Fidelity of recovery and geometric squashed entanglement

where H(F)σ ≡ −Tr{σF log σF} is the von Neumann entropy of a state σF on system F and we unambiguously let ρC ≡ TrAB{ρABC} denote the reduced density operator on system C, for example. The CQMI captures the correlations present between Alice and Bob from the perspective of Charlie in the independent and identically distributed (i.i.d.) resource limit, where an asymptotically large number of copies of the state ρABC are shared between the three parties. In an attempt to develop a version of the CQMI, which would be relevant for the “one-shot” or finite resource regimes, we along with Berta [3] recently proposed Rényi generalizations of the CQMI. We proved that these Rényi generalizations of the CQMI retain many of the properties of the original CQMI in (1). We used them to define a Rényi squashed entanglement and a Rényi quantum discord [12], which retain several properties of the respective, original, von Neumann entropy-based quantities. One contribution of [3] was the conjecture that the proposed Rényi CQMIs are monotone increasing in the Rényi parameter, as is known to be the case for other Rényi entropic quantities. That is, for a tripartite state ρABC, and for a Rényi conditional mutual information Ĩα (A; B|C)ρ defined as [3, Section 6]