Manuscript XXII-C@1 · XXII-C.tex · section · b9a05a24

Fixed parties, dimensions, and Choi convention

The frozen process class › Fixed parties, dimensions, and Choi convention · lines 183–234

XXII-C@1#eq:choi, XXII-C@1#eq:local-sets

The party set is fixed once and for all:

J = {A, B, C} .

Each party has one qubit input and one qubit output:

H_{a_{I}} = H_{a_{O}} = C^{2}, a \in J .

Let

H = a \in J ⨂ (H_{a_{I}} \otimes H_{a_{O}}), dim H = 64.

A written tensor-factor ordering is bookkeeping only. It is not an operational or temporal order of the parties.

Fix bases and use the transposed, unnormalized Choi convention

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

where $^{T}$ denotes the full matrix transpose. This is the convention used in the original process-matrix probability formula OCB.

For each party set

C_{a} = {M \geq 0 : Tr_{a_{O}} M = I_{a_{I}}}, B_{a} = {B \geq 0 : Tr_{a_{O}} B \leq I_{a_{I}}} .

The first set represents channels. The second represents completely positive trace-nonincreasing instrument elements in the convention (eq:choi) Watrous (Chapter 2).

Displayed formulas in this passage

unlabeled · line 186
$J = {A, B, C} .$
unlabeled · line 190
$H_{a_{I}} = H_{a_{O}} = C^{2}, a \in J .$
unlabeled · line 195
$H = a \in J ⨂ (H_{a_{I}} \otimes H_{a_{O}}), dim H = 64.$
XXII-C@1#eq:choi · line 206
$J (M) = i, j = 0 \sum^{1} ∣ i ⟩ ⟨ j ∣ \otimes M (∣ i ⟩ ⟨ j ∣), M = J (M)^{T},$
XXII-C@1#eq:local-sets · line 220
$C_{a} = {M \geq 0 : Tr_{a_{O}} M = I_{a_{I}}}, B_{a} = {B \geq 0 : Tr_{a_{O}} B \leq I_{a_{I}}} .$

Fixed parties, dimensions, and Choi convention

Displayed formulas in this passage

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