Two- and three-qubit geometry, quaternionic and octonionic conformal maps, and intertwining stereographic projection

Najarbashi, G.; Seifi, B.; Mirzaei, S.

doi:10.1007/s11128-015-1172-0

Cited by 11 publications

(13 citation statements)

References 29 publications

(42 reference statements)

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“…As already pointed out in the introduction, this topic in the octonionic setting attracted in the recent time a lot of interest from physicists in the context of extensions of the standard model of particle physics and super gravity, see for instance [5,12,21].…”

Section: Resultsmentioning

confidence: 99%

Conformal Mappings Revisited in the Octonions and Clifford Algebras of Arbitrary Dimension

Kraußhar

2020

Adv. Appl. Clifford Algebras

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In this paper we revisit the concept of conformality in the sense of Gauss in the context of octonions and Clifford algebras. We extend a characterization of conformality in terms of a system of partial differential equations and differential forms using special orthonormal sets of continuous functions that have been used before in the particular quaternionic setting. The aim is to describe to which higher dimensional algebras this characterization can exactly be extended and under which circumstances. It turns out to be crucial that this characterization requires a domain of definition that lies in a subalgebra that has the norm composition property and that is either associative (Clifford algebra case) or at least alternative (octonionic case). The orthonormal frames are elements of the spin group Spin$$(n+1)$$ ( n + 1 ) . We round off by relating the nature of the orthonormal frames to the associated Möbius transformation which are related to SO(9, 1) in the octonionic case and to the Ahlfors–Vahlen group in the case of a Clifford algebra.

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Section: Resultsmentioning

confidence: 99%

Conformal Mappings Revisited in the Octonions and Clifford Algebras of Arbitrary Dimension

Kraußhar

2020

Adv. Appl. Clifford Algebras

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“…The polynomials 0 ( ) and 1 ( ) in this pair of formulas are the same as the difference equation (19) polynomials 0 0 ( ) and 1 0 ( ), given in (25) and (26). At this point in our research, we were able to employ the Mathematica-based HolonomicFunctions package of Christoph Koutschan of the Research Institute for Symbolic Computation (RISC) of Johannes Kepler University.…”

Section: (Concise Formulas) For ( )mentioning

confidence: 99%

Formulas for Generalized Two-Qubit Separability Probabilities

Slater

2018

Advances in Mathematical Physics

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To begin, we find certain formulas ( , ) = 1 ( ) 2 ( ), for = −1, 0, 1, . . . , 9. These yield that part of the total separability probability, ( , ), for generalized (real, complex, quaternionic, etc.) two-qubit states endowed with random induced measure, for which the determinantal inequality | PT | > | | holds. Here denotes a 4×4 density matrix, obtained by tracing over the pure states in 4×(4+ )-dimensions, and PT denotes its partial transpose. Further, is a Dyson-index-like parameter with = 1 for the standard (15-dimensional) convex set of (complex) two-qubit states. For = 0, we obtain the previously reported Hilbert-Schmidt formulas, with (0, 1/2) = 29/128 (the real case), (0, 1) = 4/33 (the standard complex case), and (0, 2) = 13/323 (the quaternionic case), the three simply equalling (0, )/2. The factors 2 ( ) are sums of polynomial-weighted generalized hypergeometric functions −1 , ≥ 7, all with argument = 27/64 = (3/4) 3 . We find number-theoretic-based formulas for the upper ( ) and lower ( ) parameter sets of these functions and, then, equivalently express 2 ( ) in terms of first-order difference equations. Applications of Zeilberger's algorithm yield "concise" forms of (−1, ), (1, ), and (3, ), parallel to the one obtained previously (Slater 2013) for (0, ) = 2 (0, ). For nonnegative half-integer and integer values of , ( , ) (as well as ( , )) has descending roots starting at = − − 1. Then, we (Dunkl and I) construct a remarkably compact (hypergeometric) form for ( , ) itself. The possibility of an analogous "master" formula for ( , ) is, then, investigated, and a number of interesting results are found.

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“…The geometric phase is the magnetic flux due to magnetic monopoles located at the level crossing points [6,7]. In quaternionic representation of quantum state [3,4,8,9], Levay provided a elegant interpretation of the geometric phase as the parallel transformation of quaternionic spinors due to Mannoury-Fubini-Study metric in Hilbert space of two qubit states [11,10]. The relation between geometric phases, phase transition and level crossings for the Heisenberg XY model with transverse magnetic field has been investigated by Oh et al [12].…”

Section: Introductionmentioning

confidence: 99%

Quantum phase transition in the Dzyaloshinskii–Moriya interaction with inhomogeneous magnetic field: Geometric approach

Najarbashi

Seifi

2016

Quantum Inf Process

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Two- and three-qubit geometry, quaternionic and octonionic conformal maps, and intertwining stereographic projection

Cited by 11 publications

References 29 publications

Conformal Mappings Revisited in the Octonions and Clifford Algebras of Arbitrary Dimension

Conformal Mappings Revisited in the Octonions and Clifford Algebras of Arbitrary Dimension

Formulas for Generalized Two-Qubit Separability Probabilities

Quantum phase transition in the Dzyaloshinskii–Moriya interaction with inhomogeneous magnetic field: Geometric approach

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