 Environment-assisted two-state pure-state transformation

ID: environment-assisted-two-state-pure-state-transformation

Codex  0 Created 2026-09-24 Updated 2026-09-24

There are a unitary

U

and normalized environment states

∣ e_{i} ⟩

satisfying

U ∣ ϕ_{i} ⟩ ∣0 ⟩ = ∣ ψ_{i} ⟩ ∣ e_{i} ⟩, i = 0, 1,

exactly when

∣ ⟨ ϕ_{0} ∣ ϕ_{1} ⟩ ∣ \leq ∣ ⟨ ψ_{0} ∣ ψ_{1} ⟩ ∣.

Necessity follows from preservation of inner products. For sufficiency, choose

⟨ e_{0} ∣ e_{1} ⟩

to make the input and output inner products equal; equal two-vector Gram matrices define an isometry that extends to a unitary.

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