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

Corollary: Measurement channels

Exact channel, subsystem, and algebraic interfaces · lines 1194–1216 · proof included or linked

En01-A@1#cor:measurement

For a POVM $M = {M_{o}}$ , let

Λ_{M} (Y) = o \sum tr (M_{o} Y) ∣ o ⟩ ⟨ o ∣.

Then

δ_{J} (Λ_{M}) = \frac{1}{2} o \sum ∣ tr (M_{o} X_{J}) ∣.

For the binary POVM ${M, 1 - M}$ , denoted by $Λ_{M}$ ,

δ_{J} (Λ_{M}) = ∣ tr (M X_{J}) ∣.

Thus a fixed question is itself a particular interface channel.

Proof. The output difference is diagonal. For the binary case its two entries are $tr (M X_{J})$ and its negative.

Displayed formulas in this passage

unlabeled · line 1196
$Λ_{M} (Y) = o \sum tr (M_{o} Y) ∣ o ⟩ ⟨ o ∣.$
unlabeled · line 1200
$δ_{J} (Λ_{M}) = \frac{1}{2} o \sum ∣ tr (M_{o} X_{J}) ∣.$
unlabeled · line 1206
$δ_{J} (Λ_{M}) = ∣ tr (M X_{J}) ∣.$