Recoupling coefficients and quantum entropies

Abstract

We prove that the asymptotic behavior of the recoupling coefficients of the symmetric group is characterized by a quantum marginal problem -- namely, by the existence of quantum states of three particles with given eigenvalues for their reduced density operators. This generalizes Wigner's observation that the semiclassical behavior of the 6j-symbols for SU(2) -- fundamental to the quantum theory of angular momentum -- is governed by the existence of Euclidean tetrahedra. As a corollary, we deduce solely from symmetry considerations the strong subadditivity property of the von Neumann entropy. Lastly, we show that the problem of characterizing the eigenvalues of partial sums of Hermitian matrices arises as a special case of the quantum marginal problem. We establish a corresponding relation between the recoupling coefficients of the unitary and symmetric groups, generalizing a classical result of Littlewood and Murnaghan.

25 pages

References in corpus (12)

• N-representability is QMA-complete
• Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information
• A Matrix Convexity Approach to Some Celebrated Quantum Inequalities
• The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group
• Quantum state transformations and the Schubert calculus
• A new inequality for the von Neumann entropy
• Convexity of momentum map, Morse index, and quantum entanglement
• Critical sets of the total variance of state detect all SLOCC entanglement classes
• Stabilizer information inequalities from phase space distributions
• Changepoint Problem in Quantumn Setting
• The Ponzano-Regge asymptotic of the 6j symbol: an elementary proof
• A quantum information-theoretic proof of the relation between Horn's problem and the Littlewood-Richardson coefficients

Cited by in corpus (7)

• Asymptotic performance of port-based teleportation
• A Quantum Multiparty Packing Lemma and the Relay Channel
• Tip of the Quantum Entropy Cone
• All Kronecker coefficients are reduced Kronecker coefficients
• Lower bound on entanglement in subspaces defined by Young diagrams
• Refuting spectral compatibility of quantum marginals
• Refining ensemble $N$-representability of one-body density matrices from partial information