$$\epsilon_L(t) \;=\; \min_{\mathcal R}\Big[\,1 - F_e\big(\mathcal R \circ \mathcal N_t,\; \mathrm{id}\big)\,\Big], \qquad t_L(\epsilon) \;=\; \min\{\,t : \epsilon_L(t) \le \epsilon\,\}$$

Equation 23 · Evolutionary Physics

The Bit Comes Back Before the Bearing

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

For the qubit, returned means: there exists a recovery channel $\mathcal R$ , built from quantum operations an agent holding both the full early radiation R and a further batch of radiation $C_t$ collected up to time t can actually perform, whose entanglement fidelity to the identity channel on A is within $\epsilon$ of perfect, $\epsilon_L(t) \;=\; \min_{\mathcal R}\Big[\,1 - F_e\big(\mathcal R \circ \mathcal N_t,\; \mathrm{id}\big)\,\Big], \qquad t_L(\epsilon) \;=\; \min\{\,t : \epsilon_L(t) \le \epsilon\,\}$. $\mathcal N_t$ is the effective channel from A to R\!$\cup$\!$C_t$ induced by the hole’s own scrambling dynamics up to time t ; $\epsilon_L(t)$ is a dimensionless number in [0,1] , and $t_L$ has units of time. This is a yes-or-no-shaped promise sharpened into a number: either a decoder within $\epsilon$ of perfect exists on the stated resources, or it does not,…

Meanings in this article

• $\epsilon_L$: a dimensionless number in [0,1] , and $t_L$ has units of time.
• $N_t$: the effective channel from A to R\!

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