Manuscript Hawk02-B@2 · Hawk02-B.tex · section · 95ea744f

Many-cell recovery and extensive energy release

Many-cell recovery and extensive energy release · lines 700–756

Hawk02-B@2#thm:B5, Hawk02-B@2#eq:manyexact, Hawk02-B@2#eq:manycommon, Hawk02-B@2#eq:manyleak

Take $K$ memory cells with distinct radiation cells $R_{ℓ, j}$ . The code is

H_{C^{(K)}} = (C^{d})^{\otimes K} .

Allow arbitrary entanglement between cells and with a reference.

Ideal interactions act on distinct cell mechanisms. Their complete map factorizes, up to output ordering:

W^{(K)} = ℓ ⨂ W_{ℓ}, N^{(K)} = ℓ ⨂ N_{ℓ}, C^{(K)} = ℓ ⨂ C_{ℓ} .

This factorization of maps does not assume a product input. Write $q_{ℓ}$ for the individual survival probabilities.

The product of the declared single-cell decoders is optimal:

ε_{rec} (N^{(K)}) = 1 - ℓ = 1 \prod^{K} (1 - c_{d} q_{ℓ}) .

For identical mechanisms,

ε_{rec} (N_{N}^{\otimes K}) c_{d} q_{N} = 1 - (1 - c_{d} q_{N})^{K}, \leq ε_{rec} (N_{N}^{\otimes K}) \leq 1 - (1 - q_{N})^{K} \leq K q_{N} .

The complete residual has the sufficient bound

δ_{leak} (C^{(K)}) \leq 1 - ℓ \prod (1 - c_{d} q_{ℓ}) .

No exact joint leakage optimization is required or asserted.

For identical mechanisms, the initial bare energy is $K ω$ , the probability of full discharge is $(1 - q_{N})^{K}$ , and

E E_{rad} = K ω (1 - q_{N}), E E_{memory} = K ω q_{N} .

The emitted-excitation count is binomial with parameters $K$ and $1 - q_{N}$ , for every input code state.

Displayed formulas in this passage

Hawk02-B@2#eq:manyexact · line 722
$ε_{rec} (N^{(K)}) = 1 - ℓ = 1 \prod^{K} (1 - c_{d} q_{ℓ}) .$
Hawk02-B@2#eq:manycommon · line 730
$ε_{rec} (N_{N}^{\otimes K}) c_{d} q_{N} = 1 - (1 - c_{d} q_{N})^{K}, \leq ε_{rec} (N_{N}^{\otimes K}) \leq 1 - (1 - q_{N})^{K} \leq K q_{N} .$
Hawk02-B@2#eq:manyleak · line 740
$δ_{leak} (C^{(K)}) \leq 1 - ℓ \prod (1 - c_{d} q_{ℓ}) .$
unlabeled · line 704
$H_{C^{(K)}} = (C^{d})^{\otimes K} .$
unlabeled · line 711
$W^{(K)} = ℓ ⨂ W_{ℓ}, N^{(K)} = ℓ ⨂ N_{ℓ}, C^{(K)} = ℓ ⨂ C_{ℓ} .$
unlabeled · line 749
$E E_{rad} = K ω (1 - q_{N}), E E_{memory} = K ω q_{N} .$