Séminaire de Mathématique

Quantum Entropies and Representation Theory

by Prof. Michael WALTER (ETH, Zürich)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette
(joint work with M. Christandl and B. Sahinoglu, http://arxiv.org/abs/1210.0463 [5]) The von Neumann entropy is an extension of the classical Shannon entropy to quantum theory. It plays a fundamental role in quantum statistical mechanics and quantum information theory. Mathematically, given a quantum state described by a positive-semidefinite "density matrix", the Neumann entropy equals the Shannon entropy of the eigenvalues. In this talk I will describe an approach to studying eigenvalues and entropies of quantum states that is based on the representation theory of the symmetric group. Its irreducible representations are labeled by Young diagrams, which can be understood as discretizations, or "quantizations", of the spectra of quantum states. In this spirit, I will show that the existence of a quantum state of three particles with given eigenvalues for its reduced density matrices is determined by the asymptotic behavior of a representation-theoretic quantity: the recoupling coefficient, which measures the overlap between two "incompatible" decompositions of a triple tensor product of irreducible representations. As an application, the strong subadditivity of the von Neumann entropy can be deduced solely from symmetry properties of this coefficient. If time permits, I will also discuss the connection of our work to Wigner's observation that the semiclassical behavior of the 6j-symbols for SU(2) -- basic to the quantum theory of angular momentum -- is governed by the existence of Euclidean tetrahedra, and more generally to Horn's problem.