Quantum entanglement is a crucial resource in quantum information processing. However, quantifying the entanglement required to prepare quantum states and implement quantum processes remains challenging. This paper proposes computable and faithful lower bounds for the entanglement cost of general quantum states and quantum channels. We introduce the concept of logarithmic $k$-negativity, a generalization of logarithmic negativity, to establish a general lower bound for the entanglement cost of quantum states under quantum operations that completely preserve the positivity of partial transpose (PPT). This bound is efficiently computable via semidefinite programming and is non-zero for any entangled state that is not PPT, making it faithful in the entanglement theory with non-positive partial transpose. Furthermore, we delve into specific and general examples to demonstrate the advantages of our proposed bounds compared with previously known computable ones. Notably, we affirm the irreversibility of asymptotic entanglement manipulation under PPT operations for full-rank entangled states and the irreversibility of channel manipulation for amplitude damping channels. We also establish the best-known lower bound for the entanglement cost of arbitrary dimensional isotropic states. These findings push the boundaries of understanding the structure of entanglement and the fundamental limits of entanglement manipulation.

翻译：量子纠缠是量子信息处理中的关键资源。然而，量化制备量子态和实现量子过程所需的纠缠量仍然具有挑战性。本文针对一般量子态和量子通道的纠缠代价提出了可计算且保真的下界。我们引入了对数$k$-负性的概念（这是对数负性的推广），为在完全保持部分转置正性（PPT）的量子操作下，量子态的纠缠代价建立了一般下界。该下界可通过半定规划高效计算，且对于任何非PPT纠缠态非零，因此在非正部分转置的纠缠理论中具有保真性。此外，我们深入探讨了具体和一般示例，以证明与已知可计算下界相比，本文提出的下界具有的优势。值得注意的是，我们证实了在PPT操作下，满秩纠缠态的渐进纠缠操纵不可逆性，以及振幅阻尼通道的通道操纵不可逆性。我们还为任意维度的各向同性态建立了已知最佳的纠缠代价下界。这些发现推动了理解纠缠结构以及纠缠操纵基本极限的边界。