$$D_{\rm irr}(t_c)= \inf_{u\in\mathcal A(t_c)} \frac12\left\|\rho^{u}_{0,H}-\rho^{u}_{1,H}\right\|_1$$

Equation 42 · Evolutionary Physics

How Fast Can a Horizon Learn Which Path You Took?

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Define $\mathcal A(t_c)$ as a preregistered set of physically admissible future protocols after the cutoff. It must specify which controls Alice retains, their duration, energy, acceleration, localization, and boundary conditions. For each u$\in$$\mathcal A(t_c)$ , let $\rho^{u}_{0,H}$ and $\rho^{u}_{1,H}$ be the completed conditional states on the declared horizon algebra. Then define an irreducible distinguishability envelope $D_{\rm irr}(t_c)= \inf_{u\in\mathcal A(t_c)} \frac12\left\|\rho^{u}_{0,H}-\rho^{u}_{1,H}\right\|_1$. The infimum asks how little path distinguishability remains after the best allowed attempt to erase it. For equal priors, an associated unavoidable decision information is

Meanings in this article

• $D_{\rm irr}$: monotone.
• $t_c$: the become irrevocable at an intermediate cutoff.

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