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 $\kappa$ 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 $\alpha$-logarithmic negativity entanglement measures, each of which is identified by a parameter $\alpha \in \left[1,\infty \right]$. In this family, the original logarithmic negativity is recovered as the smallest with $\alpha =1$, and the $\kappa$ entanglement is recovered as the largest with $\alpha =\infty$. We prove that the $\alpha$-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 $\alpha$-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.

中文翻译：

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α-对数负性

二分体量子态的对数负性是量子信息理论中广泛使用的纠缠度，这是因为它易于计算，并且是可蒸馏纠缠的上限。最近，$\kappa$二元态的纠缠被证明是一种易于计算且具有精确的信息理论意义的纠缠量度，当自由操作完全保持正态性时，等于二元量子态的精确纠缠成本。部分移调[Xin Wang和Mark M. Wilde，Phys。牧师 125，040502（2020）]。在本文中，我们通过证明这是两个有序家族的极端，提供了这两个纠缠度量之间的重要联系。$\alpha$对数负性纠缠度量，每个度量都由一个参数标识 $\alpha \in \left[\mathrm{1个}，\infty \right]$。在这个家族中，原始对数负值恢复为最小$\alpha =\mathrm{1个}$和 $\kappa$ 纠缠被回收为最大 $\alpha =\infty$。我们证明$\alpha$对数负性满足以下属性：纠缠单调，规范化，忠实性和次可加性。我们还证明它既不是凸面的也不是一夫一妻制的。最后，我们定义$\alpha$量子通道的对数负性作为量子状态概念的泛化，并且我们展示了如何将许多概念泛化为任意资源理论。