2005/11/03 by Matthias Christandl, Aram W. Harrow, Aram Harrow +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #quant-ph
paper · pdf · doi:10.1007/s00220-006-0157-3
published as Commun. Math. Phys., 270, 575-585 (2007) · 13 pages
arxiv created 2005/11/03 · openalex publication_date 2007/01/08 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
A triple of spectra (rA, rB, rAB) is said to be admissible if there is a density operator rhoAB with (Spec rhoA, Spec rhoB, Spec rhoAB)=(rA, rB, rAB). How can we characterise such triples? It turns out that the admissible spectral triples correspond to Young diagrams (mu, nu, lambda) with nonzero Kronecker coefficient [M. Christandl and G. Mitchison, to appear in Comm. Math. Phys., quant-ph/0409016; A. Klyachko, quant-ph/0409113]. This means that the irreducible representation Vlambda is contained in the tensor product of Vmu and Vnu. Here, we show that such triples form a finitely generated semigroup, thereby resolving a conjecture of Klyachko. As a consequence we are able to obtain stronger results than in [M. Ch. and G. M. op. cit.] 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.