Optimal universal 1 → 2 qubit cloning at fidelity 5/6, the universal-NOT, and asymmetric cloning
A Lean 4 development of the optimal universal 1 -> 2 cloning of a qubit, with every figure of merit finite: a channel from one qubit to two is a finite Kraus family K_i : C^2 -> C^2 (x) C^2 with sum K_i^* K_i = 1 (the blank qubit is fixed, so this is a 1 -> 2 map; the Choi-Kraus theorem is not proved), the two clone fidelities of a pure input psi are Re and Re , and F1-bar, F2-bar are their averages over the six Pauli eigenstates (1 +- sigma)/2, Gaussian-rational density matrices (the six states form a 2-design, the octahedron being a spherical 3-design, so these equal the Bloch-sphere…
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