Wootters: exact entanglement-of-formation formula for arbitrary two-qubit states

What changed

For every two-qubit density matrix rho, its convex-roof entanglement of formation equals h2((1+sqrt(1-C^2))/2), where C=max(0,lambda1-lambda2-lambda3-lambda4) and the ordered lambda values are the square roots of the eigenvalues of rho times its spin flip. An optimal decomposition exists with at most four pure states.

Scientific scope

For every two-qubit density matrix rho, its convex-roof entanglement of formation equals h2((1+sqrt(1-C^2))/2), where C=max(0,lambda1-lambda2-lambda3-lambda4) and the ordered lambda values are the square roots of the eigenvalues of rho times its spin flip. An optimal decomposition exists with at most four pure states. This formula is for two qubits, not arbitrary higher-dimensional or multipartite states. The paper does not prove entanglement-of-formation additivity or equality with a regularized asymptotic entanglement cost. The earlier Hill-Wootters concurrence conjecture and rank-at-most-two proof are antecedents, not solely invented in this later sole-author proof.

Evidence

The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. This formula is for two qubits, not arbitrary higher-dimensional or multipartite states. The paper does not prove entanglement-of-formation additivity or equality with a regularized asymptotic entanglement cost. The earlier Hill-Wootters concurrence conjecture and rank-at-most-two proof are antecedents, not solely invented in this later sole-author proof.

Contributors

• The named authors in the research record · SCIENCE: The original named human researchers developed the exact mathematical construction or framework. Composite source families retain separate original paper bylines and predecessor roles; they are not a fictional joint publication. Individual effort shares, person identity joins and nationalities are not inferred. The source reading is not independent proof certification.
• Scott Hill · SCIENCE: Earlier joint Hill-Wootters concurrence formula conjecture and rank-at-most-two proof, explicitly credited in this sole-author extension

• Entanglement of Formation of an Arbitrary State of Two Qubits

Provisional WRS assessment, edition 2026-10-23.9. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.

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