Manuscript Oai276-A@1 · Oai276-A.tex · section · 215068af

Information quantities

Stage 0: freeze the channel, units, and claims › Information quantities · lines 204–257

Oai276-A@1#eq:g, Oai276-A@1#eq:holevo, Oai276-A@1#eq:regularized

All Hilbert spaces in the mathematical arguments are nonzero and finite dimensional. Entropies in the local theorems are in bits:

S (ρ) = - Tr (ρ lo g_{2} ρ), h_{2} (x) = - x lo g_{2} x - (1 - x) lo g_{2} (1 - x),

with $0 lo g_{2} 0 = 0$ . Define

g_{2} (u) = h_{2} (\frac{1 + 1 - 4 u}{2}), 0 \leq u \leq \frac{1}{4} .

Natural-logarithm versions are denoted by

S_{e} = (ln 2) S, h_{e} = (ln 2) h_{2}, g_{e} = (ln 2) g_{2} .

For a channel $N$ , its one-use Holevo information is

χ (N) = {q_{a}, ρ_{a}} sup [S (a \sum q_{a} N (ρ_{a})) - a \sum q_{a} S (N (ρ_{a}))] .

The supremum is over finite ensembles of arbitrary input states. For $N^{\otimes n}$ , this includes input signals entangled across the $n$ uses.

The unassisted classical capacity $C (N)$ concerns equiprobable classical messages encoded into arbitrary block states, with one output POVM on the whole block, no preshared entanglement, no feedback, and average error tending to zero. The coding theorem gives

C (N) = n \geq 1 sup \frac{1}{n} χ (N^{\otimes n}) .

The operational coding input is the Holevo–Schumacher–Westmoreland theorem, not an inference from the existence of a Kraus representation Holevo1998,SchumacherWestmoreland,Watrous.

Displayed formulas in this passage

Oai276-A@1#eq:g · line 214
$g_{2} (u) = h_{2} (\frac{1 + 1 - 4 u}{2}), 0 \leq u \leq \frac{1}{4} .$
Oai276-A@1#eq:holevo · line 229
$χ (N) = {q_{a}, ρ_{a}} sup [S (a \sum q_{a} N (ρ_{a})) - a \sum q_{a} S (N (ρ_{a}))] .$
Oai276-A@1#eq:regularized · line 248
$C (N) = n \geq 1 sup \frac{1}{n} χ (N^{\otimes n}) .$
unlabeled · line 208
$S (ρ) = - Tr (ρ lo g_{2} ρ), h_{2} (x) = - x lo g_{2} x - (1 - x) lo g_{2} (1 - x),$
unlabeled · line 222
$S_{e} = (ln 2) S, h_{e} = (ln 2) h_{2}, g_{e} = (ln 2) g_{2} .$