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Geometry of two-qubit system and hopf fibration

Wen Wei Wong

Published Mar 1, 2013• 0 citations
OpenAlex

Abstract

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.

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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.
In this paper, geometric relation of complex and quaternionic Hopf fibration is studied using base space map. Local sections of the complex Hopf fibration are mapped to the base space of quaternionic Hopf fibration in order to establish correspondence between base space CP 3 and S 4 , with respect to both entangled states or non-entangled states. It is shown that there is a symmetry of coordinate transformation corresponds to different sections of CP 3 , which is expressed in terms of transition functions having the U (1) group structure. Under this map, the phases of the transition functions in S 4 doubled over that of the phases in the transition functions for CP 3 .
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.
Standard quantum information is formulated over complex Hilbert spaces, where the pure state space of a single qubit is geometrically encoded by the first Hopf fibration S3→S2. Beyond this familiar setting, the normed division algebras C, H and O provide a hierarchy of increasingly rich algebraic and topological structures that has motivated several extensions of quantum theory and quantum computation. This review synthesizes the literature connecting quaternionic and octonionic frameworks, Clifford algebras, spinors, projective spaces and Hopf fibrations. We emphasize a central conceptual point that is often blurred in the literature: the second and third Hopf fibrations play two distinct roles, namely, (i) as kinematical descriptions of hypercomplex single-particle state spaces such as the quaternionic projective line HP1≃S4, and (ii) as entanglement-sensitive descriptions of multi-qubit complex systems, especially two- and three-qubit Hilbert spaces. On the algebraic side, Clifford and geometric algebras provide a natural language for rotations, spinors, and gate synthesis, while quaternionic Hilbert modules furnish a mathematically consistent extension of standard qubit kinematics and dynamics. By contrast, octonionic models face major obstructions due to non-associativity, which affects inner products, tensor products, spectral theory and circuit composition. We therefore distinguish carefully between robust results, partial constructions and speculative directions. The outcome is a unified geometric review of hypercomplex quantum information, together with a map of open problems at the interface of topology, noncommutative algebra, and quantum computation.
FINDING: The Hopf fibration (S³→S²) is the topological backbone of single-qubit Bloch sphere geometry, and its generalization to multi-qubit entangled states via higher-dimensional Hopf fibrations reveals a nested sphere structure where entanglement corresponds to specific fiber/base decompositions. | MATH: Single qubit: S³ ≅ SU(2) ≅ Hopf fiber bundle S¹ ↪ S³ → S² (base = Bloch sphere, fiber = U(1) phase). For 2 qubits: S⁷ → S⁴ (Hopf fibration with S³ fibers) — the arXiv paper (quant-ph/0108137v1) explicitly constructs entangled-state geometry via S⁷ → S⁴, where the S³ fiber encodes relative phase and the S⁴ base encodes the joint probability amplitudes. Key invariants: Hopf invariant H = 1 for the S³→S² map (linking number of any two fibers = 1); for S⁷→S⁴, the Hopf invariant is also 1 (the only higher-dimensional Hopf maps with fiber S³, S⁷, S¹⁵ exist in dimensions 2, 4, 8 — tied to normed division algebras ℝ, ℂ, ℍ, 𝕆). | CONNECTION: The Hopf fibration's base S² is the Riemann sphere Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We discuss a generalization of the standard Bloch sphere representation for a single qubit to two qubits, in the framework of Hopf fibrations of high-dimensional spheres by lower dimensional spheres. The single-qubit Hilbert space is the three-dimensional sphere S 3 . The S 2 base space of a suitably oriented S 3 Hopf fibration is nothing but the Bloch sphere, while the circular fibres represent the overall qubit phase degree of freedom. For the two-qubits case, the Hilbert space is a seven-dimensional sphere S 7 , which also allows for a Hopf fibration, with S 3 fibres and a S 4 base. The most striking result is that suitably oriented S 7 Hopf fibrations are entanglement sensitive. The relation with the standard Schmidt decomposition is also discussed.
We present a generalization to three qubits of the standard Bloch sphere representation for a single qubit and of the seven-dimensional sphere representation for two qubits presented in Mosseri et al (Mosseri R and Dandoloff R 2001 J. Phys. A: Math. Gen. 34 10243). The Hilbert space of the three-qubit system is the 15-dimensional sphere S 15 , which allows for a natural (last) Hopf fibration with S 8 as base and S 7 as fibre. A striking feature is, as in the case of one and two qubits, that the map is entanglement sensitive, and the two distinct ways of un-entangling three qubits are naturally related to the Hopf map. We define a quantity that measures the degree of entanglement of the three-qubit state. Conjectures on the possibility of generalizing the construction for higher qubit states are also discussed.