<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math>-logarithmic negativity
Xin Wang, Mark M. Wilde
Abstract
The logarithmic negativity of a bipartite quantum state is a widely employed entanglement measure in quantum information theory due to the fact that it is easy to compute and serves as an upper bound on distillable entanglement. More recently, the $\ensuremath{\kappa}$ entanglement of a bipartite state was shown to be an entanglement measure that is both easily computable and has a precise information-theoretic meaning, being equal to the exact entanglement cost of a bipartite quantum state when the free operations are those that completely preserve the positivity of the partial transpose [Xin Wang and Mark M. Wilde, Phys. Rev. Lett. 125, 040502 (2020)]. In this paper, we provide a nontrivial link between these two entanglement measures by showing that they are the extremes of an ordered family of $\ensuremath{\alpha}$-logarithmic negativity entanglement measures, each of which is identified by a parameter $\ensuremath{\alpha}\ensuremath{\in}\left[1,\ensuremath{\infty}\right]$. In this family, the original logarithmic negativity is recovered as the smallest with $\ensuremath{\alpha}=1$, and the $\ensuremath{\kappa}$ entanglement is recovered as the largest with $\ensuremath{\alpha}=\ensuremath{\infty}$. We prove that the $\ensuremath{\alpha}$-logarithmic negativity satisfies the following properties: entanglement monotone, normalization, faithfulness, and subadditivity. We also prove that it is neither convex nor monogamous. Finally, we define the $\ensuremath{\alpha}$-logarithmic negativity of a quantum channel as a generalization of the notion for quantum states, and we show how to generalize many of the concepts to arbitrary resource theories.