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

Theorem: Exact purity moments for the entangled preparation

The coherent and destructive entropy calculations · lines 1017–1101 · proof included or linked

Hawk06-A@1#thm:entropy, Hawk06-A@1#eq:qpurity, Hawk06-A@1#eq:cpurity, Hawk06-A@1#eq:Sbounds, Hawk06-A@1#eq:Hbounds, Hawk06-A@1#eq:ranktent

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At a cut $D = r b$ , 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_{R}^{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

S = E_{U} S (ρ_{R}^{U}), H = E_{U} H (p^{U}), t = min (r, bh),

one has

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

Proof. For the quantum purity, apply Lemma lem:twirl to $F_{R} \otimes I_{B B^{'}}$ . Its coefficients are

α_{R} = \frac{r ( b ^{2} - 1 )}{D ^{2} - 1}, β_{R} = \frac{b ( r ^{2} - 1 )}{D ^{2} - 1} .

Since the initial vault state has trace one and purity $1/ h$ , the expected purity is $α_{R} + β_{R} / h$ .

For the record, $p_{y} = Tr (Π_{y} ρ_{vault}^{U})$ , where $Π_{y} = ∣ y ⟩ ⟨ y ∣_{R} \otimes I_{B}$ has rank $b$ . The averaged squared 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 .

Therefore

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

Sum over the $r$ outcomes to obtain (eq:cpurity).

The rank of $ρ_{R}^{U}$ is at most $min (r, bh)$ , proving its entropy upper bound. The record has $r$ outcomes. For either a state or probability distribution, entropy is at least minus the logarithm of its purity. Jensen's inequality then proves the two lower bounds. Finally, direct comparison in (eq:qpurity) gives

p_{q} (r, b, h) \leq \frac{1}{r} + \frac{1}{bh},

which proves (eq:ranktent).

Displayed formulas in this passage

Hawk06-A@1#eq:qpurity · line 1020
$p_{q} (r, b, h) p_{c} (r, b, h) = \frac{r ( b ^{2} - 1 ) + b ( r ^{2} - 1 ) / h}{D ^{2} - 1}, = \frac{D b + ( D - b ) / h - 1}{D ^{2} - 1} .$
Hawk06-A@1#eq:cpurity · line 1020
$p_{q} (r, b, h) p_{c} (r, b, h) = \frac{r ( b ^{2} - 1 ) + b ( r ^{2} - 1 ) / h}{D ^{2} - 1}, = \frac{D b + ( D - b ) / h - 1}{D ^{2} - 1} .$
Hawk06-A@1#eq:Sbounds · line 1046
$- lo g_{2} p_{q} (r, b, h) - lo g_{2} p_{c} (r, b, h) 0 \leq lo g_{2} t - S \leq S \leq lo g_{2} t, \leq H \leq lo g_{2} r, \leq lo g_{2} (1 + \frac{min ( r , bh )}{max ( r , bh )}) \leq 1.$
Hawk06-A@1#eq:Hbounds · line 1046
$- lo g_{2} p_{q} (r, b, h) - lo g_{2} p_{c} (r, b, h) 0 \leq lo g_{2} t - S \leq S \leq lo g_{2} t, \leq H \leq lo g_{2} r, \leq lo g_{2} (1 + \frac{min ( r , bh )}{max ( r , bh )}) \leq 1.$
Hawk06-A@1#eq:ranktent · line 1046
$- lo g_{2} p_{q} (r, b, h) - lo g_{2} p_{c} (r, b, h) 0 \leq lo g_{2} t - S \leq S \leq lo g_{2} t, \leq H \leq lo g_{2} r, \leq lo g_{2} (1 + \frac{min ( r , bh )}{max ( r , bh )}) \leq 1.$
unlabeled · line 1031
$E_{U} Tr [(ρ_{R}^{U})^{2}] = p_{q} (r, b, h) .$
unlabeled · line 1036
$E_{U} y \sum (p_{y}^{U})^{2} = p_{c} (r, b, h) .$
unlabeled · line 1040
$S = E_{U} S (ρ_{R}^{U}), H = E_{U} H (p^{U}), t = min (r, bh),$
unlabeled · line 1065
$α_{R} = \frac{r ( b ^{2} - 1 )}{D ^{2} - 1}, β_{R} = \frac{b ( r ^{2} - 1 )}{D ^{2} - 1} .$
unlabeled · line 1076
$E_{U} (ρ_{vault}^{U})^{\otimes 2} = \frac{1 - 1/ ( h D )}{D ^{2} - 1} I + \frac{1/ h - 1/ D}{D ^{2} - 1} F .$
unlabeled · line 1084
$E p_{y}^{2} = \frac{b ^{2} [ 1 - 1/ ( h D )] + b [ 1/ h - 1/ D ]}{D ^{2} - 1} .$
unlabeled · line 1097
$p_{q} (r, b, h) \leq \frac{1}{r} + \frac{1}{bh},$

Theorem: Exact purity moments for the entangled preparation

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