Renyi generalizations of the conditional quantum mutual information

The conditional quantum mutual information I(A; B vertical bar C) of a tripartite state rho(ABC) is an information quantity which lies at the center of many problems in quantum information theory. Three of its main properties are that it is non-negative for any tripartite state, that it decreases under local operations applied to systems A and B, and that it obeys the duality relation I(A; B vertical bar C) = I(A; B vertical bar D) for a four-party pure state on systems ABCD. The conditional mutual information also underlies the squashed entanglement, an entanglement measure that satisfies all of the axioms desired for an entanglement measure. As such, it has been an open question to find Renyi generalizations of the conditional mutual information, that would allow for a deeper understanding of the original quantity and find applications beyond the traditional memoryless setting of quantum information theory. The present paper addresses this question, by defining different a-Renyi generalizations I alpha(A; B vertical bar C) of the conditional mutual information, some of which we can prove converge to the conditional mutual information in the limit alpha -> 1. Furthermore, we prove that many of these generalizations satisfy non-negativity, duality, and monotonicity with respect to local operations on one of the systems A or B (with it being left as an open question to prove that monotonicity holds with respect to local operations on both systems). The quantities defined here should find applications in quantum information theory and perhaps even in other areas of physics, but we leave this for future work. We also state a conjecture regarding the monotonicity of the Renyi conditional mutual informations defined here with respect to the Renyi parameter alpha. We prove that this conjecture is true in some special cases and when a is in a neighborhood of one. (C) 2015 AIP Publishing LLC.