Checking-time comparison under the Jm lift

PDF p. 4: Fig. 3 and code-comparison paragraph

Consider the following idealized comparison. Take a finite angular-momentum emitter with spin blocks $J=0,1,\dots ,5$, magnetic labels $m=-J,\dots ,J$, common instantaneous downward rate $\gamma >0$ for each state with $J\ge 1$, and rigid-rotor energy factors $J(J+1)$. Add the modeled energy lift $\mathrm{\Delta }Jm$, with $\mathrm{\Delta }\ge 0$. Use the downward-jump and other physical, spectral and choice hypotheses of the equal-rate model. Adopt the finite lifted-model postulate with $\mathrm{\Delta }(J,m)=\mathrm{\Delta }Jm$: its same three benchmark downward jump matrices and rate parameter are retained after adding the lift. This continuation is a separate physical assumption, not a deduction from the energy shifts. Compare the four specified two-dimensional encodings: the minimal code in $J=1$, detection code in $J=2$, Aydin–Barg code in $J=4$, and absorption–emission (Æ) code in $J=5$.

For each code and elapsed time, apply the modeled noise and its code-dependent transpose recovery. Its checking time is $\tau =\inf \{t\ge 0:F_{\mathrm{t}\mathrm{r}}(t)<0.9999\}$, with $\tau =+\mathrm{\infty }$ if the set is empty. Here $F_{\mathrm{t}\mathrm{r}}$ is the entanglement fidelity at the maximally mixed logical input; the transpose recovery is constructed from the channel and code at that time. A larger checking time means a later first threshold crossing in this definition, not by itself a repeated-cycle guarantee.

The article reports the following numerical results for this benchmark. At small lift, the checking times saturate at code-dependent values ordered ${\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{l}}^{\star }<{\tau }_{\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^{\star }<{\tau }_{\mathrm{A}\mathrm{y}\mathrm{d}\mathrm{i}\mathrm{n}\text{–}\mathrm{B}\mathrm{a}\mathrm{r}\mathrm{g}}^{\star }<{\tau }_{\text{Æ}}^{\star }$. For the compared codes retaining part of the encoded information after a decay, checking time decreases as distinguishability increases; the minimal code's checking time is independent of the lift. The Æ curve is reported highest up to approximately $\mathrm{\Delta }\mathrm{/}\gamma =50$; above that the two rank-one-correcting curves nearly coincide, with Aydin–Barg marginally ahead. Rank one here describes the angular-momentum transition family, not matrix rank. These statements are attributed numerical predictions, not independently established inequalities or measured outcomes.

Operational checking time

Equal-rate rigid-rotor benchmark parameters

Phenomenological angular-momentum degeneracy lift

Minimal single-manifold encoding

J equals two detection encoding

J equals five absorption-emission code

J equals four Aydin–Barg encoding

Lindbladian exponentiation and Choi Kraus extraction

General bounded diagonal lift for fast-checking analysis

Normalized jumps give a scalar total decay operator

The article reports direct exponentiation of the full Lindbladian and extraction of Kraus operators from the resulting Choi matrix. No solver version, precision, crossing search, certified error bounds or independent numerical replay is supplied here. In particular, a nearly coincident pair of curves cannot be ordered reliably from this item alone. Real species may have rates that depend on $J$, so the equal-rate benchmark does not by itself determine their code ordering.