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arXiv 1507.07585hep-thgr-qcmath.DGquant-ph
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变形锥、膜的紧化与量子比特的几何

Compactifications of Deformed Conifolds, Branes and the Geometry of Qubits

M. Cvetic, G. W. Gibbons, C. N. Pope

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31 pages
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摘要：

我们提出了三个家庭的确切的，一致性1爱因斯坦度规在$(2n+2)$维，这是推广的Stenzel建设的Ricci平坦度规，那些与一个积极的宇宙学常数。第一类解是复射影空间$CP^{n+1}$上的Fubini-Study度量，以Stenzel形式表示，其主轨道是Stiefel流形$V_2(R ^{n+2})=SO(n+2)/SO(n)$除以$Z_2$。第二类也是爱因斯坦-凯勒度量，现在是在格拉斯曼流形$G_2(R^{n+3})=SO(n+3)/((SO(n+1)\times SO(2))$上，其主轨道是斯蒂费尔流形$V_2(R^{n+2})$（在这种情况下没有$Z_2$因子）。第三类度量是积流形$S^{n+1}\times S^{n+1}$上的Einstein度量，并且是仅对$n=1$的Kähler度量。这些度量中的一些被认为在一致弦理论紧化的研究和AdS/CFT对应的背景下发挥了作用。我们还详细阐述了基于$CP^{n+1}$上Fubini-Study度量的Kähler几何的量子力学的几何方法，并应用该方法研究了量子比特的量子纠缠.

英文摘要：

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 $CP^{n+1}$, written in a Stenzel form, whose principal orbits are the Stiefel manifolds $V_2(R ^{n+2})=SO(n+2)/SO(n)$ divided by $Z_2$. The second family are also Einstein-Kähler metrics, now on the Grassmannian manifolds $G_2(R^{n+3})=SO(n+3)/((SO(n+1)\times SO(2))$, whose principal orbits are the Stiefel manifolds $V_2(R^{n+2})$ (with no $Z_2$ factoring in this case). The third family are Einstein metrics on the product manifolds $S^{n+1}\times S^{n+1}$, 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 $CP^{n+1}$, and we apply the formalism to study the quantum entanglement of qubits.

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