Abstract
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.
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Communicated by M.B. Ruskai
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Christandl, M., Harrow, A.W. & Mitchison, G. Nonzero Kronecker Coefficients and What They Tell us about Spectra. Commun. Math. Phys. 270, 575–585 (2007). https://doi.org/10.1007/s00220-006-0157-3
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DOI: https://doi.org/10.1007/s00220-006-0157-3