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arXiv:0708.3637 · Journal of Physics A Mathematical and Theoretical

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

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

Published May 30, 2007• 8 citations
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Abstract

In this paper the geometry of two-qubit systems under local unitary group $SO(2)\otimes SU(2)$ is discussed. It is shown that the quaternionic conformal map intertwines between this local unitary subgroup of $Sp(2)$ and the quaternionic Möbius transformation which is rather a generalization of the results of Lee et al (2002 Quantum Inf. Process. 1 129).

DOI
10.1088/1751-8113/40/24/014

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Physical Review A•Jul 17, 2007•8 citations
In this paper the geometry of two-qubit systems under local unitary group SO(2)⊗SU(2)SO(2)\otimes SU(2)SO(2)⊗SU(2) is discussed. It is shown that the quaternionic conformal map intertwines between this local unitary subgroup of Sp(2)Sp(2)Sp(2) and the quaternionic Möbius transformation which is rather a generalization of the results of Lee et al (2002 Quantum Inf. Process. 1 129).
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)SU(2)SU(2). We show that the quaternionic stereographic projection intertwines between local unitary group SU(2)⊗SU(2)SU(2)\otimes SU(2)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)Sp(2)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.
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 Riemanian sphere under the action of group SU(2)SU(2)SU(2). We show that the second Hopf fiberation intertwines between local unitary group SU(2)⊗SU(2)SU(2)\otimes SU(2)SU(2)⊗SU(2) and quaternionic M\obius transformation. The invariant term appearing in this operation is related to concurrence measure. Yet, there exists the same intertwining Hopf map for much more global group Sp(2)Sp(2)Sp(2), generalizing the familiar Bloch sphere in 2-level systems. Subsequently, we introduce third Hopf fiberation and octonionic conformal map (or octonionic M\obius maps) for three-qubit states and find evidence that they may have invariant terms under local unitary operations which shows that both maps have entanglement sensitive.
Complex Hopf fibration and quaternionic Hopf fibration provide distinct ways of describing the state space for two-qubit quantum states. In this research, we have studied the geometry of quantum states for a two-level quantum system, along with the correspondence between complex Hopf bundle and quaternionic Hopf bundle. In the first part of our study, we investigate the behaviour of local coordinates for both Hopf bundles under different degree of entanglement such as entangled states and non-entangled states. Fubini-Study metric for complex projective space is also obtained. Its form suggests that for the intermediate entangled states, complex projective space CP3 can be described as a set of flat three-tori parametrized by a three-sphere. The local inhomogeneous coordinate of CP3 (base space of complex Hopf fibration) is found to carry the description of both subsystems A and B, whereas in the case of maximally entangled state, basis elements of local coordinates CP3 is not linearly independent of each other. Next, we construct a base space map between CP3 and S4, which is denoted as η map. After the mapping, we obtained phases in the base space manifold, different sections and coordinate charts are related by transition functions. We found that there is an inherent symmetry of coordinate transformation corresponds to different sections of CP3, which is expressed in terms of transition functions having the U(1) group structure. Also under η map, phases and transition function in S4 is doubled over that of CP3, indicating subtle symmetric changes after the mapping. The base space coordinates of quaternionic Hopf bundle are consist of two parts, whereby the first part is invariant to the coordinate transformation in CP3 but sensitive to the coordinate transformation in S4.
This paper introduces the quaternionic wavefunction as a compact way to organize the standard spinor geometry of a spin one half particle. It shows how a two component complex state can be regrouped into quaternionic form in a way that makes the geometric relationship among spinors, the three sphere, and the transformation group SU(2) easier to see. The paper emphasizes that the mathematics is standard and that its contribution is expository. Its goal is to clarify how spin one half structure, unit quaternions, and rotational geometry fit together, while carefully distinguishing the full normalized state geometry from the projected Bloch sphere description. It is intended as a conceptual prelude to the broader RQM paper series, not as a claim of a new physical theory.
The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such systems for quantum information processes. In the last two decades, two different approaches, with independent starting points and motivations, have been combined for this purpose. One approach has been to develop the unitary time evolution of two or more qubits in order to study quantum correlations; by exploiting the relevant Lie algebras and, especially, sub-algebras of the Hamiltonians involved, researchers have arrived at connections to finite projective geometries and combinatorial designs. Independently, geometers, by studying projective ring lines and associated finite geometries, have come to parallel conclusions. This review brings together the Lie-algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions. Altogether, this may be seen as further development of Felix Klein’s Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives that we consider may provide further insight into quantum information problems. Extensions are considered for multiple qubits, as well as higher-spin or higher-dimensional qudits.