  It is known that the Frenet-Serret apparatus of a space curve in
three-dimensional Euclidean space determines the local geometry of curves. In
particular, the Frenet-Serret apparatus specifies important geometric
invariants, including the curvature and the torsion of a curve. It is also
acknowledged in quantum information science that low complexity and high
efficiency are essential features to achieve when cleverly manipulating quantum
states that encode quantum information about a physical system. In this paper,
we propose a geometric perspective on how to quantify the bending and the
twisting of quantum curves traced by dynamically evolving state vectors.
Specifically, we propose a quantum version of the Frenet-Serret apparatus for a
quantum trajectory in projective Hilbert space traced by a parallel-transported
pure quantum state evolving unitarily under a stationary Hamiltonian specifying
the Schrodinger equation. Our proposed constant curvature coefficient is given
by the magnitude squared of the covariant derivative of the tangent vector to
the state vector and represents a useful measure of the bending of the quantum
curve. Our proposed constant torsion coefficient, instead, is defined in terms
of the magnitude squared of the projection of the covariant derivative of the
tangent vector, orthogonal to both the tangent vector and the state vector. The
torsion coefficient provides a convenient measure of the twisting of the
quantum curve. Remarkably, we show that our proposed curvature and torsion
coefficients coincide with those existing in the literature, although
introduced in a completely different manner...
