Q25: Which irrep measurements are useful for qudits?

Working mathematical formulation

Consider first fixed- $d$ $\mathrm{SU}(d)$ fundamental systems, initially maximally mixed, with projective measurements onto specified irreducible components of two or $k$ tensor factors. Specify the measured subspaces and whether individual multiplicity copies are resolved; preserve coherence within each reported subspace. Determine the bounded-error decision power and preparation cost for fixed $d,k$ , then separate growing- $d$ and growing- $k$ regimes. Compare with physical spin- $S$ representations of $\mathrm{SU}(2)$ , which have a different tensor-product decomposition.

Scope & assumptions

Encoding a spin- $S$ system with $2S$ qubits has cost polynomial in $S$ , which may be exponential in $\log S$ . State the size convention before drawing a complexity conclusion.

Why this is interesting

The two-qubit correspondence is unusually economical. Its extensions may reveal which part of the primitive matters.

A place to start

Compare two qutrits viewed as $\mathrm{SU}(3)$ fundamentals with two physical spin-one particles and list exactly which subspaces each proposed instrument resolves.

Related work

These works provide context and established results. They need not be the origin of this exact formulation.

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