Catalysis lowers the exact asymptotic entanglement cost for preparing multiple copies of a quantum state via an explicit protocol.
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Two quantum states ρ₁ and ρ₂ commute exactly when tr(ρ₁²ρ₂²) = tr(ρ₁ ρ₂ ρ₁ ρ₂).
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Catalytic advantage in asymptotic entanglement manipulation
Catalysis lowers the exact asymptotic entanglement cost for preparing multiple copies of a quantum state via an explicit protocol.
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Commutativity from a single Bargmann invariant equality
Two quantum states ρ₁ and ρ₂ commute exactly when tr(ρ₁²ρ₂²) = tr(ρ₁ ρ₂ ρ₁ ρ₂).