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Mathematical Structures in Computer Science

Entanglement and Hilbert space geometry for systems with a few qubits

Rémy Mosseri, Pedro Ribeiro

Published Nov 23, 2007• 10 citations
Publisher

Abstract

This paper reviews recent attempts to describe the two- and three-qubit Hilbert space geometries. In the first part, this is done with the help of Hopf fibrations of hyperspheres. It is shown that the associated Hopf map is strongly sensitive to states’ entanglement content. In the two-qubit case, a generalisation of the celebrated one-qubit Bloch sphere representation is described. In the second part, we present Hilbert space discrete versions, which are comparable to polyhedral approximations of spheres in standard geometry.

DOI
10.1017/s0960129507006329

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This paper reviews recent attempts to describe the two- and three-qubit Hilbert space geometries. In the first part, this is done with the help of Hopf fibrations of hyperspheres. It is shown that the associated Hopf map is strongly sensitive to states’ entanglement content. In the two-qubit case, a generalisation of the celebrated one-qubit Bloch sphere representation is described. In the second part, we present Hilbert space discrete versions, which are comparable to polyhedral approximations of spheres in standard geometry.
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.
Based on the geometry of entangled three and two qubit states, we present the connection between the entanglement measure of the three-qubit state defined using the last Hopf fibration and the entanglement measures known as two- and three-tangle. Moreover, the generalization of the geometric representation of four qubit state and a potential entanglement measure is studied using sedenions for the simplification of the Hilbert space S^31 of the four qubit system. An entanglement measure is proposed and the degree of entanglement is calculated for specific states. The difficulties of a possible generalization are discussed.
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 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.