Differential geometry and fibre bundle theory in quantum mechanics, Dirac monopole, Aharanov-Bohm effect and Berry phase of quantum systems, geometric phases in entangled states, geometry of separable and entangled states

**Geometry of quantum states in the simplex of qubits:
**Tetrahedron of physical states in 2 x 2 dimensions spanned by the four Bell states psi+ , psi- , phi+ , phi- : The separable states form the blue double pyramid and the entangled states are located in the remaining tetrahedron cones. The unitary invariant Kus-Zyczkowski ball (shaded in green) is placed within the double pyramid and the maximal mixture 1/4

*1*is located at the origin. The local states according to a Bell inequality lie within the dark-yellow surfaces containing all separable but also some entangled states. For reference, see Walter Thirring, Reinhold A. Bertlmann, Philipp Köhler, and Heide Narnhofer, Eur. Phys. Journal D64, 181 (2011), and arXiv:1106.3047 [quant-ph].

**Geometry of Entanglement in the simplex of qubits:
**All quantum states represent a tetrahedron. The separable states form the blue double pyramid inside. The entangled states lie in the cones to the Bell states, which are at the corners of the tetrahedron. The green line from the maximal mixture 1/4

*1*at the origin to the Bell state ρ- at the corner represents the Werner states ρ_w. For reference, see Reinhold A. Bertlmann, Heide Narnhofer, and Walter Thirring, Phys.Rev. A66, 032319 (2002) and arXiv:quant-ph/0111116.

**Geometry of entanglement in the magic simplex of qutrits:**

All quantum states represent a pyramide with triangular base. The separable states form the purple cone inside. The yellow shell represents the bound entangled states and in the rest of the pyramide lie the entangled states. The blue line on the boundary plane are the Horodecki states. For reference, see Reinhold A. Bertlmann and Philipp Krammer, Annals of Physics 324 (2009) 1388, and arXiv:0901.4729 [quant-ph].

Ein geometrischer Witz , Video by Elisa Asenbaum

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