Manuscript Hawk06-A@1 · Hawk06-A.tex · section · c59fe67c

Exactly where Page–Sen is applicable

The coherent and destructive entropy calculations › Exactly where Page–Sen is applicable · lines 1107–1149

When $h = 1$ , (eq:randomprojection) is Haar pure. With $m_{0} = min (r, b)$ , $n_{0} = max (r, b)$ , the established Page–Sen evaluation gives

S = \frac{H _{D} - H _{n_{0}} - ( m _{0} - 1 ) / ( 2 n _{0} )}{ln 2}, H = \frac{H _{D} - H _{b}}{ln 2} .

The second identity follows by differentiating the Dirichlet moment of the prefix probabilities; the first is the Page–Sen integral PageAverage,Sen. These are the pure-preparation comparisons of Hawk05-B and Hawk05-C Hawk05B,Hawk05C.

For $h > 1$ , the exact means are the finite random-projector averages specified by (eq:randomprojection); the proved quantitative evaluations used here are (eq:qpurity)–(eq:ranktent). No claim of a closed Page–Sen formula for those mixed-state means is made.

At complete release $b = 1$ , included only as a comparison beyond a positive-residual cutoff,

S (R_{N}^{U}) = lo g_{2} h

for every unitary and pure diary input. For $D > h$ ,

H_{N} \geq lo g_{2} \frac{h ( D + 1 )}{h + 1} > lo g_{2} h .

Thus the coherent endpoint is not zero in the early-entangled model.

Displayed formulas in this passage

unlabeled · line 1112
$S = \frac{H _{D} - H _{n_{0}} - ( m _{0} - 1 ) / ( 2 n _{0} )}{ln 2}, H = \frac{H _{D} - H _{b}}{ln 2} .$
unlabeled · line 1136
$S (R_{N}^{U}) = lo g_{2} h$
unlabeled · line 1141
$H_{N} \geq lo g_{2} \frac{h ( D + 1 )}{h + 1} > lo g_{2} h .$