C061 derivation

Claim

Spontaneous emission from $|e\rangle$ with optical-qubit lifetime $T_1$ limits an MS gate of length $t$ to a Bell-state error $\approx 0.5\, t/T_1$, and the amplified error gives a lower bound on the protocol steady-state infidelity of $\approx 3\, t/T_1$.

Part 1 — MS-gate spontaneous-emission error $\approx 0.5\, t/T_1$

During a maximally-entangling MS gate the spins evolve $|\!\downarrow\downarrow\rangle \to (|\!\downarrow\downarrow\rangle - i|ee\rangle)/\sqrt2$. The expected number of decay events over the gate is $$ p_{\rm dec} = \frac{1}{T_1}\int_0^t \langle \hat n_e(s)\rangle\, ds = \frac{\langle n_e\rangle_{\rm avg}\, t}{T_1}, $$ where $\hat n_e$ is the number of ions in $|e\rangle$ and $\langle n_e\rangle_{\rm avg}$ is its time-average over the gate.

Simple physical estimate (paper's wording, "population in $e$ is about half the gate time"): the final state has total excited population $\langle \hat n_e\rangle = 1$ (half of two ions in $|ee\rangle$), and the population rises monotonically from $0$ to $1$ over the gate, so the time-average is roughly $\tfrac12 \times 1 = 0.5$ ions. Hence $p_{\rm dec}\approx 0.5\, t/T_1$, and a single decay from a Bell component collapses the entanglement, giving Bell-state error $\approx 0.5\, t/T_1$. This reproduces the paper's coefficient.

Numerical sanity check (area swept linearly in time, sympy_check.py): $\langle n_e\rangle_{\rm avg}=0.36$, the same order of magnitude as $0.5$. The exact coefficient depends on the assumed time-profile of the excited population (linear-in-area gives $0.36$; a profile weighted toward the end gives $0.5$); the $0.5$ value is an order-of-magnitude estimate, which is how the paper states it ("$\approx$").

Part 2 — amplification to $\approx 3\, t/T_1$

A spontaneous-emission event during the gate acts like a (correlated) bit-flip in the protocol subspace. The protocol amplifies a per-cycle correlated bit-flip of probability $p$ into a steady-state singlet error. Using the self-contained model (optimal $\gamma=0.221\pi$, $\theta=0.716\pi$, correlated bit-flip $U_x(\sqrt{2p})$ inserted after drive A), the steady-state singlet error is $$ \varepsilon_{\rm ss} \approx 3.18\, p \quad (\text{small } p), $$ matching the main-text correlated-bit-flip amplification factor ($16$-cycle error $=3.2p$). With the per-gate (per-cycle) spontaneous-emission bit-flip probability $p \approx 0.5\, t/T_1$: $$ \varepsilon_{\rm ss} \approx 3.18 \times 0.5\, t/T_1 \approx 1.6\, t/T_1. $$ The paper quotes $\approx 3\, t/T_1$. The amplification factor $3.18$ is reproduced exactly, but recovering the full $3\, t/T_1$ requires the per-cycle injected error to be $\approx 1\, t/T_1$ (i.e. counting the full integrated excited-population $\langle n_e\rangle\to 1$ over the gate, not the $0.5$ time-average) rather than $0.5\, t/T_1$. In other words, $0.5\,t/T_1$ (gate Bell error) and $3\,t/T_1$ (steady-state lower bound) are consistent in form ($\varepsilon_{\rm ss}\sim 3.2\times(\text{per-cycle error})$) but the two coefficients are not simultaneously pinned by a single clean derivation: the gate error uses the $0.5$ time-average while the steady-state bound effectively uses the full $\sim 1\times t/T_1$ per-cycle decay probability with the same $\sim 3.2$ amplification. Both are order-of-magnitude ("$\approx$") statements.

Conclusion

• The $\approx 0.5\, t/T_1$ MS-gate error is reproduced as a correct order-of-magnitude estimate (half the gate time in $|e\rangle$); the linear-area numeric gives $0.36$, same order.
• The amplification factor $\approx 3.2$ is reproduced exactly from the self-contained model, matching the main text.
• The $\approx 3\, t/T_1$ steady-state lower bound follows from (amplification $\approx 3.2$) $\times$ (per-cycle decay $\approx t/T_1$), i.e. it is consistent with the model, but the factor-of-3 vs the gate's $0.5$ requires the amplification model plus an order-unity choice of the per-cycle decay probability. Both coefficients are stated as approximate.

Verdict: partial — the structure and the amplification factor are confirmed, and both numbers are reproduced to within order-of-magnitude; the exact factor of 3 needs the amplification model and is not a clean closed-form identity.