Manuscript Em03-C@1 · Em03-C-v1.tex · theorem · 47dd71ec

Theorem: Joint referenced destructive record

C1 interface: joint carriers and referenced records › Exact record identity and tail · lines 705–746 · proof included or linked

Em03-C@1#thm:jointrecord, Em03-C@1#eq:jointtail

Back to manuscript

For every normal input,

(P_{Z}^{S} \otimes id_{R}) (B_{a} \otimes id_{S R}) (ρ) = Ω_{a, S R} (ρ) .

Let $Ω_{a, S R}$ be its decoded version and $Ω_{\infty, S R}$ the unsaturated joint number record. In the common countable record space,

d_{tr} (Ω_{a, S R}, Ω_{\infty, S R}) = p_{a} (ρ) := ρ Pr (N_{1} > a_{1} or N_{2} > a_{2}) .

For the baseline,

p_{a} (ρ_{0}) \leq min {1, q_{1}^{a_{1} + 1} + q_{2}^{a_{2} + 1}} .

No independence assumption is needed.

Proof. Tensor the two Kraus families. Destructive readout selects the number-diagonal input blocks and sums exactly the blocks with the same clipped pair. Positive block traces sum to one, so the series converge in trace norm.

The two decoded records agree on the rectangular interior. Outside that rectangle, the unsaturated record has positive blocks that are absent from the clipped record. The clipped record adds their positive mass to boundary records inside the rectangle. These inside and outside differences occupy orthogonal classical records. Their total trace norm is twice the outside probability, proving (eq:jointtail). The union bound and the geometric marginal tails give the last assertion.

Displayed formulas in this passage

Em03-C@1#eq:jointtail · line 716
$d_{tr} (Ω_{a, S R}, Ω_{\infty, S R}) = p_{a} (ρ) := ρ Pr (N_{1} > a_{1} or N_{2} > a_{2}) .$
unlabeled · line 708
$(P_{Z}^{S} \otimes id_{R}) (B_{a} \otimes id_{S R}) (ρ) = Ω_{a, S R} (ρ) .$
unlabeled · line 725
$p_{a} (ρ_{0}) \leq min {1, q_{1}^{a_{1} + 1} + q_{2}^{a_{2} + 1}} .$

Theorem: Joint referenced destructive record

Displayed formulas in this passage

Nearby in this section