Manuscript Psy03-D@1 · Psy03-D-v1.tex · theorem · 3e18951d

Theorem: Cue distinguishability and rule execution

A subordinate finite quantum cue extension › Two observed rule cues · lines 1940–1969 · proof included or linked

Psy03-D@1#thm:qexecution, Psy03-D@1#eq:qexecution

The optimal current-label accuracy is

Q_{two cued rules}^{*} = 1 - \frac{d}{2} + \frac{d D}{2} .

A complete readout first distinguishes the two cue states on informative points and then executes the corresponding label function. On uninformative points it returns the common label.

Proof. On a point where the functions agree, correctness one is available without distinguishing the cue.

Where they disagree, label classification is equal-prior binary state discrimination. For an effect $F$ declaring cue 0,

Q = \frac{1}{2} + \frac{1}{2} Tr [F (Λ (ρ_{0}) - Λ (ρ_{1}))] .

The maximum effect expectation is the positive-part trace, equal to $D$ . Average over informative and uninformative points Watrous.

Displayed formulas in this passage

Psy03-D@1#eq:qexecution · line 1943
$Q_{two cued rules}^{*} = 1 - \frac{d}{2} + \frac{d D}{2} .$
unlabeled · line 1961
$Q = \frac{1}{2} + \frac{1}{2} Tr [F (Λ (ρ_{0}) - Λ (ρ_{1}))] .$

Nearby in this section

section: Two observed rule cues