Renyi squashed entanglement, discord, and relative entropy differences

The squashed entanglement quantifies the amount of entanglement in a bipartite quantum state, and it satisfies all of the axioms desired for an entanglement measure. The quantum discord is a measure of quantum correlations that are different from those due to entanglement. What these two measures have in common is that they are both based upon the conditional quantum mutual information. In Berta et al (2015 J. Math. Phys. 56 022205), we recently proposed Renyi generalizations of the conditional quantum mutual information of a tripartite state on ABC (with C being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system B (with it being left open to show that the Renyi quantity is monotone with respect to local operations on system A). Here we define a Renyi squashed entanglement and a Renyi quantum discord based on a Renyi conditional quantum mutual information and investigate these quantities in detail. Taking as a conjecture that the Renyi conditional quantum mutual information is monotone with respect to local operations on both systems A and B, we prove that the Renyi squashed entanglement and the Renyi quantum discord satisfy many of the properties of the respective original von Neumann entropy based quantities. In our prior work (Berta et al 2015 Phys. Rev. A 91 022333), we also detailed a procedure to obtain Renyi generalizations of any quantum information measure that is equal to a linear combination of von Neumann entropies with coefficients chosen from the set {-1, 0, 1}. Here, we extend this procedure to include differences of relative entropies. Using the extended procedure and a conjectured monotonicity of the Renyi generalizations in the Renyi parameter, we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy, joint convexity of the relative entropy, and the Holevo bound.