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

Theorem: Exact delayed rule-cue retention

A subordinate finite quantum cue extension › An exact delayed cue pilot · lines 1991–2023 · proof included or linked

Psy03-D@1#thm:qpilot

After $T$ delay steps,

ρ_{T}^{\pm} = \frac{1}{2} [I \pm (η λ)^{T} X], D_{T} = (η λ)^{T} .

The rule-execution accuracy is

Q_{T}^{*} = 1 - \frac{d}{2} + \frac{d}{2} (η λ)^{T} .

For the pair $f_{0}, f_{3}$ , $d = 3/4$ , giving

Q_{T}^{*} = \frac{5}{8} + \frac{3}{8} (η λ)^{T} .

The exact global carrier trace-distance contraction coefficient of one step is $η$ . It is not generally $η λ$ .

Proof. The Bloch map scales the two transverse components by $η λ$ and the longitudinal component by $η$ . Iterate the transverse phase inputs. Their trace distance is the transverse separation factor. Apply Theorem thm:qexecution.

The largest singular value of the Bloch map is $η$ , attained by opposite longitudinal-axis states. Qubit trace distance is half Euclidean Bloch distance.

Displayed formulas in this passage

unlabeled · line 1994
$ρ_{T}^{\pm} = \frac{1}{2} [I \pm (η λ)^{T} X], D_{T} = (η λ)^{T} .$
unlabeled · line 2000
$Q_{T}^{*} = 1 - \frac{d}{2} + \frac{d}{2} (η λ)^{T} .$
unlabeled · line 2004
$Q_{T}^{*} = \frac{5}{8} + \frac{3}{8} (η λ)^{T} .$

Theorem: Exact delayed rule-cue retention

Displayed formulas in this passage

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