scholarly journals A general framework for quantum splines

2018 ◽  
Vol 15 (09) ◽  
pp. 1850147 ◽  
Author(s):  
L. Abrunheiro ◽  
M. Camarinha ◽  
J. Clemente-Gallardo ◽  
J. C. Cuchí ◽  
P. Santos
Keyword(s):  
Optimal Control ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Nonlinear System ◽  
General Framework ◽  
Hamiltonian Approach ◽  
Density Matrices ◽  
Quantum Domain ◽  
Riemannian Cubic ◽  
Geometrical Formulation

Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this concept to general density matrices with a Hamiltonian approach and using a geometrical formulation of quantum mechanics. Our main goal is to formulate an optimal control problem for a nonlinear system on [Formula: see text] which corresponds to the variational problem of quantum splines. The corresponding Hamiltonian equations and interpolation conditions are derived. The results are illustrated with some examples and the corresponding quantum splines are computed with the implementation of a suitable iterative algorithm.

scholarly journals QUANTUM INFORMATION, ENTANGLEMENT AND ENTROPY

2020 ◽  
Author(s):  
Siddharth Sharma
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Information ◽  
Density Matrix ◽  
Pythagorean Theorem ◽  
Quantum Entropy ◽  
Density Matrices ◽  
Relative Quantum ◽  
Axiomatic Quantum Mechanics ◽  
Measure Entropy

In this review I had given an introduction to axiomatic Quantum Mechanics, Quantum Information and its measure entropy. Also, I had given an introduction to Entanglement and its measure as the minimum of relative quantum entropy of a density matrix of a Hilbert space with respect to all separable density matrices of the same Hilbert space. I also discussed geodesic in the space of density matrices, their orthogonality and Pythagorean theorem of density matrix.

On the Heisenberg algebra in Bargmann–Fock space with natural cutoffs

2016 ◽  
Vol 13 (05) ◽  
pp. 1650054 ◽  
Author(s):  
K. Nozari ◽  
M. Roushan
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
General Framework ◽  
Fock Space ◽  
Heisenberg Algebra ◽  
Minimal Length ◽  
Space Representation ◽  
Maximal Momentum

We construct a generalized Hilbert space representation of quantum mechanics in [Formula: see text]-dimensions based on Bargmann–Fock space with natural cutoffs as a minimal length, minimal momentum and maximal momentum in order to have a general framework for Heisenberg algebra in Bargmann–Fock space.

Nonlinear quantum mechanics in a q-deformed Hilbert space

2019 ◽  
Vol 383 (23) ◽  
pp. 2729-2738 ◽  
Author(s):  
Bruno G. da Costa ◽  
Ernesto P. Borges
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Nonlinear Quantum ◽  
Nonlinear Quantum Mechanics

On the error of approximations in quantum mechanics. I. General theory

1965 ◽  
Vol 257 (1081) ◽  
pp. 309-326 ◽  
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Computer Applications ◽  
Quantum Mechanical ◽  
Quantum Mechanical Calculations ◽  
Density Matrices ◽  
Single Error

General formulas for estimating the errors in quantum-mechanical calculations are given in the formalism of density matrices. Some properties of the traces of matrices are used to simplify the estimating and to indicate a way of obtaining a better approximation. It is shown that the simultaneous correction of all the equations to be fulfilled leads in most cases to a faster convergence than the exact fulfilment of some of the equations and approximating stepwise to some of the others. The corrective formulas contain only direct operations of the matrices occurring and so they are advantageous in computer applications. In the last section a ‘subjective error’ definition is given and by taking into account the weight of the errors of the several equations a faster convergence and a single error quantity is obtained. Some special applications of the method will be published later.

THE UNREASONABLE EFFECTIVENESS OF EXPONENTIALLY SUPPRESSED CORRECTIONS IN PRESERVING INFORMATION

2013 ◽  
Vol 22 (12) ◽  
pp. 1342030 ◽  
Author(s):  
KYRIAKOS PAPADODIMAS ◽  
SUVRAT RAJU
Keyword(s):  
Hilbert Space ◽  
Black Hole ◽  
Quantum Mechanics ◽  
Nonperturbative Effects ◽  
Von Neumann Entropy ◽  
Von Neumann ◽  
Information Theoretic ◽  
Black Hole Evaporation ◽  
Strong Subadditivity ◽  
Unreasonable Effectiveness

We point out that nonperturbative effects in quantum gravity are sufficient to reconcile the process of black hole evaporation with quantum mechanics. In ordinary processes, these corrections are unimportant because they are suppressed by e-S. However, they gain relevance in information-theoretic considerations because their small size is offset by the corresponding largeness of the Hilbert space. In particular, we show how such corrections can cause the von Neumann entropy of the emitted Hawking quanta to decrease after the Page time, without modifying the thermal nature of each emitted quantum. Second, we show that exponentially suppressed commutators between operators inside and outside the black hole are sufficient to resolve paradoxes associated with the strong subadditivity of entropy without any dramatic modifications of the geometry near the horizon.

The optimal control software environment for a nonlinear system class via quadratic criteria

2015 ◽  
Vol 38 ◽  
pp. 19-21
Author(s):  
А.П. Афанасьев ◽  
Afanasiev A.P. ◽  
И.И. Емельянова ◽  
Emelyanova I.I. ◽  
С.М. Дзюба ◽  
...  
Keyword(s):  
Optimal Control ◽  
Nonlinear System ◽  
Control Software ◽  
Software Environment ◽  
Quadratic Criteria
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