Manuscript Hawk02-A@1 · Hawk02-A.tex · section · 7fbae745

Two conserving isometries

Thermal diagnostics do not determine assisted recovery › Two conserving isometries · lines 1053–1090

Hawk02-A@1#eq:V1, Hawk02-A@1#eq:V0

Choose mutually orthogonal residual subspaces: one normalized vector $∣ s_{n, k} ⟩$ for every paired sector, and a two-dimensional isometric embedding $I_{n} : H_{C} \to H_{B}$ for every even- $n$ failure sector. Define on the prepared input subspace

V_{1} (∣ ψ ⟩_{C} ∣ r ⟩_{K}) = ∣ r ⟩_{K} \otimes [n = 0 \sum^{L} k = 0 \sum^{m_{n} - 1} 2 d_{n} ∣ s_{n, k} ⟩_{B} \otimes J_{n, k} P_{r} ∣ ψ ⟩_{L} + 0 \leq n \leq L n even \sum d_{n} I_{n} ∣ ψ ⟩_{B} \otimes ∣ n, n ⟩_{L}] .

For a second model, choose mutually orthogonal embeddings $I_{n, j}^{(0)} : H_{C} \to H_{B}$ in the same declared energy sectors and set

V_{0} (∣ ψ ⟩_{C} ∣ r ⟩_{K}) = ∣ r ⟩_{K} \otimes n = 0 \sum^{L} j = 0 \sum^{n} d_{n} I_{n, j}^{(0)} ∣ ψ ⟩_{B} \otimes ∣ n, j ⟩_{L} .

A common enlarged residual space contains both sets of subspaces. Apply either isometry to the fixed resource $Ω$ . The accessible output is $A = E L$ , and the complete residual is $B K$ .

Displayed formulas in this passage

Hawk02-A@1#eq:V1 · line 1060
$V_{1} (∣ ψ ⟩_{C} ∣ r ⟩_{K}) = ∣ r ⟩_{K} \otimes [n = 0 \sum^{L} k = 0 \sum^{m_{n} - 1} 2 d_{n} ∣ s_{n, k} ⟩_{B} \otimes J_{n, k} P_{r} ∣ ψ ⟩_{L} + 0 \leq n \leq L n even \sum d_{n} I_{n} ∣ ψ ⟩_{B} \otimes ∣ n, n ⟩_{L}] .$
Hawk02-A@1#eq:V0 · line 1078
$V_{0} (∣ ψ ⟩_{C} ∣ r ⟩_{K}) = ∣ r ⟩_{K} \otimes n = 0 \sum^{L} j = 0 \sum^{n} d_{n} I_{n, j}^{(0)} ∣ ψ ⟩_{B} \otimes ∣ n, j ⟩_{L} .$

Two conserving isometries

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