Manuscript VII-J@1 · VII-J.tex · example · d8c01b04

Example: Strict causality fails for a generic local Hamiltonian

The Hamiltonian route: tails and a constructive error › An explicit finite-volume tail · lines 1247–1260

For two sites separated by a positive assigned geometric distance, take $H = J X_{A} X_{B}$ , $J > 0$ . Prepare $B$ in $∣ 0 ⟩$ , and use replacement interventions preparing $A$ in $∣ + ⟩$ or $∣ - ⟩$ . The two $B$ outputs are

e^{- i J tX} ∣ 0 ⟩, e^{i J tX} ∣ 0 ⟩ .

Their trace distance is $∣ sin (2 J t) ∣$ , nonzero for arbitrarily small positive $t$ . A finite-dimensional interaction of this type therefore does not have an exact geometric propagation cutoff.

Displayed formulas in this passage

unlabeled · line 1253
$e^{- i J tX} ∣ 0 ⟩, e^{i J tX} ∣ 0 ⟩ .$

Nearby in this section

section: An explicit finite-volume tail
theorem: Theorem: Reference-stable finite-volume influence tail
section: An explicit finite-volume tail