Manuscript Hawk05-A@1 · Hawk05-A.tex · lemma · 449f85bb

Lemma: Complete scheduled evolution and conservation

An explicit conserving transfer schedule › Memory, cells, and transfer unitaries · lines 947–993 · proof included or linked

Hawk05-A@1#lem:schedule, Hawk05-A@1#eq:flaggedisometry

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Each $U_{j}$ is unitary and commutes with $H_{0}$ . After $n$ transfers, let

q_{n} = j = 1 \prod^{n} cos^{2} θ_{j}, q_{0} = 1.

For $q_{n} < 1$ , there is an isometry $J_{n} : C \to R_{\leq n}$ such that the complete prepared evolution is

V_{n} ∣ ψ ⟩ = 1 - q_{n} J_{n} ∣ ψ ⟩_{R} ∣ \emptyset ⟩_{M} + q_{n} ∣ \emptyset ⟩_{R} ∣ ψ ⟩_{M} .

Here $∣ \emptyset ⟩_{R}$ means that all emitted cells are vacant. Future vacant cells are fixed spectators in the complete residual.

Proof. The rotations are orthogonal unitary blocks. Their two basis vectors have the same $H_{0}$ eigenvalue $ω$ , so the rotations commute with $H_{0}$ .

Let $∣ j, i ⟩_{R}$ denote label $i$ in cell $j$ , with every other emitted cell vacant. Induction gives the emitted amplitudes

a_{j} = k < j \prod cos θ_{k} sin θ_{j} .

They satisfy

j = 1 \sum^{n} a_{j}^{2} = j = 1 \sum^{n} (q_{j - 1} - q_{j}) = 1 - q_{n} .

Thus

J_{n} ∣ i ⟩ = \frac{1}{1 - q _{n}} j = 1 \sum^{n} a_{j} ∣ j, i ⟩_{R}

has orthonormal columns and proves (eq:flaggedisometry). At $q_{n} = 1$ , the emitted term is zero and no $J_{n}$ is required.

Displayed formulas in this passage

Hawk05-A@1#eq:flaggedisometry · line 957
$V_{n} ∣ ψ ⟩ = 1 - q_{n} J_{n} ∣ ψ ⟩_{R} ∣ \emptyset ⟩_{M} + q_{n} ∣ \emptyset ⟩_{R} ∣ ψ ⟩_{M} .$
unlabeled · line 951
$q_{n} = j = 1 \prod^{n} cos^{2} θ_{j}, q_{0} = 1.$
unlabeled · line 976
$a_{j} = k < j \prod cos θ_{k} sin θ_{j} .$
unlabeled · line 980
$j = 1 \sum^{n} a_{j}^{2} = j = 1 \sum^{n} (q_{j - 1} - q_{j}) = 1 - q_{n} .$
unlabeled · line 986
$J_{n} ∣ i ⟩ = \frac{1}{1 - q _{n}} j = 1 \sum^{n} a_{j} ∣ j, i ⟩_{R}$

Lemma: Complete scheduled evolution and conservation

Displayed formulas in this passage

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