Manuscript En02-C@1 · En02-C-v1.tex · lemma · 7549c6df

Lemma: First-carrier consumption destroys recoverable reference entanglement

C3: an optimal approximate memory obstruction · lines 769–832 · proof included or linked

En02-C@1#lem:firstmeasurement, En02-C@1#eq:conditional

For every positive measurement effect $E$ on carrier $1$ , the conditional state of $R 23$ obtained from $ω_{R 123}$ is separable across $R ∣ 23$ , whenever that outcome has positive probability.

Proof. Write

E = \frac{tr E}{2} (I + v_{x} X + v_{y} Y + v_{z} Z), v_{x}^{2} + v_{y}^{2} + v_{z}^{2} \leq 1.

Its outcome probability is $tr E /2$ . The normalized conditional state is

ω_{R 23 ∣ E} = \frac{1}{8} (I - v_{z} X_{R} A + v_{x} Y_{R} B - v_{y} Z_{R} C),

where

A = X_{2} Z_{3}, B = Y_{2} X_{3}, C = Z_{2} Y_{3} .

The first two operators commute, are independent, and satisfy $A B = - C$ . Their four rank-one joint projectors are

P_{α β} = \frac{1}{4} (I + α A) (I + β B), α, β \in {+ 1, - 1} .

Equation (eq:conditional) becomes

ω_{R 23 ∣ E} = \frac{1}{4} α, β \sum ρ_{α β}^{R} \otimes P_{α β},

with

ρ_{α β}^{R} = \frac{1}{2} [I - α v_{z} X_{R} + β v_{x} Y_{R} + α β v_{y} Z_{R}] .

Each Bloch vector has length $∥ v ∥_{2} \leq 1$ , so every $ρ_{α β}^{R}$ is a density operator. This is an explicit separable decomposition.

Displayed formulas in this passage

En02-C@1#eq:conditional · line 788
$ω_{R 23 ∣ E} = \frac{1}{8} (I - v_{z} X_{R} A + v_{x} Y_{R} B - v_{y} Z_{R} C),$
unlabeled · line 780
$E = \frac{tr E}{2} (I + v_{x} X + v_{y} Y + v_{z} Z), v_{x}^{2} + v_{y}^{2} + v_{z}^{2} \leq 1.$
unlabeled · line 798
$A = X_{2} Z_{3}, B = Y_{2} X_{3}, C = Z_{2} Y_{3} .$
unlabeled · line 803
$P_{α β} = \frac{1}{4} (I + α A) (I + β B), α, β \in {+ 1, - 1} .$
unlabeled · line 811
$ω_{R 23 ∣ E} = \frac{1}{4} α, β \sum ρ_{α β}^{R} \otimes P_{α β},$
unlabeled · line 819
$ρ_{α β}^{R} = \frac{1}{2} [I - α v_{z} X_{R} + β v_{x} Y_{R} + α β v_{y} Z_{R}] .$

Lemma: First-carrier consumption destroys recoverable reference entanglement

Displayed formulas in this passage

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