Mehmet Burak Şahinoğlu

arXiv2016

Recoupling coefficients and quantum entropies

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.

quant-ph↗

arXiv2014

Characterizing Topological Order with Matrix Product Operators

One of the most striking features of quantum phases that exhibit topological order is the presence of long range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of the corresponding ground state wavefunctions, in which the full wavefunction is encoded in terms of local tensors. Topological order is reflected in the symmetries of these tensors, and we give a characterization of those symmetries in terms of matrix product operators acting on the virtual level. This leads to a set of algebraic rules characterizing states with topological quantum order. The corresponding matrix product operators fully encode all topological features of the theory, and provide a systematic way of constructing topological states. We generalize the conditions of $\mathsf{G}$ and twisted injectivity to the matrix product operator case, and provide a complete picture of the ground state manifold on the torus. As an example, we show how all string-net models of Levin and Wen fit within this formalism, and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of certain matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string-nets, and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.

quant-ph↗