Schmidt-Correlated States, Weak Schmidt Decomposition and Generalized Bell Bases Related to Hadamard Matrices

Introduction: The Hadamard conjecture about the existence of Hadamard matrices in all orders multiple of 4, and the Gauss problem about the number of points in a circle are among the most important turning points in the development of mathematics. They both stimulated the development of scientific schools around the world with an immense amount of works. There are substantiations that these scientific problems are deeply connected. The number of Gaussian points (Z3 lattice points) on a spheroid, cone, paraboloid or parabola, along with their location, determines the number and types of Hadamard matrices.Purpose: Specification of the upper and lower bounds for the number of Gaussian points (with odd coordinates) on a spheroid depending on the problem size, in order to specify the Gauss theorem (about the solvability of quadratic problems in triangular numbers by projections onto the Liouville plane) with estimates for the case of Hadamard matrices. Methods: The authors, in addition to their previous ideas about proving the Hadamard conjecture on the base of a one-to-one correspondence between orthogonal matrices and Gaussian points, propose one more way, using the properties of generalized circles on Z3 .Results: It is proved that for a spheroid, the lower bound of all Gaussian points with odd coordinates is equal to the equator radius R, the upper limit of the points located above the equator is equal to the length of this equator L=2πR, and the total number of points is limited to 2L. Due to the spheroid symmetry in the sector with positive coordinates (octant), this gives the values of R/8 and L/4. Thus, the number of Gaussian points with odd coordinates does not exceed the border perimeter and is no less than the relative share of the sector in the total volume of the figure.Practical significance: Hadamard matrices associated with lattice points have a direct practical significance for noise-resistant coding, compression and masking of video information.

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Quantum Information in Neural Systems

Symmetry ◽

10.3390/sym13050773 ◽

2021 ◽

Vol 13 (5) ◽

pp. 773

Author(s):

Danko D. Georgiev

Keyword(s):

Quantum Entanglement ◽

Quantum Coherence ◽

Quantum Physics ◽

Quantitative Measure ◽

Quantum States ◽

Quantum Evolution ◽

Einstein Relation ◽

Schmidt Decomposition ◽

The Central Nervous System ◽

The Individual

Identifying the physiological processes in the central nervous system that underlie our conscious experiences has been at the forefront of cognitive neuroscience. While the principles of classical physics were long found to be unaccommodating for a causally effective consciousness, the inherent indeterminism of quantum physics, together with its characteristic dichotomy between quantum states and quantum observables, provides a fertile ground for the physical modeling of consciousness. Here, we utilize the Schrödinger equation, together with the Planck–Einstein relation between energy and frequency, in order to determine the appropriate quantum dynamical timescale of conscious processes. Furthermore, with the help of a simple two-qubit toy model we illustrate the importance of non-zero interaction Hamiltonian for the generation of quantum entanglement and manifestation of observable correlations between different measurement outcomes. Employing a quantitative measure of entanglement based on Schmidt decomposition, we show that quantum evolution governed only by internal Hamiltonians for the individual quantum subsystems preserves quantum coherence of separable initial quantum states, but eliminates the possibility of any interaction and quantum entanglement. The presence of non-zero interaction Hamiltonian, however, allows for decoherence of the individual quantum subsystems along with their mutual interaction and quantum entanglement. The presented results show that quantum coherence of individual subsystems cannot be used for cognitive binding because it is a physical mechanism that leads to separability and non-interaction. In contrast, quantum interactions with their associated decoherence of individual subsystems are instrumental for dynamical changes in the quantum entanglement of the composite quantum state vector and manifested correlations of different observable outcomes. Thus, fast decoherence timescales could assist cognitive binding through quantum entanglement across extensive neural networks in the brain cortex.

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Hermitian congruence and the existence and completion of generalized Hadamard matrices

Journal of Combinatorial Theory Series A ◽

10.1016/0097-3165(88)90054-4 ◽

1988 ◽

Vol 49 (2) ◽

pp. 233-261 ◽

Cited By ~ 15

Author(s):

Bradley W Brock

Keyword(s):

Hadamard Matrices

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Hadamard Matrices with Cocyclic Core

Mathematics ◽

10.3390/math9080857 ◽

2021 ◽

Vol 9 (8) ◽

pp. 857

Author(s):

Víctor Álvarez ◽

José Andrés Armario ◽

María Dolores Frau ◽

Félix Gudiel ◽

María Belén Güemes ◽

...

Keyword(s):

Hadamard Matrices ◽

Combinatorial Designs ◽

Powerful Technique ◽

Uniform Approach ◽

Structured Approach

Since Horadam and de Launey introduced the cocyclic framework on combinatorial designs in the 1990s, it has revealed itself as a powerful technique for looking for (cocyclic) Hadamard matrices. Ten years later, the series of papers by Kotsireas, Koukouvinos and Seberry about Hadamard matrices with one or two circulant cores introduced a different structured approach to the Hadamard conjecture. This paper is built on both strengths, so that Hadamard matrices with cocyclic cores are introduced and studied. They are proved to strictly include usual Hadamard matrices with one and two circulant cores, and therefore provide a wiser uniform approach to a structured Hadamard conjecture.

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Automorphism groups of Hadamard matrices

Journal of Combinatorial Theory ◽

10.1016/s0021-9800(69)80088-8 ◽

1969 ◽

Vol 6 (3) ◽

pp. 279-281 ◽

Cited By ~ 24

Author(s):

William M. Kantor

Keyword(s):

Automorphism Groups ◽

Hadamard Matrices

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Construction of symmetric Hadamard matrices of order 4v for v = 47, 73, 113

Special Matrices ◽

10.1515/spma-2018-0002 ◽

2018 ◽

Vol 6 (1) ◽

pp. 11-22 ◽

Cited By ~ 5

Author(s):

N. A. Balonin ◽

D. Ž. Ðokovic ◽

D. A. Karbovskiy

Keyword(s):

Hadamard Matrices ◽

Systematic Search

Abstract We continue our systematic search for symmetric Hadamard matrices based on the so called propus construction. In a previous paper this search covered the orders 4v with odd v ≤ 41. In this paper we cover the cases v = 43, 45, 47, 49, 51. The odd integers v < 120 for which no symmetric Hadamard matrices of order 4v are known are the following: 47, 59, 65, 67, 73, 81, 89, 93, 101, 103, 107, 109, 113, 119. By using the propus construction, we found several symmetric Hadamard matrices of order 4v for v = 47, 73, 113.

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