This paper presents a method for purifying noisy entanglement and faithfully teleporting quantum states through noisy channels. The authors show that two separated observers can distill pure entangled pairs (e.g., near-perfect singlets) from mixed 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 paper discusses the concept of quantum data compression and teleportation, which exemplify a new goal of quantum information theory: understanding the kind and quantity of channel resources needed for the transmission of intact quantum states, rather than classical information. It introduces the idea of quantum data compression, where quantum data can be transmitted with asymptotically perfect fidelity using a number of qubits approaching the source's von Neumann entropy. Quantum teleportation achieves faithful transmission by substituting classical communication and prior entanglement for a direct quantum channel. Using teleportation, an arbitrary unknown qubit can be faithfully transmitted via a pair of maximally-entangled qubits (e.g., two spin-1/2 particles in a pure singlet state) previously shared between sender and receiver, and a 2-bit classical message from the sender to the receiver. The paper describes a purification protocol for entangled pairs that results in almost perfectly entangled states from mixed entangled states that result from transmission through the noisy channel. This purification is achieved by performing local unitary operations and measurements on the shared entangled pairs, coordinating their actions through classical messages, and sacrificing some of the entangled pairs to increase the purity of the remaining ones. The authors show that, given two Werner pairs of fidelity F > 1/2, Alice and Bob can use local operations and two-way classical communication to obtain, with probability greater than 1/4, one Werner pair of fidelity F' > F, where F' satisfies a recurrence relation. They also describe a breeding method that uses previously purified Φ+ pairs to produce more pure pairs than consumed if the mixed state's von Neumann entropy is less than 1. The paper concludes that the optimal asymptotic yield of purified singlets distillable from general mixed states is not yet known, but for Werner states, the yield is positive for F > 0.8107. The paper also discusses the entanglement of formation and distillable entanglement as two alternative extensions of the definition of entanglement from pure to mixed states.This paper presents a method for purifying noisy entanglement and faithfully teleporting quantum states through noisy channels. The authors show that two separated observers can distill pure entangled pairs (e.g., near-perfect singlets) from mixed 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 paper discusses the concept of quantum data compression and teleportation, which exemplify a new goal of quantum information theory: understanding the kind and quantity of channel resources needed for the transmission of intact quantum states, rather than classical information. It introduces the idea of quantum data compression, where quantum data can be transmitted with asymptotically perfect fidelity using a number of qubits approaching the source's von Neumann entropy. Quantum teleportation achieves faithful transmission by substituting classical communication and prior entanglement for a direct quantum channel. Using teleportation, an arbitrary unknown qubit can be faithfully transmitted via a pair of maximally-entangled qubits (e.g., two spin-1/2 particles in a pure singlet state) previously shared between sender and receiver, and a 2-bit classical message from the sender to the receiver. The paper describes a purification protocol for entangled pairs that results in almost perfectly entangled states from mixed entangled states that result from transmission through the noisy channel. This purification is achieved by performing local unitary operations and measurements on the shared entangled pairs, coordinating their actions through classical messages, and sacrificing some of the entangled pairs to increase the purity of the remaining ones. The authors show that, given two Werner pairs of fidelity F > 1/2, Alice and Bob can use local operations and two-way classical communication to obtain, with probability greater than 1/4, one Werner pair of fidelity F' > F, where F' satisfies a recurrence relation. They also describe a breeding method that uses previously purified Φ+ pairs to produce more pure pairs than consumed if the mixed state's von Neumann entropy is less than 1. The paper concludes that the optimal asymptotic yield of purified singlets distillable from general mixed states is not yet known, but for Werner states, the yield is positive for F > 0.8107. The paper also discusses the entanglement of formation and distillable entanglement as two alternative extensions of the definition of entanglement from pure to mixed states.