Manuscript Psy03-A@1 · Psy03-A.tex · theorem · b30eb6e9

Theorem: Operational-budget ranking counterexample

Exact coherence-sensitive pilots › Equal operational distance is not equal utility · lines 1702–1736 · proof included or linked

Psy03-A@1#thm:equalop

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For $0 < θ < π /2$ , let

U_{Z} = e^{- i θ Z}, U_{X} = e^{- i θ X} .

Both unitary channels have distance $sin θ$ from identity:

d_{⋄} (Ad_{U_{Z}}, id) = d_{⋄} (Ad_{U_{X}}, id) = sin θ .

For the phase classification task of Theorem thm:phase, their successes are nevertheless

Q_{Z} = cos^{2} θ, Q_{X} = 1.

Proof. For either Pauli $P$ ,

⟨ U_{P} ⟩_{ψ} = cos θ - i sin θ ⟨ P ⟩_{ψ} .

Its modulus is at least $cos θ$ , with equality at a state of zero $P$ -expectation. The same expression holds for the reduced state of a referenced pure input. The pure-output trace distance is the square root of one minus the squared overlap, giving exactly $sin θ$ .

The $X$ -eigenstates acquire only phases under $U_{X}$ . Under $U_{Z}$ , the overlap with their correct $X$ -eigenstate has squared modulus $cos^{2} θ$ .

Displayed formulas in this passage

unlabeled · line 1705
$U_{Z} = e^{- i θ Z}, U_{X} = e^{- i θ X} .$
unlabeled · line 1709
$d_{⋄} (Ad_{U_{Z}}, id) = d_{⋄} (Ad_{U_{X}}, id) = sin θ .$
unlabeled · line 1715
$Q_{Z} = cos^{2} θ, Q_{X} = 1.$
unlabeled · line 1722
$⟨ U_{P} ⟩_{ψ} = cos θ - i sin θ ⟨ P ⟩_{ψ} .$

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section: Equal operational distance is not equal utility