Manuscript XXII-F@1 · XXII-F.tex · section · 67821705

Targets and endpoint coordinates

Separate quantum transport and resource contracts › Targets and endpoint coordinates · lines 984–1019

XXII-F@1#eq:arrival-target

Let

\Null_{D} = \partial K_{D} ∖ {0} .

A target is $ζ = (p, v)$ , where $p \in E_{D}$ and $v \in \Null_{D}$ . Write

T_{D} ι_{D} (p) = (s, y), T_{D} v = (τ, x), τ = ∥ x ∥_{2} > 0.

Its desired source is $(s, y)$ , its endpoint is $(s + τ, y + x)$ , and its absolute arrival coordinate is

t_{ζ} = s + τ .

The endpoint need not be represented by an old event.

Let $T_{D}$ be a supplied nonempty family of such targets. For protocol $(n, ζ)$ , declare coordinates $a_{n, ζ}$ , $b_{n, ζ}$ of its realizing source and receiver and write

b_{n, ζ} - a_{n, ζ} = (τ_{n, ζ}, x_{n, ζ}) .

These coordinate assignments require (R3)–(R4). They do not follow from numbering experiments or circuit instructions.

Use

∥ (t, x) ∥_{*} = ∣ t ∣ + ∥ x ∥_{2} .

This is a frame-dependent mathematical comparison norm.

Displayed formulas in this passage

unlabeled · line 987
$\Null_{D} = \partial K_{D} ∖ {0} .$
unlabeled · line 992
$T_{D} ι_{D} (p) = (s, y), T_{D} v = (τ, x), τ = ∥ x ∥_{2} > 0.$
XXII-F@1#eq:arrival-target · line 998
$t_{ζ} = s + τ .$
unlabeled · line 1007
$b_{n, ζ} - a_{n, ζ} = (τ_{n, ζ}, x_{n, ζ}) .$
unlabeled · line 1015
$∥ (t, x) ∥_{*} = ∣ t ∣ + ∥ x ∥_{2} .$