Manuscript XXII-D@1 · XXII-D.tex · section · 17889d18

The unchanged XXII-C interface

The unchanged XXII-C interface · lines 196–271

XXII-D@1#eq:process-validity, XXII-D@1#eq:process-law, XXII-D@1#eq:process-family

For completeness, the relevant frozen objects are stated without using predecessor auxiliary files.

Let

J = {A, B, C}, H_{a_{I}} = H_{a_{O}} = C^{2}, H = a \in J ⨂ (H_{a_{I}} \otimes H_{a_{O}}) .

Use the transposed unnormalized Choi convention

J (M) = i, j = 0 \sum^{1} ∣ i ⟩ ⟨ j ∣ \otimes M (∣ i ⟩ ⟨ j ∣), M = J (M)^{T} .

The set $W_{3}$ consists of positive operators $W$ on $H$ satisfying

Tr [W (M_{A} \otimes M_{B} \otimes M_{C})] = 1 whenever M_{a} \geq 0, Tr_{a_{O}} M_{a} = I_{a_{I}} .

Local instrument insertions give the complete probability law

P_{W} (x ∣ s) = Tr [W a \in J ⨂ M^{a}_{x_{a} ∣ s_{a}}] .

This is the process-matrix language of Oreshkov, Costa, and Brukner OCB, with the exact access restrictions of XXII-C CheungXXIIC (spec:access).

The frozen family is

P_{3q} = (W_{3})^{N}, W = (W_{e})_{e \in N} .

Each experiment is tested separately. Its identifier is not time, a refinement scale, or spatial position.

The access $A_{0}$ permits one use of each party's process interface in a selected experiment. The receiver datum is its local classical instrument record. Other records are summed; postselection and undeclared interparty feed-forward are excluded. Local auxiliary implementations are allowed only insofar as they implement the declared local instrument on that one-use interface.

Let $Δ_{e} (a, b)$ be the supremum of the total-variation distance between $b$ 's unconditional laws when a deterministic insertion at $a$ is replaced, with all other insertions fixed. The direct relation on $N \times J$ is

(e, a) R_{inf}_{D} (f, b) ⟺ e = f, a \neq = b, Δ_{e} (a, b) > 0.

There are no cross-experiment edges. Taking strongly connected components and then reachability gives

(E_{D}, ⪯_{D}) .

Displayed formulas in this passage

unlabeled · line 202
$J = {A, B, C}, H_{a_{I}} = H_{a_{O}} = C^{2}, H = a \in J ⨂ (H_{a_{I}} \otimes H_{a_{O}}) .$
unlabeled · line 211
$J (M) = i, j = 0 \sum^{1} ∣ i ⟩ ⟨ j ∣ \otimes M (∣ i ⟩ ⟨ j ∣), M = J (M)^{T} .$
XXII-D@1#eq:process-validity · line 220
$Tr [W (M_{A} \otimes M_{B} \otimes M_{C})] = 1 whenever M_{a} \geq 0, Tr_{a_{O}} M_{a} = I_{a_{I}} .$
XXII-D@1#eq:process-law · line 228
$P_{W} (x ∣ s) = Tr [W a \in J ⨂ M^{a}_{x_{a} ∣ s_{a}}] .$
XXII-D@1#eq:process-family · line 241
$P_{3q} = (W_{3})^{N}, W = (W_{e})_{e \in N} .$
unlabeled · line 261
$(e, a) R_{inf}_{D} (f, b) ⟺ e = f, a \neq = b, Δ_{e} (a, b) > 0.$
unlabeled · line 268
$(E_{D}, ⪯_{D}) .$

The unchanged XXII-C interface

Displayed formulas in this passage

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