Manuscript V-B@1 · V-B-v1.tex · proposition · 3fadc1ad

Proposition: Clock limitations

Finite compression and the complete Enter interface › Destructive clock readout · lines 1842–1861 · proof included or linked

V-B@1#prop:clocklimits

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The clock channel has no universal left inverse. Its record marginal does not determine the residual. No matrices on the finite clock satisfy $[T_{C}, H_{C}] = i I_{C}$ .

Proof. The two inputs

\frac{∣ τ _{0} ⟩ + ∣ τ _{1} ⟩}{2}, \frac{∣ τ _{0} ⟩ - ∣ τ _{1} ⟩}{2}

with the same residual state have identical clock outputs. A left inverse cannot recover both. Inputs $P_{0} \otimes ρ_{X}$ and $P_{0} \otimes ρ_{X}^{'}$ have the same deterministic record and different residuals. Finally the trace of a commutator is zero, while $Tr (i I_{C}) \neq = 0$ .

Displayed formulas in this passage

unlabeled · line 1851
$\frac{∣ τ _{0} ⟩ + ∣ τ _{1} ⟩}{2}, \frac{∣ τ _{0} ⟩ - ∣ τ _{1} ⟩}{2}$

Nearby in this section

section: Destructive clock readout
theorem: Theorem: Complete clock and response instrument
section: Destructive clock readout