as quantum superposition and quantum tunneling are worth studying because they may overcome the weakness of gradient descent method in classical neural networks. Also, the technique established for neural networks can be useful for developing a quantum algorithm. In this chapter, first the authors show the effectiveness of incorporating quantum dynamics and then propose neuromorphic adiabatic quantum computation algorithm based on the adiabatic change of Hamiltonian. The proposed method can be viewed as one of complex-valued neural networks because a qubit operates like a neuron. Next, the performance of the proposed algorithm is studied by applying it to a combinatorial optimization problem. Finally, they discuss the learning ability and hardware implementation.

scholarly journals The Measure Aspect of Quantum Uncertainty, of Entanglement, and the Associated Entropies

2021 ◽  
Vol 3 (3) ◽  
pp. 534-548
Author(s):  
Ivan Horváth
Keyword(s):  
Number Theory ◽  
Density Matrix ◽  
Quantum State ◽  
Degrees Of Freedom ◽  
Minimal Amount ◽  
Effective Number ◽  
Quantum Uncertainty ◽  
Commuting Operators ◽  
Central Result

Indeterminacy associated with the probing of a quantum state is commonly expressed through spectral distances (metric) featured in the outcomes of repeated experiments. Here, we express it as an effective amount (measure) of distinct outcomes instead. The resulting μ-uncertainties are described by the effective number theory whose central result, the existence of a minimal amount, leads to a well-defined notion of intrinsic irremovable uncertainty. We derive μ-uncertainty formulas for arbitrary set of commuting operators, including the cases with continuous spectra. The associated entropy-like characteristics, the μ-entropies, convey how many degrees of freedom are effectively involved in a given measurement process. In order to construct quantum μ-entropies, we are led to quantum effective numbers designed to count independent, mutually orthogonal states effectively comprising a density matrix. This concept is basis-independent and leads to a measure-based characterization of entanglement.

scholarly journals Quantum Computation and Arrows of Time

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 49
Author(s):  
Nathan Argaman
Keyword(s):  
Quantum Computation ◽  
Quantum Physics ◽  
Bell’S Theorem ◽  
Arrow Of Time ◽  
Bell's Theorem ◽  
Classical Domain ◽  
Further Development

Quantum physics is surprising in many ways. One surprise is the threat to locality implied by Bell’s Theorem. Another surprise is the capacity of quantum computation, which poses a threat to the complexity-theoretic Church-Turing thesis. In both cases, the surprise may be due to taking for granted a strict arrow-of-time assumption whose applicability may be limited to the classical domain. This possibility has been noted repeatedly in the context of Bell’s Theorem. The argument concerning quantum computation is described here. Further development of models which violate this strong arrow-of-time assumption, replacing it by a weaker arrow which is yet to be identified, is called for.

scholarly journals Asset–liability modelling in the quantum era

2021 ◽  
Vol 26 ◽  
Author(s):  
T. Berry ◽  
J. Sharpe
Keyword(s):  
Quantum Mechanics ◽  
Quantum Computer ◽  
Quantum Algorithm ◽  
Capital Requirements ◽  
Quantum Computers ◽  
Investment Management ◽  
Asset Liability Management ◽  
Liability Management ◽  
Insight Into ◽  
Current Practices

Abstract This paper introduces and demonstrates the use of quantum computers for asset–liability management (ALM). A summary of historical and current practices in ALM used by actuaries is given showing how the challenges have previously been met. We give an insight into what ALM may be like in the immediate future demonstrating how quantum computers can be used for ALM. A quantum algorithm for optimising ALM calculations is presented and tested using a quantum computer. We conclude that the discovery of the strange world of quantum mechanics has the potential to create investment management efficiencies. This in turn may lead to lower capital requirements for shareholders and lower premiums and higher insured retirement incomes for policyholders.

LOCAL ANALYSIS BY WAVELETS VERSUS NONLOCAL FOURIER ANALYSIS

2007 ◽  
Vol 05 (01n02) ◽  
pp. 89-95
Author(s):  
J. R. CROCA
Keyword(s):  
Quantum Mechanics ◽  
Fourier Analysis ◽  
Quantum Physics ◽  
Plane Waves ◽  
Local Analysis ◽  
Mathematical Tool ◽  
Space Harmonic ◽  
New Approach ◽  
Spatial Frequencies ◽  
Temporal And Spatial

Orthodox quantum mechanics has another implicit postulate stating that temporal and spatial frequencies of the Planck–Einstein and de Broglie formulas can only be linked with the infinite, in time and space, harmonic plane waves of Fourier analysis. From this assumption, nonlocality either in space and time follows directly. This is what is called Fourier Ontology. In order to build nonlinear causal and local quantum physics, it is necessary to reject Fourier ontology and accept that in certain cases a finite wave may have a well defined frequency. Now the mathematical tool to describe this new approach is wavelet local analysis. This more general nonlinear local and causal quantum physics, in the limit of the linear approximation, contains formally orthodox quantum mechanics as a particular case.

Quantum algorithm for measuring the energy of n qubits with unknown pair-interactions

2002 ◽  
Vol 2 (3) ◽  
pp. 198-207
Author(s):  
D. Janzing
Keyword(s):  
Quantum State ◽  
Quantum Computer ◽  
Quantum Algorithm ◽  
Pair Interaction ◽  
Phase Estimation ◽  
Solid States ◽  
Energy Gaps ◽  
Large N ◽  
Conditional Transformation ◽  
Essential Assumption

The well-known algorithm for quantum phase estimation requires that the considered unitary is available as a conditional transformation depending on the quantum state of an ancilla register. We present an algorithm converting an unknown n-qubit pair-interaction Hamiltonian into a conditional one such that standard phase estimation can be applied to measure the energy. Our essential assumption is that the considered system can be brought into interaction with a quantum computer. For large n the algorithm could still be applicable for estimating the density of energy states and might therefore be useful for finding energy gaps in solid states.

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