A Geometric View on Quantum Tensor Networks

Tensor network states and algorithms play a key role in understanding the structure of complex quantum systems and their entanglement properties. This report is devoted to the problem of the construction of an arbitrary quantum state using the differential geometric scheme of covariant series in Riemann normal coordinates. The building blocks of the scheme are polynomials in the Pauli operators with the coefficients specified by the curvature, torsion, and their covariant derivatives on some base manifold. The problem of measuring the entanglement of multipartite mixed states is shortly discussed.