  Entanglement distillation, an essential quantum information processing task,
refers to the conversion from multiple copies of noisy entangled states to a
smaller number of highly entangled states. In this work, we study the
non-asymptotic fundamental limits for entanglement distillation. We investigate
the optimal tradeoff between the distillation rate, the number of prepared
states, and the error tolerance. First, we derive the one-shot distillable
entanglement under completely positive partial transpose preserving operations
as a semidefinite program and demonstrate an exact characterization via the
quantum hypothesis testing relative entropy. Second, we establish efficiently
computable second-order estimations of the distillation rate for general
quantum states. In particular, we provide explicit as well as approximate
evaluations for various quantum states of practical interest, including pure
states, mixture of Bell states, maximally correlated states and isotropic
states.
