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Register a new userSuperadditivity of quantum relative entropy for general states
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The property of superadditivity of the quantum relative entropy states that, in a bipartite system $mathcal{H}_{AB}=mathcal{H}_A otimes mathcal{H}_B$, for every density operator $rho_{AB}$ one has $ D( rho_{AB} || sigma_A otimes sigma_B ) ge D( rho_A || sigma_A ) +D( rho_B || sigma_B) $. In this work, we provide an extension of this inequality for arbitrary density operators $ sigma_{AB} $. More specifically, we prove that $ alpha (sigma_{AB})cdot D({rho_{AB}}||{sigma_{AB}}) ge D({rho_A}||{sigma_A})+D({rho_B}||{sigma_B})$ holds for all bipartite states $rho_{AB}$ and $sigma_{AB}$, where $alpha(sigma_{AB})= 1+2 || sigma_A^{-1/2} otimes sigma_B^{-1/2} , sigma_{AB} , sigma_A^{-1/2} otimes sigma_B^{-1/2} - mathbb{1}_{AB} ||_infty$.
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In this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. The latter inequality, which we call approximate tensorization of the relative entropy, can be expressed as a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.
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The scattering amplitude in simple quantum graphs is a well-known process which may be highly complex. In this work, motivated by the Shannon entropy, we propose a methodology that associates to a graph a scattering entropy, which we call the average scattering entropy. It is defined by taking into account the period of the scattering amplitude which we calculate using the Greens function procedure. We first describe the methodology on general grounds, and then exemplify our findings considering several distinct groups of graphs. We go on and investigate other possibilities, one that contains groups of graphs with the same number of vertices, with the same degree, and the same number of edges, with the same length, but with distinct topologies and with different entropies. And the other, which contains graphs of the fishbone type, where the scattering entropy depends on the boundary conditions on the vertices of degree $1$, with the corresponding values decreasing and saturating very rapidly, as we increase the number of elementary structures in the graphs.
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