  We take inspiration from the Okounkov-Vershik approach to the representation
theory of the symmetric groups to develop a new way of understanding how the
Schur-Weyl duality can be used to perform the Quantum Schur Transform. The
Quantum Schur Transform is a unitary change of basis transformation between the
computational basis of $(\mathbb{C}^d)^{\otimes n}$ and the Schur-Weyl basis of
$(\mathbb{C}^d)^{\otimes n}$. We describe a new multigraph, which we call the
Schur-Weyl-Young graph, that represents both standard Weyl tableaux and
standard Young tableaux in the same diagram. We suggest a major improvement on
Louck's formula for calculating the transition amplitudes between two standard
Weyl tableaux appearing in adjacent levels of the Schur-Weyl-Young graph for
the case $d=2$, merely by looking at the entries in the two tableaux. The key
theoretical component that underpins our results is the discovery of a
branching rule for the Schur-Weyl states, which we call the Schur-Weyl
branching rule. This branching rule allows us to perform the change of basis
transformation described above in a straightforward manner for any $n$ and $d$.
