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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Exact Mixed State Cloning Fidelity Criterion

Statement

Let $ρ_{0}, ρ_{1}$ be finite-dimensional density operators. If one channel $Φ$ satisfies $Φ (ρ_{i}) = ρ_{i} \otimes ρ_{i}$ for $i = 0, 1$ , then $F (ρ_{0}, ρ_{1}) \in {0, 1}$ . Therefore the inputs are identical or have orthogonal supports.

Facts & Assumptions

Given: States $ρ_{0}, ρ_{1}$ and a channel exactly cloning both.

[L1]

Fidelity cannot decrease under a channel (Fidelity Data Processing); the QC06 proof is draft.

[L2]

Fidelity is multiplicative on tensor products, lies in $[0, 1]$ , equals one exactly for equal states, and equals zero exactly for orthogonal supports (Fidelity range equality and products); its QC06 proof is draft.

[L3]

The output states of the exact cloning channel are $ρ_{i} \otimes ρ_{i}$ (Finite-dimensional quantum channel, given hypothesis).

Proof

1.1givenL1L2L3algebra

Apply data processing [L1] to the channel and the two inputs, then use its exact outputs and the tensor-product formula [L2]: $F (ρ_{0}, ρ_{1}) \leq F (ρ_{0} \otimes ρ_{0}, ρ_{1} \otimes ρ_{1}) = F (ρ_{0}, ρ_{1})^{2} .$ Writing $f = F (ρ_{0}, ρ_{1}) \in [0, 1]$ , the inequality $f \leq f^{2}$ forces $f = 0$ or $f = 1$ , because $f (1 - f) \leq 0$ and both factors are nonnegative.

2.1step 1.1L2∎

If $f = 1$ , [L2] gives $ρ_{0} = ρ_{1}$ . If $f = 0$ , [L2] gives orthogonality of their supports. These are the asserted alternatives.

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)
Howard Barnum, Carlton M. Caves, Christopher A. Fuchs, Richard Jozsa, and Benjamin Schumacher, Noncommuting Mixed States Cannot Be Broadcast (standard reference, not scraped)