Manuscript VII-J-ST@1 · VII-J-ST.tex · proposition · f1f1d82c

Proposition: Sampled covariance and stationarity

Destructive clocks, programme records, and calibration › Optional relational interpretation · lines 1354–1383 · proof included or linked

VII-J-ST@1#prop:relational

Back to manuscript

Suppose finite Hermitian $H_{C}, H_{S}$ and $ρ_{C S}$ satisfy

[H_{C} \otimes I + I \otimes H_{S}, ρ] = 0, P_{a} = U_{C} (t_{a}) P_{0} U_{C} (t_{a})^{†} .

Then

D_{a} ρ D_{a}^{†} = U_{S} (t_{a}) (D_{0} ρ D_{0}^{†}) U_{S} (t_{a})^{†} .

For a complete orthonormal reading basis all probabilities equal $2^{- m}$ . For an extended $S E$ assertion, stationarity of the full extension is required.

Proof. Stationarity gives

(U_{C}^{†} \otimes I) ρ (U_{C} \otimes I) = (I \otimes U_{S}) ρ (I \otimes U_{S}^{†}) .

Insert the covariant projector into the clock partial trace and use unitary invariance of that trace. The resulting blocks are unitarily related, so their traces are equal and sum to one. The same calculation applies to an extension exactly under the extended stationarity hypothesis.

Displayed formulas in this passage

unlabeled · line 1357
$[H_{C} \otimes I + I \otimes H_{S}, ρ] = 0, P_{a} = U_{C} (t_{a}) P_{0} U_{C} (t_{a})^{†} .$
unlabeled · line 1362
$D_{a} ρ D_{a}^{†} = U_{S} (t_{a}) (D_{0} ρ D_{0}^{†}) U_{S} (t_{a})^{†} .$
unlabeled · line 1374
$(U_{C}^{†} \otimes I) ρ (U_{C} \otimes I) = (I \otimes U_{S}) ρ (I \otimes U_{S}^{†}) .$

Proposition: Sampled covariance and stationarity

Displayed formulas in this passage

Nearby in this section