Interests in Quantum Physics
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, Bell inequalities and rigorous inequalities of quantum systems
Geometry of Quantum Systems
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.
Tetrahetron of Weyl states:
All states in the set W, the Well states, can (up to local unitaries) be geometrically represented as a tetrahedron spanned by the four Bell states. The singular values (t1; t2; t3) of the correlation tensor serve as coordinates. The set of separable states S forms a double pyramid (blue) and the entangled states are located in the remaining corners (green) outside. The maximally mixed state is located at the origin and the maximal ball of absolutely separable states (purple) around the maximally mixed state is contained within S, but touches the double pyramid at the central points of its eight faces. The states located at the tips of the double pyramid, for instance, the Narnhofer state N at the coordinates (1; 0; 0), are the separable states with maximal purity. All states that cannot violate the CHSH-Bell inequality lie within the dark-yellow, curved surfaces drawn outside the double pyramid, which includes also some entangled states, whereas all states with positive conditional entropy lie within the outermost curved surfaces (red) and the states with negative conditional entropy are located from these surfaces to the tips of the tetrahedron, the Bell states. The solid red line indicates the Werner states, while the dashed red line represents the subset of Gisin states with θ = π/4. For reference, see Nicolai Friis, Sridhar Bulusu and Reinhold A. Bertlmann, Journal of Physics A: Mathematical and General 50, 125301 (2017) and arXiv:1609.04144.
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
Rigorous Inequalities on Radius, Kinetic Energy and Green Functions of Quantum Systems – Selected Papers
Inequalities on radius, kinetic energy, masses and Green functions of quantum systems
