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Communications in Mathematical Physics

, Volume 270, Issue 3, pp 575–585 | Cite as

Nonzero Kronecker Coefficients and What They Tell us about Spectra

  • Matthias ChristandlEmail author
  • Aram W. Harrow
  • Graeme Mitchison
Article

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.

Keywords

Irreducible Representation Symmetric Group Density Operator Young Diagram Convex Polytope 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Matthias Christandl
    • 1
    Email author
  • Aram W. Harrow
    • 2
  • Graeme Mitchison
    • 1
  1. 1.Centre for Quantum Computation, DAMTPUniversity of CambridgeCambridgeUK
  2. 2.Department of Computer ScienceUniversity of BristolBristolUK

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