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.
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Fuchs Van De Graaf

Statement

For density operators $ρ, σ$ on a nonzero finite-dimensional complex Hilbert space, $1 - F (ρ, σ) \leq D (ρ, σ) \leq \sqrt{1 - F (ρ, σ)^{2}},$ where $F$ is root fidelity and $D (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}$ .

Facts & Assumptions

Given: Density operators $ρ, σ$ on a nonzero finite-dimensional $H$ .

[L1]

Some finite rank-one PVM has classical overlap $\sum_{a} \sqrt{p_{a} q_{a}} = F (ρ, σ)$ , and every such measurement has total variation at most $D$ (Fidelity classical measurement minimum, Trace Distance, Measurement Variation At Most Trace Distance).

[L2]

Taking the fixed auxiliary space $R = H$ , there are purifications with overlap $F$ (Fidelity Purification Characterization); the QC06 theorem proves this as part of its every- $R$ statement.

[L3]

For unit vectors $x, y$ , $D (∣ x ⟩ ⟨ x ∣, ∣ y ⟩ ⟨ y ∣) = \sqrt{1 - ∣ ⟨ x, y ⟩ ∣^{2}}$ (Pure state trace distance).

[L4]

Partial trace is a channel (Partial trace over a finite tensor factor, An isometry followed by partial trace is a channel) and trace distance is contractive under channels (Trace Distance Channel Contractivity).

Proof

1.1givenL1algebra

Let $(E_{a})$ be the rank-one PVM from [L1], with probabilities $p_{a} = tr (ρ E_{a})$ and $q_{a} = tr (σ E_{a})$ . For nonnegative $p, q$ , $\sqrt{p q} \geq \min (p, q)$ because if $p \leq q$ then $p q \geq p^{2}$ , and the other case is symmetric. Since $\sum_{a} p_{a} = \sum_{a} q_{a} = 1$ , $1 - F = 1 - \sum_{a} \sqrt{p_{a} q_{a}} \leq 1 - \sum_{a} \min (p_{a}, q_{a}) = \frac{1}{2} \sum_{a} ∣ p_{a} - q_{a} ∣ \leq D (ρ, σ) .$ The last inequality is [L1].

2.1givenL2L3L4∎

Choose purifications $x, y$ attaining the overlap $∣ ⟨ x, y ⟩ ∣ = F (ρ, σ)$ by [L2]. By [L3], their pure-state trace distance is $\sqrt{1 - F (ρ, σ)^{2}}$ . Taking the partial trace over the auxiliary factor maps their rank-one projectors to $ρ$ and $σ$ ; contractivity [L4] therefore gives $D (ρ, σ) \leq \sqrt{1 - F (ρ, σ)^{2}}$ . This proves the upper bound.

Depends on

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Sources

John Watrous, The Theory of Quantum Information (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 2: Foundations I: States and Ensembles (updated July 2015) (standard reference, not scraped)