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

Truncated thermal occupation sectors

Thermal diagnostics do not determine assisted recovery › Truncated thermal occupation sectors · lines 994–1052

Hawk02-A@1#eq:thermaltoy, Hawk02-A@1#eq:successp

Let $C$ be a qubit, let $ω > 0$ , let $L \geq 1$ be an integer, and let $0 \leq q < 1$ . The late-radiation space has basis

{∣ n, j ⟩ : 0 \leq n \leq L, 0 \leq j \leq n}, H_{L} ∣ n, j ⟩ = nω ∣ n, j ⟩ .

These are the occupation sectors of two ideal equal-frequency oscillators truncated by total occupation. They are not asserted to be exact finite-duration wavepackets of definite Killing energy.

Set

Z_{L} (q) = n = 0 \sum^{L} (n + 1) q^{n}, d_{n} = \frac{q ^{n}}{Z _{L} ( q )}, σ_{L} (q) = n = 0 \sum^{L} j = 0 \sum^{n} d_{n} ∣ n, j ⟩ ⟨ n, j ∣ .

At $q = 0$ the state is the vacuum. For $q > 0$ , it is the normalized truncation of the two-oscillator Gibbs state with $q = e^{- β ω}$ .

Let

m_{n} = ⌊ \frac{n + 1}{2} ⌋ .

For $0 \leq k < m_{n}$ , define the equal-energy rail isometry

J_{n, k} ∣ 0 ⟩ = ∣ n, 2 k ⟩, J_{n, k} ∣ 1 ⟩ = ∣ n, 2 k + 1 ⟩ .

When $n$ is even, the unpaired vector is $∣ n, n ⟩$ . Define

p = n = 0 \sum^{L} 2 m_{n} d_{n} = 1 - 0 \leq n \leq L n even \sum d_{n} .

The early resource is

∣ Ω ⟩_{E K} = \frac{1}{2} r = 0 \sum^{3} ∣ r ⟩_{E} ∣ r ⟩_{K},

where $E$ is accessible and $K$ is inaccessible. Let $P_{r}$ be the four qubit Pauli operators, including $I$ .

Fix an initial energy $M_{⋆}$ and require

M_{⋆} - L ω \geq M_{cut} > 0.

The code and key registers are energy-degenerate. Residual basis vectors in sector $n$ have energy $M_{⋆} - nω$ .

Displayed formulas in this passage

Hawk02-A@1#eq:thermaltoy · line 1008
$Z_{L} (q) = n = 0 \sum^{L} (n + 1) q^{n}, d_{n} = \frac{q ^{n}}{Z _{L} ( q )}, σ_{L} (q) = n = 0 \sum^{L} j = 0 \sum^{n} d_{n} ∣ n, j ⟩ ⟨ n, j ∣ .$
Hawk02-A@1#eq:successp · line 1030
$p = n = 0 \sum^{L} 2 m_{n} d_{n} = 1 - 0 \leq n \leq L n even \sum d_{n} .$
unlabeled · line 998
${∣ n, j ⟩ : 0 \leq n \leq L, 0 \leq j \leq n}, H_{L} ∣ n, j ⟩ = nω ∣ n, j ⟩ .$
unlabeled · line 1020
$m_{n} = ⌊ \frac{n + 1}{2} ⌋ .$
unlabeled · line 1024
$J_{n, k} ∣ 0 ⟩ = ∣ n, 2 k ⟩, J_{n, k} ∣ 1 ⟩ = ∣ n, 2 k + 1 ⟩ .$
unlabeled · line 1038
$∣ Ω ⟩_{E K} = \frac{1}{2} r = 0 \sum^{3} ∣ r ⟩_{E} ∣ r ⟩_{K},$
unlabeled · line 1047
$M_{⋆} - L ω \geq M_{cut} > 0.$