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We take inspiration from the Okounkov-Vershik approach to the representationtheory of the symmetric groups to develop a new way of understanding how theSchur-Weyl duality can be used to perform the Quantum Schur Transform. TheQuantum Schur Transform is a unitary change of basis transformation between thecomputational 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 theSchur-Weyl-Young graph, that represents both standard Weyl tableaux andstandard Young tableaux in the same diagram. We suggest a major improvement onLouck's formula for calculating the transition amplitudes between two standardWeyl tableaux appearing in adjacent levels of the Schur-Weyl-Young graph forthe case $d=2$, merely by looking at the entries in the two tableaux. The keytheoretical component that underpins our results is the discovery of abranching rule for the Schur-Weyl states, which we call the Schur-Weylbranching rule. This branching rule allows us to perform the change of basistransformation described above in a straightforward manner for any $n$ and $d$.

A Multigraph Approach for Performing the Quantum Schur Transform