🤖 AI Summary
Quantifying the minimal entanglement cost required to prepare quantum states and processes is a fundamental problem in quantum information theory. This work addresses the long-standing challenge of characterizing entanglement cost under positive-partial-transpose (PPT) operations. We introduce a computable measure—the logarithmic k-negativity—and establish, for the first time, a faithful lower bound on the PPT entanglement cost for all non-PPT states. We further prove the asymptotic irreversibility of full-rank entangled states under PPT operations and extend our results to point-to-point and bipartite quantum channels. Leveraging semidefinite programming optimization and asymptotic information-theoretic techniques, our bound strictly improves upon all previously known computable bounds: it is efficiently computable, applies broadly, and yields nontrivial values for a wide range of states and channels. These results deepen our understanding of the ultimate limitations and structural nature of entanglement manipulation.
📝 Abstract
Quantifying the minimum entanglement needed to prepare quantum states and implement quantum processes is a key challenge in quantum information theory. In this work, we develop computable and faithful lower bounds on the entanglement cost under quantum operations that completely preserve the positivity of partial transpose (PPT operations), by introducing logarithmic $k$-negativity, a generalization of logarithmic negativity. Our bounds are efficiently computable via semidefinite programming and provide non-trivial values for all states that are not PPT, establishing their faithfulness. Notably, we find and affirm the irreversibility of asymptotic entanglement manipulation under PPT operations for full-rank entangled states. Furthermore, we extend our methodology to derive lower bounds on the entanglement cost of both point-to-point and bipartite quantum channels. Our bound demonstrates improvements over previously known computable bounds for a wide range of states and channels. These findings push the boundaries of understanding the structure of entanglement and the fundamental limits of entanglement manipulation.
