Manuscript Oai272-A@1 · Oai272-A.tex · section · a40828a0

States, maps, and distances

Stage 0: conventions and the operational package › States, maps, and distances · lines 274–308

Oai272-A@1#eq:choi

All finite matrix spaces are complex and have fixed orthonormal bases. Transpose $T_{d} (X) = X^{T}$ is complex-linear. A map $F : M_{a} (C) \to M_{b} (C)$ is PPT if $F$ and $T_{b} F$ are completely positive. No trace condition is included.

We use the unnormalized Choi convention

J (F) = i, j = 1 \sum^{a} E_{ij} \otimes F (E_{ij}) = (id_{a} \otimes F) (∣ Ω_{a} ⟩ ⟨ Ω_{a} ∣), ∣ Ω_{a} ⟩ = i = 1 \sum^{a} ∣ i, i ⟩ .

A channel has $Tr J (F) = a$ , and its normalized Choi state is $j (F) = J (F) / a$ . A nonzero CP map without trace preservation instead has normalized Choi state $J (F) / Tr J (F)$ .

For states and compatible channels,

d_{tr} (α, β) = \frac{1}{2} ∥ α - β ∥_{1}, d_{⋄} (F, G) = \frac{1}{2} ∥ F - G ∥_{⋄} .

The diamond norm includes an independent reference Watrous. Natural logarithms are used unless a base is displayed; key lengths are measured in bits.

A positive bipartite matrix is separable if it is a finite sum $\sum_{r} A_{r} \otimes B_{r}$ with $A_{r}, B_{r} ⪰ 0$ . A CP map is entanglement breaking at the CP-cone level if every positive referenced input has separable output. This definition also applies to maps that are not trace preserving.

Displayed formulas in this passage

Oai272-A@1#eq:choi · line 282
$J (F) = i, j = 1 \sum^{a} E_{ij} \otimes F (E_{ij}) = (id_{a} \otimes F) (∣ Ω_{a} ⟩ ⟨ Ω_{a} ∣), ∣ Ω_{a} ⟩ = i = 1 \sum^{a} ∣ i, i ⟩ .$
unlabeled · line 294
$d_{tr} (α, β) = \frac{1}{2} ∥ α - β ∥_{1}, d_{⋄} (F, G) = \frac{1}{2} ∥ F - G ∥_{⋄} .$