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It is known that the Frenet-Serret apparatus of a space curve inthree-dimensional Euclidean space determines the local geometry of curves. Inparticular, the Frenet-Serret apparatus specifies important geometricinvariants, including the curvature and the torsion of a curve. It is alsoacknowledged in quantum information science that low complexity and highefficiency are essential features to achieve when cleverly manipulating quantumstates that encode quantum information about a physical system. In this paper,we propose a geometric perspective on how to quantify the bending and thetwisting of quantum curves traced by dynamically evolving state vectors.Specifically, we propose a quantum version of the Frenet-Serret apparatus for aquantum trajectory in projective Hilbert space traced by a parallel-transportedpure quantum state evolving unitarily under a stationary Hamiltonian specifyingthe Schrodinger equation. Our proposed constant curvature coefficient is givenby the magnitude squared of the covariant derivative of the tangent vector tothe state vector and represents a useful measure of the bending of the quantumcurve. Our proposed constant torsion coefficient, instead, is defined in termsof the magnitude squared of the projection of the covariant derivative of thetangent vector, orthogonal to both the tangent vector and the state vector. Thetorsion coefficient provides a convenient measure of the twisting of thequantum curve. Remarkably, we show that our proposed curvature and torsioncoefficients coincide with those existing in the literature, althoughintroduced in a completely different manner...

From the classical Frenet-Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part I. Stationary Hamiltonians