  In this article, we present a novel approach to investigating entanglement in
the context of quantum computing. Our methodology involves analyzing reduced
density matrices at different stages of a quantum algorithm's execution and
representing the dominant eigenvalue and von Neumann entropy on a graph,
creating an "entanglement trajectory." To establish the trajectory's
boundaries, we employ random matrix theory. Through the examination of examples
such as quantum adiabatic computation, the Grover algorithm, and the Shor
algorithm, we demonstrate that the entanglement trajectory remains within the
established boundaries, exhibiting unique characteristics for each example.
Moreover, we show that these boundaries and features can be extended to
trajectories defined by alternative entropy measures. The entanglement
trajectory serves as an invariant property of a quantum system, maintaining
consistency across varying situations and definitions of entanglement.
Numerical simulations accompanying this research are available via open access.
