Manuscript VII-E@1 · VII-E.tex · section · b713cf3c

Finite-dimensional conventions

Types, states, and external geometry › Finite-dimensional conventions · lines 344–388

VII-E@1#eq:adjoint, VII-E@1#eq:functional, VII-E@1#eq:distance-conventions

For a finite-dimensional Hilbert space $H$ , write $L (H)$ for its operators and $D (H)$ for its density operators. A channel is a completely positive, trace-preserving linear map. Its adjoint satisfies

Tr (A Φ (X)) = Tr (Φ^{*} (A) X) .

The adjoint of a channel is unital and completely positive. A Kraus representation is

Φ (X) = α \sum K_{α} X K_{α}^{†}, α \sum K_{α}^{†} K_{α} = I .

These are the standard finite-dimensional conventions Watrous2018.

A classical register has a distinguished orthonormal basis. Admissible joint states are block diagonal in that basis, including their correlations with other registers. A diagonal classical marginal alone is insufficient. Operations declared to preserve classicality preserve this joint block-diagonal state space.

A unital observable algebra means a unital $^{*}$ -subalgebra $A \subseteq L (H)$ . Its accessible state functional is

ω_{ρ} ∣_{A} : A ⟼ Tr (ρ A) .

Agreement on $A$ means equality of these functionals.

Our norm and distance conventions are

∥ X ∥_{1} = Tr X^{†} X, d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{TV} (p, q) = \frac{1}{2} w \sum ∣ p (w) - q (w)∣ .

The operator norm is $∥ \cdot ∥_{\infty}$ . No ancillary reference is included in an optimization unless explicitly stated.

Displayed formulas in this passage

VII-E@1#eq:adjoint · line 349
$Tr (A Φ (X)) = Tr (Φ^{*} (A) X) .$
VII-E@1#eq:functional · line 371
$ω_{ρ} ∣_{A} : A ⟼ Tr (ρ A) .$
VII-E@1#eq:distance-conventions · line 378
$∥ X ∥_{1} = Tr X^{†} X, d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{TV} (p, q) = \frac{1}{2} w \sum ∣ p (w) - q (w)∣ .$
unlabeled · line 355
$Φ (X) = α \sum K_{α} X K_{α}^{†}, α \sum K_{α}^{†} K_{α} = I .$