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


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.


Irreducible Representation Symmetric Group Density Operator Young Diagram Convex Polytope 
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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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