Manuscript VII-G@1 · VII-G.tex · section · 54f561e8

Direct-sum conventions

Hybrid states, controllers, and transition instruments › Direct-sum conventions · lines 259–308

VII-G@1#eq:state-algebra, VII-G@1#eq:pairing, VII-G@1#eq:hybrid-norms, VII-G@1#eq:cq-state, VII-G@1#eq:conditional

Fix a nonempty finite label set $G$ and a Hilbert space $H$ of dimension $d < \infty$ . Set

X_{G} = g \in G ⨁ L (H) .

For $X = (X_{g})_{g}$ and $B = (B_{g})_{g}$ , use

Tr_{G} X = g \sum Tr X_{g}, ⟨ B, X ⟩ = g \sum Tr (B_{g} X_{g}) .

The identity is $1 = (I)_{g}$ . The relevant norms are

∥ X ∥_{1, G} = g \sum ∥ X_{g} ∥_{1}, ∥ B ∥_{\infty, G} = g max ∥ B_{g} ∥_{\infty} .

A hybrid state is a positive element $ρ = (ρ_{g})_{g}$ with $Tr_{G} ρ = 1$ . Equivalently,

σ_{ρ} = g \sum ∣ g ⟩ ⟨ g ∣ \otimes ρ_{g} .

The blocks are subnormalized:

p_{g} = Tr ρ_{g}, ρ_{g} = ρ_{g} / p_{g} only if p_{g} > 0.

Conditional normalization is not used in the unconditional evolution law.

We use

d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1, G}, d_{TV} (p, q) = \frac{1}{2} x \sum ∣ p_{x} - q_{x} ∣, d_{⋄} (Φ, Ξ) = \frac{1}{2} ∥ Φ - Ξ ∥_{⋄} .

The classical-input meaning of the diamond norm is made precise in Section sec:stability.

Displayed formulas in this passage

VII-G@1#eq:state-algebra · line 263
$X_{G} = g \in G ⨁ L (H) .$
VII-G@1#eq:pairing · line 268
$Tr_{G} X = g \sum Tr X_{g}, ⟨ B, X ⟩ = g \sum Tr (B_{g} X_{g}) .$
VII-G@1#eq:hybrid-norms · line 275
$∥ X ∥_{1, G} = g \sum ∥ X_{g} ∥_{1}, ∥ B ∥_{\infty, G} = g max ∥ B_{g} ∥_{\infty} .$
VII-G@1#eq:cq-state · line 284
$σ_{ρ} = g \sum ∣ g ⟩ ⟨ g ∣ \otimes ρ_{g} .$
VII-G@1#eq:conditional · line 289
$p_{g} = Tr ρ_{g}, ρ_{g} = ρ_{g} / p_{g} only if p_{g} > 0.$
unlabeled · line 299
$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1, G}, d_{TV} (p, q) = \frac{1}{2} x \sum ∣ p_{x} - q_{x} ∣, d_{⋄} (Φ, Ξ) = \frac{1}{2} ∥ Φ - Ξ ∥_{⋄} .$