The Hyperbolic Geometric Structure of the Density Matrix for Mixed State Qubits

Sergey N. Filippov, Vladimir I. Man’ko

Abraham Albert Ungar

Density matrices for mixed state qubits, parametrized by the Bloch vector in the open unit ball of the Euclidean 3-space, are well known in quantum information and computation theory. By presenting new identities for the qubit density matrix we indicate its intimate relationship with M\"obius addition and scalar multiplication. The latter, in turn, form the algebraic setting for the Poincar\'e ball model of hyperbolic geometry so that, as a result, the qubit density matrix is linked to hyperbolic geometry.

Lin Zhang, Jiamei Wang, Zhihua Chen

Vladimir N. Chernega, Olga V. Man’ko, Vladimir I. Man’ko

Péter Lévay

It is shown that Uhlmann's parallel transport of purifications along a path of mixed states represented by 2 × 2 density matrices is just the path-ordered product of Thomas rotations. These rotations are invariant under hyperbolic translations inside the Bloch sphere that can be regarded as the Poincaré ball model of hyperbolic geometry. A general expression for the mixed state geometric phase for an arbitrary geodesic triangle in terms of the Bures fidelities is derived. The formula gives back the solid angle result well known from studies of the pure state geometric phase. It is also shown that this mixed state anholonomy can be reinterpreted as the pure state non-Abelian anholonomy of entangled states living in a suitable restriction of the quaternionic Hopf bundle. In this picture Uhlmann's parallel transport is just the Pancharatnam transport of quaternionic spinors.

Chinnaraji Annamalai