Manuscript Em03-E@1 · Em03-E-v1.tex · section · c4b07e6a

Microscopic source and encoding

E0: message, source, environments, and emitted cells › Microscopic source and encoding · lines 251–306

Em03-E@1#eq:encoding

The message $M$ is a qubit. Its external reference $R$ is not controlled by the source or receiver.

The source contains:

B = Q F_{1} F_{2} K_{a} K_{b} A_{1} A_{2} .

Here $Q$ is the source message qubit, $F_{1}, F_{2}$ are fuel qubits, $K_{a}, K_{b}$ are key qubits, and $A_{1}, A_{2}$ are finite occupation-source registers.

Choose integers $J_{i} \geq 1$ , positive energy spacings $ω_{i}^{cell} > 0$ , and source energies

E_{i} \geq J_{i} ω_{i}^{cell} .

The register $A_{i}$ has basis

∣ e ⟩_{i}, ∣ d_{0} ⟩_{i}, \dots, ∣ d_{J_{i}} ⟩_{i} .

Its Hamiltonian is

H_{A_{i}} = E_{i} ∣ e ⟩ ⟨ e ∣ + n = 0 \sum^{J_{i}} (E_{i} - n ω_{i}^{cell}) ∣ d_{n} ⟩ ⟨ d_{n} ∣ .

The initial and the $n = 0$ daughter states are distinct degenerate states.

For a payload energy $ε_{cell} > 0$ , set

H_{B} = ε_{cell} (∣ 1 ⟩ ⟨ 1 ∣_{F_{1}} + ∣ 1 ⟩ ⟨ 1 ∣_{F_{2}}) + H_{A_{1}} + H_{A_{2}} .

The source factors $Q, K_{a}, K_{b}$ are energetically degenerate.

The encoding is

V_{enc} ∣ x ⟩_{M} = ∣ x ⟩_{Q} ∣ 11 ⟩_{F_{1} F_{2}} ∣ 00 ⟩_{K_{a} K_{b}} ∣ e, e ⟩_{A_{1} A_{2}}, x = 0, 1.

Its image lies in the common energy eigenspace

E_{init} = 2 ε_{cell} + E_{1} + E_{2} .

Thus different messages do not begin with different source energy.

Displayed formulas in this passage

Em03-E@1#eq:encoding · line 294
$V_{enc} ∣ x ⟩_{M} = ∣ x ⟩_{Q} ∣ 11 ⟩_{F_{1} F_{2}} ∣ 00 ⟩_{K_{a} K_{b}} ∣ e, e ⟩_{A_{1} A_{2}}, x = 0, 1.$
unlabeled · line 257
$B = Q F_{1} F_{2} K_{a} K_{b} A_{1} A_{2} .$
unlabeled · line 266
$E_{i} \geq J_{i} ω_{i}^{cell} .$
unlabeled · line 270
$∣ e ⟩_{i}, ∣ d_{0} ⟩_{i}, \dots, ∣ d_{J_{i}} ⟩_{i} .$
unlabeled · line 275
$H_{A_{i}} = E_{i} ∣ e ⟩ ⟨ e ∣ + n = 0 \sum^{J_{i}} (E_{i} - n ω_{i}^{cell}) ∣ d_{n} ⟩ ⟨ d_{n} ∣ .$
unlabeled · line 286
$H_{B} = ε_{cell} (∣ 1 ⟩ ⟨ 1 ∣_{F_{1}} + ∣ 1 ⟩ ⟨ 1 ∣_{F_{2}}) + H_{A_{1}} + H_{A_{2}} .$
unlabeled · line 302
$E_{init} = 2 ε_{cell} + E_{1} + E_{2} .$