We discuss a generalization of the standard Bloch sphere representation for a single qubit to two qubits, in the framework of Hopf fibrations of highdimensional 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 an exact calculation of the effective geometry-induced quantum potential for a particle confined on a helicoidal ribbon. This potential leads to the appearance of localized states at the rim of the helicoid. In this geometry the twist of the ribbon plays the role of an effective transverse electric field on the surface and thus this is reminiscent of the Hall effect.The interplay of geometry and topology is a recurring theme in physics, particularly when these effects manifest themselves in unusual electronic and magnetic properties of materials. Specifically, helical ribbons provide a fertile playground for such effects. Both the helicoid ͑a minimal surface͒ and helical ribbons are ubiquitous in nature: they occur in biology, e.g., as beta sheets in protein structures, 1 macromolecules ͑such as DNA͒, 2 and tilted chiral lipid bilayers. 3 Many structural motifs of biomolecules result from helical arrangements: 4 cellulose fibrils in cell walls of plants, chitin in arthropod cuticles, and collagen protein in skeletal tissue.Condensed matter examples include screw dislocations in smectic A liquid crystals 5 and certain ferroelectric liquid crystals. 6 A helicoid to spiral ribbon transition 7 and geometrically induced bifurcations from the helicoid to the catenoid 8 have also been studied.In this context, our goal is to answer the following questions: what kind of an effective quantum potential does a charge ͑or electron͒ experience on a helicoid or a helical ribbon due to its geometry ͑i.e., curvature and twist͒? If the outer edge of the helicoid is charged, how is this potential modified and if there are any bound states? Our main findings are: the twist will push the electrons in vanishing angular-momentum state toward the inner edge of the ribbon and push the electrons in nonvanishing angular-momentum states to the outer edge, thus creating an inhomogeneous effective electric field between the inner and outer rims of the helicoidal ribbon. This is reminiscent of the Hall effect; only here it is geometrically induced. We expect our results to lead to new experiments on related twisted materials where the predicted effect can be verified. In a related context we note that de Gennes 9 had explained the buckling of a flat solid ribbon in terms of the ferroelectric polarization charges on the edges.In order to answer the questions posed above, here we study the helicoidal surface to gain a broader understanding of the interaction between quantum particles and curvature, and the resulting possible physical effects. The properties of free electrons on this geometry have been considered before. 10 The results of this Brief Report are based on the Schrödinger equation for a confined quantum particle on a submanifold of R 3 . Following da Costa 11 an effective potential appears in the two-dimensional Schrödinger equation which has the following form:where m ‫ء‬ is the effective mass of the particle, ប is the Planck's constant, and M and K are the mean and the Gaussian curvature, respectively.To describe the geo...