Manuscript I@2 · I.tex · proposition · 85631fa7

Proposition: Exact record compatibility

After Enter: records, instruments, decoders, and cutoffs › D2: record pullback versus instrument compatibility · lines 2267–2293 · proof included or linked

I@2#prop:pullback, I@2#eq:pullback

Back to manuscript

Let $T$ be a finite-dimensional CPTP map from a fine input space to a coarse space, let $(E_{w})_{w}$ be a coarse POVM, and let $(Π_{w})_{w}$ be a fine POVM with the same labels. The record laws

tr (T (ρ) E_{w}), tr (ρ Π_{w})

agree for every input density operator and every $w$ if and only if

T^{*} (E_{w}) = Π_{w} for every w .

Here $T^{*}$ is the trace-dual map.

Proof. The difference is

tr (ρ [T^{*} (E_{w}) - Π_{w}]) .

The bracket is Hermitian. Vanishing expectation on all density operators, in particular its eigenvector states, is equivalent to the bracket being zero. The reverse implication is immediate.

Displayed formulas in this passage

I@2#eq:pullback · line 2277
$T^{*} (E_{w}) = Π_{w} for every w .$
unlabeled · line 2273
$tr (T (ρ) E_{w}), tr (ρ Π_{w})$
unlabeled · line 2286
$tr (ρ [T^{*} (E_{w}) - Π_{w}]) .$

Proposition: Exact record compatibility

Displayed formulas in this passage

Nearby in this section