This paper by William K. Wootters presents a proof of an explicit formula for the entanglement of formation of a pair of qubits. The entanglement of formation of a mixed state ρ is defined as the minimum average entanglement of an ensemble of pure states that represents ρ. The paper extends a previous conjecture that an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) can be expressed as a function of their density matrix. The formula is proven for arbitrary states of two qubits. Entanglement is a fundamental feature of quantum mechanics that enables phenomena such as quantum teleportation and dense coding. A pure state of a pair of quantum systems is entangled if it does not factorize, meaning each system does not have a pure state of its own. The entanglement of formation is a measure of the resources needed to create a given entangled state. It is defined as the minimum average entanglement of a pure-state decomposition of the mixed state ρ. The paper introduces the concept of "spin flip" transformation and uses it to define the concurrence of a pure state, which is a measure of entanglement. The entanglement of formation for a mixed state is then given by a function of the concurrence of the state. The formula for the entanglement of formation is derived using the concurrence and is shown to be valid for arbitrary states of two qubits. The paper also discusses the properties of the concurrence and the entanglement of formation, including their convexity and their relationship to the average concurrence of a decomposition. It shows that the entanglement of formation can always be achieved by a decomposition of the mixed state into four or fewer pure states, each with the same entanglement. The paper concludes by proving that no decomposition of the mixed state can achieve a lower average entanglement than the formula predicts. This result confirms the validity of the formula for the entanglement of formation of a pair of qubits.This paper by William K. Wootters presents a proof of an explicit formula for the entanglement of formation of a pair of qubits. The entanglement of formation of a mixed state ρ is defined as the minimum average entanglement of an ensemble of pure states that represents ρ. The paper extends a previous conjecture that an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) can be expressed as a function of their density matrix. The formula is proven for arbitrary states of two qubits. Entanglement is a fundamental feature of quantum mechanics that enables phenomena such as quantum teleportation and dense coding. A pure state of a pair of quantum systems is entangled if it does not factorize, meaning each system does not have a pure state of its own. The entanglement of formation is a measure of the resources needed to create a given entangled state. It is defined as the minimum average entanglement of a pure-state decomposition of the mixed state ρ. The paper introduces the concept of "spin flip" transformation and uses it to define the concurrence of a pure state, which is a measure of entanglement. The entanglement of formation for a mixed state is then given by a function of the concurrence of the state. The formula for the entanglement of formation is derived using the concurrence and is shown to be valid for arbitrary states of two qubits. The paper also discusses the properties of the concurrence and the entanglement of formation, including their convexity and their relationship to the average concurrence of a decomposition. It shows that the entanglement of formation can always be achieved by a decomposition of the mixed state into four or fewer pure states, each with the same entanglement. The paper concludes by proving that no decomposition of the mixed state can achieve a lower average entanglement than the formula predicts. This result confirms the validity of the formula for the entanglement of formation of a pair of qubits.