The paper discusses the purification of noisy entanglement and faithful teleportation through noisy channels. Two separated observers, Alice and Bob, can prepare a smaller number of arbitrarily high-purity entangled pairs (e.g., near-perfect singlets) from a supply of not-too-impure entangled states (e.g., singlets shared through a noisy channel). These pure entangled pairs can then be used to teleport unknown quantum states from one observer to the other, achieving faithful transmission of quantum information through a noisy channel. The authors provide upper and lower bounds on the yield \( D(M) \) of pure singlets distillable from mixed states \( M \), showing that \( D(M) > 0 \) if \( \langle \Psi^+ | M | \Psi^- \rangle > \frac{1}{2} \). They describe a purification protocol involving local unitary operations and measurements, coordinated through classical messages, to convert impure entangled pairs into almost-perfectly entangled states. This process involves random bilateral rotations, unilateral Pauli rotations, bilateral \( \pi/2 \) rotations, and the quantum XOR operation. The protocol can distill Werner states of arbitrarily high purity from any input mixed state with purity \( F_{\text{in}} > \frac{1}{2} \). The paper also introduces a "breeding" method that uses prepurified pairs as targets to simplify the analysis and increase the yield. The breeding method has a yield of \( 1 - S(W) \), producing more pure pairs than consumed if the mixed state's von Neumann entropy \( S(W) \) is less than 1. The authors compare the yields of several purification methods for Werner states and provide an upper bound on the entanglement distillable from Werner states. The paper concludes by discussing the threshold \( F = \frac{1}{2} \) for Werner states, where they can be made from unentangled ingredients or used to create pure singlets, and the distinction between distillable entanglement and entanglement of formation.The paper discusses the purification of noisy entanglement and faithful teleportation through noisy channels. Two separated observers, Alice and Bob, can prepare a smaller number of arbitrarily high-purity entangled pairs (e.g., near-perfect singlets) from a supply of not-too-impure entangled states (e.g., singlets shared through a noisy channel). These pure entangled pairs can then be used to teleport unknown quantum states from one observer to the other, achieving faithful transmission of quantum information through a noisy channel. The authors provide upper and lower bounds on the yield \( D(M) \) of pure singlets distillable from mixed states \( M \), showing that \( D(M) > 0 \) if \( \langle \Psi^+ | M | \Psi^- \rangle > \frac{1}{2} \). They describe a purification protocol involving local unitary operations and measurements, coordinated through classical messages, to convert impure entangled pairs into almost-perfectly entangled states. This process involves random bilateral rotations, unilateral Pauli rotations, bilateral \( \pi/2 \) rotations, and the quantum XOR operation. The protocol can distill Werner states of arbitrarily high purity from any input mixed state with purity \( F_{\text{in}} > \frac{1}{2} \). The paper also introduces a "breeding" method that uses prepurified pairs as targets to simplify the analysis and increase the yield. The breeding method has a yield of \( 1 - S(W) \), producing more pure pairs than consumed if the mixed state's von Neumann entropy \( S(W) \) is less than 1. The authors compare the yields of several purification methods for Werner states and provide an upper bound on the entanglement distillable from Werner states. The paper concludes by discussing the threshold \( F = \frac{1}{2} \) for Werner states, where they can be made from unentangled ingredients or used to create pure singlets, and the distinction between distillable entanglement and entanglement of formation.