Manuscript I-B@1 · I-B-v1.tex · section · 5357688f

The clock instrument

Finite names, access, control, and destructive Enter › The clock instrument · lines 420–449

Fix disjoint clock and matter names, with

H_{C} ≅ C^{D}, D = 2^{q}, q \geq 1,

and a finite-dimensional surviving space $H_{X}$ . We consistently order the input factors as $C \otimes X$ .

Choose an orthonormal clock basis

{∣ τ_{k} ⟩ : 0 \leq k < D}, P_{k} = ∣ τ_{k} ⟩ ⟨ τ_{k} ∣,

and the unitary

B_{C} ∣ τ_{k} ⟩ = ∣ k ⟩ .

The program applies $B_{C}$ , then computational-basis Enter to all clock names. It returns the $q$ -bit code $k$ , not the names $C$ .

Define

D_{k} = ⟨ τ_{k} ∣_{C} \otimes I_{X} : H_{C} \otimes H_{X} ⟶ H_{X} .

The record has diagonal observable algebra

A_{R} = {k = 0 \sum^{D - 1} a_{k} ∣ k ⟩ ⟨ k ∣_{R} : a_{k} \in C} .

Displayed formulas in this passage

unlabeled · line 423
$H_{C} ≅ C^{D}, D = 2^{q}, q \geq 1,$
unlabeled · line 430
${∣ τ_{k} ⟩ : 0 \leq k < D}, P_{k} = ∣ τ_{k} ⟩ ⟨ τ_{k} ∣,$
unlabeled · line 434
$B_{C} ∣ τ_{k} ⟩ = ∣ k ⟩ .$
unlabeled · line 441
$D_{k} = ⟨ τ_{k} ∣_{C} \otimes I_{X} : H_{C} \otimes H_{X} ⟶ H_{X} .$
unlabeled · line 446
$A_{R} = {k = 0 \sum^{D - 1} a_{k} ∣ k ⟩ ⟨ k ∣_{R} : a_{k} \in C} .$