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

Lemma: Finite-dimensional conditional expectation

Exact channel, subsystem, and algebraic interfaces · lines 1134–1165 · proof included or linked

En01-A@1#lem:expectation

For a unital $*$ -subalgebra $A \subseteq B (H_{Q})$ , the Hilbert–Schmidt orthogonal projection $E_{A}$ onto $A$ is CPTP, unital, and self-adjoint. Hermitian elements of $A$ have spectral projectors in $A$ .

Proof. After a unitary change of coordinates, write

H_{Q} = ℓ ⨁ C^{n_{ℓ}} \otimes C^{m_{ℓ}}, A = ℓ ⨁ B (C^{n_{ℓ}}) \otimes 1_{m_{ℓ}} .

If $P_{ℓ}$ are the block projectors, then

E_{A} (Y) = ℓ ⨁ tr_{m_{ℓ}} (P_{ℓ} Y P_{ℓ}) \otimes \frac{1 _{m_{ℓ}}}{m _{ℓ}} .

Block pinching, partial trace, and replacement by a maximally mixed factor are completely positive. The formula preserves trace and the identity and is the Hilbert–Schmidt orthogonal projection. Spectral projectors belong to $A$ by finite-dimensional functional calculus.

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unlabeled · line 1144
$H_{Q} = ℓ ⨁ C^{n_{ℓ}} \otimes C^{m_{ℓ}}, A = ℓ ⨁ B (C^{n_{ℓ}}) \otimes 1_{m_{ℓ}} .$
unlabeled · line 1152
$E_{A} (Y) = ℓ ⨁ tr_{m_{ℓ}} (P_{ℓ} Y P_{ℓ}) \otimes \frac{1 _{m_{ℓ}}}{m _{ℓ}} .$

Lemma: Finite-dimensional conditional expectation

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