Manuscript IV-B@1 · IV-B-v1.tex · proposition · dfb66408

Proposition: Finite covariant conditional evolution

The finite destructive clock instrument › Conditional evolution requires additional hypotheses · lines 1539–1567 · proof included or linked

IV-B@1#prop:relational

Back to manuscript

Let $U_{C} (t), U_{X} (t)$ be prescribed unitary groups, and assume

(U_{C} (t) \otimes U_{X} (t)) ρ (U_{C} (t)^{*} \otimes U_{X} (t)^{*}) = ρ

at the relevant times. Suppose a complete rank-one reading basis obeys

P_{k} = U_{C} (t_{k}) P_{0} U_{C} (t_{k})^{*} .

Then its subnormalized residuals satisfy

σ_{k} = U_{X} (t_{k}) σ_{0} U_{X} (t_{k})^{*}, p_{k} = 1/ d .

Hence the normalized residuals have the corresponding unitary relation.

Proof. Stationarity gives

(U_{C} (t)^{*} \otimes I) ρ (U_{C} (t) \otimes I) = (I \otimes U_{X} (t)) ρ (I \otimes U_{X} (t)^{*}) .

Substitute the covariance formula for $P_{k}$ into the partial trace. Unitary invariance of the clock trace gives the displayed relation. All traces $p_{k}$ are equal and sum to one.

Displayed formulas in this passage

unlabeled · line 1542
$(U_{C} (t) \otimes U_{X} (t)) ρ (U_{C} (t)^{*} \otimes U_{X} (t)^{*}) = ρ$
unlabeled · line 1547
$P_{k} = U_{C} (t_{k}) P_{0} U_{C} (t_{k})^{*} .$
unlabeled · line 1551
$σ_{k} = U_{X} (t_{k}) σ_{0} U_{X} (t_{k})^{*}, p_{k} = 1/ d .$
unlabeled · line 1560
$(U_{C} (t)^{*} \otimes I) ρ (U_{C} (t) \otimes I) = (I \otimes U_{X} (t)) ρ (I \otimes U_{X} (t)^{*}) .$

Nearby in this section

section: Conditional evolution requires additional hypotheses