  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.
