Manuscript Oai212-A@1 · Oai212-A.tex · section · 79323841

Independent geometric-pair quench

Bridge 2: an engineered entanglement-cost membrane › Independent geometric-pair quench · lines 849–888

For each edge $e = {u, v}$ of a finite lattice graph, reserve one mode at $u$ and one mode at $v$ , distinct from the modes for every other edge. Let $w_{e} \sim Exp (λ)$ independently and set

z_{e} = tanh (w_{e} /2) \in (0, 1) .

Prepare

∣ ψ_{e} ⟩ = 1 - z_{e} r = 0 \sum^{\infty} z_{e}^{r /2} ∣ r ⟩_{u} ∣ r ⟩_{v}, ∣Ψ ⟩ = e \in E ⨂ ∣ ψ_{e} ⟩ .

This preparation is a local finite-time quench in a precise bounded-Hamiltonian model. Put

c_{e} = 1 - z_{e}, ∣ v_{e} ⟩ = \frac{∣00 ⟩ - ∣ ψ _{e} ⟩}{2 ( 1 - c _{e} )} .

For a prescribed duration $t_{q} > 0$ , use

H_{e} = \frac{π ℏ}{t _{q}} ∣ v_{e} ⟩ ⟨ v_{e} ∣.

Then

e^{- i t_{q} H_{e} /ℏ} = I - 2∣ v_{e} ⟩ ⟨ v_{e} ∣, e^{- i t_{q} H_{e} /ℏ} ∣00 ⟩ = ∣ ψ_{e} ⟩ .

The edge Hamiltonians commute because they act on disjoint mode factors, even when their geometric edges meet at a vertex. Each is bounded and self-adjoint; the finite product evolution is therefore well defined.

Displayed formulas in this passage

unlabeled · line 854
$z_{e} = tanh (w_{e} /2) \in (0, 1) .$
unlabeled · line 858
$∣ ψ_{e} ⟩ = 1 - z_{e} r = 0 \sum^{\infty} z_{e}^{r /2} ∣ r ⟩_{u} ∣ r ⟩_{v}, ∣Ψ ⟩ = e \in E ⨂ ∣ ψ_{e} ⟩ .$
unlabeled · line 868
$c_{e} = 1 - z_{e}, ∣ v_{e} ⟩ = \frac{∣00 ⟩ - ∣ ψ _{e} ⟩}{2 ( 1 - c _{e} )} .$
unlabeled · line 874
$H_{e} = \frac{π ℏ}{t _{q}} ∣ v_{e} ⟩ ⟨ v_{e} ∣.$
unlabeled · line 878
$e^{- i t_{q} H_{e} /ℏ} = I - 2∣ v_{e} ⟩ ⟨ v_{e} ∣, e^{- i t_{q} H_{e} /ℏ} ∣00 ⟩ = ∣ ψ_{e} ⟩ .$

Independent geometric-pair quench

Displayed formulas in this passage

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