Nonzero Kronecker Coefficients and What They Tell us about Spectra |
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Authors: | Matthias Christandl Aram W. Harrow Graeme Mitchison |
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Affiliation: | (1) Centre for Quantum Computation, DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK;(2) Department of Computer Science, University of Bristol, Bristol, BS8 1UB, UK |
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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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