Compactifications of Deformed Conifolds, Branes and the Geometry of Qubits

We present three families of exact, cohomogeneity-one Einstein metrics in (2n+2) dimensions, which are generalizations of the Stenzel construction of Ricci-flat metrics to those with a positive cosmological constant. The first family of solutions are Fubini-Study metrics on the complex projective spaces CPⁿ⁺¹, written in a Stenzel form, whose principal orbits are the Stiefel manifolds V₂\(R ⁿ⁺²\)=SO(n+2)/SO(n) divided by Z₂. The second family are also Einstein-Kähler metrics, now on the Grassmannian manifolds G₂\(Rⁿ⁺³\)=SO(n+3)/((SO(n+1)times SO(2)), whose principal orbits are the Stiefel manifolds V₂\(Rⁿ⁺²\) with no $Z₂$ factoring in this case. The third family are Einstein metrics on the product manifolds Sⁿ⁺¹times Sⁿ⁺¹, and are Kähler only for n=1. Some of these metrics are believed to play a role in studies of consistent string theory compactifications and in the context of the AdS/CFT correspondence. We also elaborate on the geometric approach to quantum mechanics based on the Kähler geometry of Fubini-Study metrics on CPⁿ⁺¹, and we apply the formalism to study the quantum entanglement of qubits.