Nonzero Kronecker Coefficients and What They Tell us about Spectra

A triple of spectra (r^A, r^B, r^AB) is said to be admissible if there is a density operator ρ^AB with $$({\rm Spec} \rho^{A}, {\rm Spec} \rho^{B}, {\rm Spec} \rho^{AB})=(r^A, r^B, r^{AB})$$. How can we characterise such triples? It turns out that the admissible spectral triples correspond to Young diagrams (μ, ν, λ) with nonzero Kronecker coefficient g_μνλ [5, 14]. This means that the irreducible representation of the symmetric group V_λ is contained in the tensor product of V_μ and V_ν. Here, we show that such triples form a finitely generated semigroup, thereby resolving a conjecture of Klyachko [14]. As a consequence we are able to obtain stronger results than in [5] and give a complete information-theoretic proof of the correspondence between triples of spectra and representations. Finally, we show that spectral triples form a convex polytope.