TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Optimal Universal Pure State Cloning Bound

Statement

Assume the choice branch used by the normalized Haar moment theorem. For $d \geq 1$ and $1 \leq n \leq m$ , the largest possible worst-case squared root fidelity of a channel from $n$ identical $d$ -dimensional input registers to $m$ output registers is $\frac{N_{n}}{N_{m}}, N_{r} = (\binom{r + d - 1}{d - 1}) .$ For $n = 1, m = 2$ this value is $2 / (d + 1)$ .

Facts & Assumptions

Given: $d \geq 1$ , $1 \leq n \leq m$ , and the choice-dependent symmetric-subspace/Haar results.

[L1]

Every such channel has worst-case squared fidelity at most $N_{n} / N_{m}$ (Optimal Cloning Worst Case Upper Bound, The Axiom of Choice). The lemma is a draft and depends on the Haar-moment branch.

[L2]

The displayed measure-and-prepare correction channel is CPTP and has squared fidelity exactly $N_{n} / N_{m}$ on every pure input (Optimal Cloner Attains Bound). Its supplier is a draft.

Proof

1.1givenL1L2algebra

The upper bound [L1] applies to every admissible channel. The explicit channel in [L2] belongs to this same class and attains the bound for every unit input, so its worst-case value is at least $N_{n} / N_{m}$ . Combining the two inequalities shows that the supremum over channels is exactly $N_{n} / N_{m}$ , and the construction attains it.

2.1givenstep 1.1algebra∎

For $n = 1, m = 2$ , $N_{1} = (\binom{d}{d - 1}) = d$ and $N_{2} = (\binom{d + 1}{d - 1}) = d (d + 1) / 2$ . Their ratio is $2 / (d + 1)$ , including $d = 1$ , where it equals one and the one-dimensional state is cloned trivially.

Depends on

Used by

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Dependency tree · two levels

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Sources

John Watrous, The Theory of Quantum Information (standard reference, not scraped)