Annales Henri Poincaré

, Volume 19, Issue 2, pp 385–410 | Cite as

Recoupling Coefficients and Quantum Entropies

  • Matthias Christandl
  • M. Burak ŞahinoğluEmail author
  • Michael Walter


We prove that the asymptotic behavior of the recoupling coefficients of the symmetric group \(S_k\) is characterized by a quantum marginal problem: they decay polynomially in k if there exists a quantum state of three particles with given eigenvalues for their reduced density operators and exponentially otherwise. As an application, we deduce solely from symmetry considerations of the coefficients the strong subadditivity property of the von Neumann entropy, first proved by Lieb and Ruskai (J Math Phys 14:1938–1941, 1973). Our work may be seen as a non-commutative generalization of the representation-theoretic aspect of the recently found connection between the quantum marginal problem and the Kronecker coefficient of the symmetric group, which has applications in quantum information theory and algebraic complexity theory. This connection is known to generalize the correspondence between Weyl’s problem on the addition of Hermitian matrices and the Littlewood–Richardson coefficients of SU(d). In this sense, our work may also be regarded as a generalization of Wigner’s famous observation of the semiclassical behavior of the recoupling coefficients (here also known as 6j or Racah coefficients), which decay polynomially whenever a tetrahedron with given edge lengths exists. More precisely, we show that our main theorem contains a characterization of the possible eigenvalues of partial sums of Hermitian matrices thus presenting a representation-theoretic characterization of a generalization of Weyl’s problem. The appropriate geometric objects to SU(d) recoupling coefficients are thus tuples of Hermitian matrices and to \(S_k\) recoupling coefficients they are three-particle quantum states.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Matthias Christandl
    • 1
  • M. Burak Şahinoğlu
    • 2
    Email author
  • Michael Walter
    • 3
    • 4
  1. 1.QMATH, Department of Mathematical SciencesUniversity of CopenhagenCopenhagen ØDenmark
  2. 2.Institute for Quantum Information and MatterCalifornia Institute of TechnologyPasadenaUSA
  3. 3.QuSoftKorteweg-de Vries Institute for Mathematics Institute of Physics and Institute for Logic, Language and Computation University of AmsterdamAmsterdamThe Netherlands
  4. 4.Stanford Institute for Theoretical PhysicsStanford UniversityStanfordUSA

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