Stability of optimal 1→2 qubit cloning

How close must a nearly optimal 1→2 qubit cloning channel be to the Bužek–Hillery channel? For finite-Kraus channels with mean fidelity at least 5/6−ε over the six Pauli eigenstates, the target bounds squared Kraus-space distance by 6ε and output Frobenius error by 200√ε‖ρ‖F.

Why it matters and what counts as progress

Why it matters: It asks for quantitative rigidity of near-optimal cloning, beyond identifying the unique exact optimum.

What counts as progress: Sharp stability bounds forcing nearly optimal cloning channels toward the Bužek–Hillery channel, or stronger norms and constants under the same six-state fidelity test.

How the problem developed

1 Sept 1996 · Construction [1]
Vladimir Bužek and Mark Hillery Introduced a universal approximate cloning machine for arbitrary spin-1/2 states, the channel underlying the later optimal 1→2 qubit benchmark.
1998 · Optimality theorem [2]
Dagmar Bruß, David P. DiVincenzo, Artur Ekert, Christopher A. Fuchs, Chiara Macchiavello and John A. Smolin Established the optimal symmetric universal 1→2 qubit cloning fidelity of 5/6.
13 Oct 2026 · Stability theorem [3]
JD Jones Reported quantitative bounds placing near-optimal finite-Kraus channels close to the span of the Bužek–Hillery Kraus operators and close in Frobenius norm to the optimal channel.

AI contribution

Claude-assisted proof gives quantitative stability for optimal qubit cloning

For any finite-Kraus 1→2 qubit channel whose mean fidelity over the six Pauli eigenstates is at least 5/6−ε, the Kraus operators have total squared Frobenius distance at most 6ε from the span of the two Bužek–Hillery Kraus operators. For every input matrix ρ, the channel output is within 200√ε‖ρ‖F of the Bužek–Hillery channel in Frobenius norm.

AI involvementAI-led

ImportanceModest

Technical verificationFormal (Lean)

Human understandingFew