Manuscript Hawk07-A@1 · Hawk07-A.tex · theorem · f9c0f5da

Theorem: Entropy estimates for the declared preparation

Coherent entropy, classical records, and the dimension neck · lines 1121–1190 · proof included or linked

Hawk07-A@1#thm:entropy, Hawk07-A@1#eq:purities, Hawk07-A@1#eq:entropybound

At $D = r b$ , define

p_{q} (r, b, h) p_{c} (r, b, h) = \frac{r ( b ^{2} - 1 ) + b ( r ^{2} - 1 ) / h}{D ^{2} - 1}, = \frac{D b + ( D - b ) / h - 1}{D ^{2} - 1} .

Then

E_{U} Tr [(ρ_{P}^{U})^{2}] = p_{q} (r, b, h) .

For computational-basis Enter on the released cells, with prefix probabilities $p_{y}^{U}$ ,

E_{U} y \sum (p_{y}^{U})^{2} = p_{c} (r, b, h) .

Writing $t = min (r, bh)$ , one has

- lo g_{2} p_{q} - lo g_{2} p_{c} 0 \leq lo g_{2} t - E_{U} S (P) \leq E_{U} S (P) \leq lo g_{2} t, \leq E_{U} H (Y) \leq lo g_{2} r, \leq lo g_{2} (1 + \frac{min ( r , bh )}{max ( r , bh )}) \leq 1.

Proof. Twirl $F_{P} \otimes I_{B B^{'}}$ . Its identity and swap coefficients are

\frac{r ( b ^{2} - 1 )}{D ^{2} - 1}, \frac{b ( r ^{2} - 1 )}{D ^{2} - 1} .

The initial vault purity is $1/ h$ , giving $p_{q}$ .

For record probabilities, use $Π_{y} = ∣ y ⟩ ⟨ y ∣_{P} \otimes I_{B}$ , of rank $b$ . The averaged two-copy vault state is

E_{U} (ρ_{vault}^{U})^{\otimes 2} = \frac{1 - 1/ ( h D )}{D ^{2} - 1} I + \frac{1/ h - 1/ D}{D ^{2} - 1} F .

Since $p_{y}^{U} = Tr (Π_{y} ρ_{vault}^{U})$ ,

E (p_{y}^{U})^{2} = \frac{b ^{2} [ 1 - 1/ ( h D )] + b [ 1/ h - 1/ D ]}{D ^{2} - 1} .

Sum over $r$ outcomes.

The late marginal has rank at most $min (r, bh)$ , and the record has at most $r$ outcomes. These prove the upper bounds. Entropy is at least minus the logarithm of purity; Jensen's inequality proves the lower bounds. Direct comparison gives $p_{q} \leq 1/ r + 1/ (bh)$ , proving the final estimate.

Displayed formulas in this passage

Hawk07-A@1#eq:purities · line 1124
$p_{q} (r, b, h) p_{c} (r, b, h) = \frac{r ( b ^{2} - 1 ) + b ( r ^{2} - 1 ) / h}{D ^{2} - 1}, = \frac{D b + ( D - b ) / h - 1}{D ^{2} - 1} .$
Hawk07-A@1#eq:entropybound · line 1143
$- lo g_{2} p_{q} - lo g_{2} p_{c} 0 \leq lo g_{2} t - E_{U} S (P) \leq E_{U} S (P) \leq lo g_{2} t, \leq E_{U} H (Y) \leq lo g_{2} r, \leq lo g_{2} (1 + \frac{min ( r , bh )}{max ( r , bh )}) \leq 1.$
unlabeled · line 1134
$E_{U} Tr [(ρ_{P}^{U})^{2}] = p_{q} (r, b, h) .$
unlabeled · line 1139
$E_{U} y \sum (p_{y}^{U})^{2} = p_{c} (r, b, h) .$
unlabeled · line 1159
$\frac{r ( b ^{2} - 1 )}{D ^{2} - 1}, \frac{b ( r ^{2} - 1 )}{D ^{2} - 1} .$
unlabeled · line 1169
$E_{U} (ρ_{vault}^{U})^{\otimes 2} = \frac{1 - 1/ ( h D )}{D ^{2} - 1} I + \frac{1/ h - 1/ D}{D ^{2} - 1} F .$
unlabeled · line 1177
$E (p_{y}^{U})^{2} = \frac{b ^{2} [ 1 - 1/ ( h D )] + b [ 1/ h - 1/ D ]}{D ^{2} - 1} .$

Theorem: Entropy estimates for the declared preparation

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