  In the field of quantum information theory, the concept of quantum fidelity
is employed to quantify the similarity between two quantum states. It has been
observed that the fidelity between two states describing a bipartite quantum
system $A \otimes B$ is always less than or equal to the quantum fidelity
between the states in subsystem $A$ alone. While this fidelity inequality is
well understood, determining the conditions under which the inequality becomes
an equality remains an open question. In this paper, we present the necessary
and sufficient conditions for the equality of fidelities between a bipartite
system $A \otimes B$ and subsystem $A$, considering pure quantum states.
Moreover, we provide explicit representations of quantum states that satisfy
the fidelity equality, based on our derived results.
