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Manuscript En01-A@1 · En01-A.tex · corollary · 558b972b

Corollary: Exact subsystem interface theorem

Exact channel, subsystem, and algebraic interfaces · lines 1114–1132 · proof included or linked

En01-A@1#thm:interface

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Take $Λ_{S} = tr_{S}$ . Then

δ_{J} (S) = \frac{1}{2} ∥ tr_{S} X_{J} ∥_{1} = 0 \leq M_{S} \leq 1_{S} sup ∣ tr ((M_{S} \otimes 1_{S}) X_{J}) ∣.

All effects on $S$ agree if and only if $tr_{S} X_{J} = 0$ . A Helstrom projector on $S$ attains the maximum.

Proof. The available effects after $tr_{S}$ are exactly $M_{S} \otimes 1_{S}$ . Apply Theorem thm:channel.

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unlabeled · line 1117
$δ_{J} (S) = \frac{1}{2} ∥ tr_{S} X_{J} ∥_{1} = 0 \leq M_{S} \leq 1_{S} sup ∣ tr ((M_{S} \otimes 1_{S}) X_{J}) ∣.$

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