Geometry of a two-qubit state and intertwining quaternionic conformal mapping under local unitary transformations

Najarbashi, G.; Ahadpour, Sodeif; Fasihi, M. A.; Tavakoli, Yaser

doi:10.1088/1751-8113/40/24/014

Cited by 6 publications

(15 citation statements)

References 19 publications

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“…We show that the quaternionic stereographic projection of two-qubit states intertwines between the local unitary SU (2) ⊗ SU (2) and corresponding quaternionic Möbius transformations [18,19], which can be useful in theoretical physics such as quaternionic quantum mechanics [20], quantum conformal field theory [17,18,19,20,21] and quaternionic computations [22]. This generalizes our early work restricted to group SO(2) ⊗ SU (2) [15].…”

Section: Introductionsupporting

confidence: 59%

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

Najarbashi

Seifi

Mirzaei

2015

Quantum Inf Process

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In this paper the geometry of two and three-qubit states under local unitary groups is discussed. We first review the one qubit geometry and its relation with Riemannian sphere under the action of group SU (2). We show that the quaternionic stereographic projection intertwines between local unitary group SU (2) ⊗ SU (2) and quaternionic Möbius transformation. The invariant term appearing in this operation is related to concurrence measure. Yet, there exists the same intertwining stereographic projection for much more global group Sp(2), generalizing the familiar Bloch sphere in 2-level systems. Subsequently, we introduce octonionic stereographic projection and octonionic conformal map (or octonionic Möbius maps) for three-qubit states and find evidence that they may have invariant terms under local unitary operations which shows that both maps are entanglement sensitive.PACs Index: 03.67.a 03.65.Ud

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Section: Introductionsupporting

confidence: 59%

“…However, it seems that there is also another geometrical approach to describe one two and three qubit states called Möbius transformation [15,16]. As is typical in physics, the local properties are more immediately useful than the global properties, and the local unitary transformation is of great importance.…”

Section: Introductionmentioning

confidence: 99%

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

Najarbashi

Seifi

Mirzaei

2015

Quantum Inf Process

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“…This is not a novel idea: representations of single qubit operations on the complex plane have been suggested by J. Lee et al 9 while G. Najarbashi et al have investigated further generalizations. 10 However, to the best of our knowledge, this is the¯rst time that such an idea is used for the analysis of a quantum algorithm. In fact, ðQÞ is an elliptic fractional linear transformation acting on the complex projective line in analogy with the Grover algorithm 11 as a rotation on the complex plane.…”

Section: Of Qmentioning

confidence: 99%

“…Let À=2 ' and =2, 1 and 2 be the¯xed points of the M€ obius transformation ðRÞ as in (10), be the argument of the complex number ð1 À 2 Þ=ð1 À 1 Þ with À < and…”

mentioning

confidence: 99%

Möbius Transformations in Quantum Amplitude Amplification With Generalized Phases

Bautista-Ramos

Castillo-Tépox

2010

Int. J. Quantum Inform.

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The iteration of the operators employed in quantum amplitude ampli¯cation with generalized phases is analyzed by using elementary properties (geometric and algebraic) of the M€ obius transformations (fractional linear transformations). It is shown that, for a given quantum algorithm without measurement, which produces a good state with probability a of success, if the phase angles ' and which mark the good and initial states respectively satisfy ' ¼ with a small enough, then, for a number n of iterations with n 2 Âð1= ffiffiffi a p Þ we get an error probability that is at most Oða 2 Þ.

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“…The Hopf fibration has a wide variety of physical applications including magnetic monopoles [1], string theory [2,3,4], new solution of Maxwell equations [5] and quantum information theory, [6,7,8,9,10,11]. An interesting approach to study the entanglement measure in spin systems, is geometric structure of multipartite qubit systems [6,12,13,14,15,16,17,18,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

Relation Between Stereographic Projection and Concurrence Measure in Bipartite Pure States

Najarbashi

Seifi

2016

Int J Theor Phys

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One-qubit pure states, living on the surface of Bloch sphere, can be mapped onto the usual complex plane by using stereographic projection. In this paper, after reviewing the entanglement of two-qubit pure state, it is shown that the quaternionic stereographic projection is related to concurrence measure. This is due to the fact that every two-qubit state, in ordinary complex field, corresponds to the one-qubit state in quaternionic skew field, called quaterbit. Like the one-qubit states in complex field, the stereographic projection maps every quaterbit onto a quaternion number whose complex and quaternionic parts are related to Schmidt and concurrence terms respectively. Rather, the same relation is established for three-qubit state under octonionic stereographic projection which means that if the state is bi-separable then, quaternionic and octonionic terms vanish.Finally, we generalize recent consequences to 2 ⊗ N and 4 ⊗ N dimensional Hilbert spaces (N ≥ 2) and show that, after stereographic projection, the quaternionic and octonionic terms are entanglement sensitive. These trends are easily confirmed by direct computation for general multi-particle W-and GHZ-states.

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Geometry of a two-qubit state and intertwining quaternionic conformal mapping under local unitary transformations

Abstract: In this paper the geometry of two-qubit systems under local unitary group SO(2) ⊗ SU (2) is discussed. It is shown that the quaternionic conformal map intertwines between this local unitary subgroup of Sp (2)

Cited by 6 publications

References 19 publications

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

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

Möbius Transformations in Quantum Amplitude Amplification With Generalized Phases

Relation Between Stereographic Projection and Concurrence Measure in Bipartite Pure States

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