Here, we also allow backward time evolution, i.e., $|\alpha ⟩=\mathcal{U}(-t)|a⟩$. Throughout the text, we implicitly assume $t\ge 0$, except for a brief discussion in Sec. 3. Nonetheless, our results remain valid if the time order is reversed or extended to the limit $t\to -\infty$.

Let us use the eigenstate expansion for the both terms of the generalized fidelity in Eq. (3), and then time averaged these terms. Since it is assumed that there are no degeneracies in the spectrum and the time evolution operator is $\mathcal{U}(t)=\exp [i(\mathcal{H}/\hslash )t]$, the Riemann-Lesbegue lemma yields us that $\overline{|⟨a|\mathcal{U}(t)|b⟩{|}^2}={\sum }_n{p}_n^a{p}_n^b$ and $\overline{⟨a|\mathcal{U}(t)|a⟩⟨b|{\mathcal{U}}^{†}(t)|b⟩}={\sum }_n{p}_n^a{p}_n^b$. Therefore, we see $\overline{P_{\alpha \beta }(t)}={\sum }_n{p}_n^a{p}_n^b$ that is the same as the time-averaged probability ${\overline{P}}_{ab}$ defined in Eq. (4).

This determination of a priori matched quantum states is formally expressed as the infinite time average of a trajectory becoming, with infinitesimal coarse graining, a uniform phase distribution of $${\rho }_{\infty }(\mathbf{p},\mathbf{q})=\frac{\delta \mathbf{(}E-H(\mathbf{p},\mathbf{q})\mathbf{)}}{\mathrm{Tr}[\delta \mathbf{(}E-H(\mathbf{p},\mathbf{q})\mathbf{)}]},$$(40)where $H$ is the classical Hamiltonian and $E$ the energy of the particle exploring the phase space.

This is justified because we here focus on a time-independent Hamiltonian of a closed system.

In our numerical studies, we used significantly larger values of $T^{*}$ and verified the stability of the resulting estimates; nevertheless, a systematic determination of the optimal cutoff time remains an open problem.

It is possible to generalize Eq. (31), so as not to be biased toward coordinate space; this can be achieved as follows: $${\mathrm{\Gamma }}_{\gamma }(t)=\frac{1}{\int d\mathit{\gamma }}\int |⟨\gamma |\mathrm{\Psi }(t)⟩{|}^{4}d\mathit{\gamma },$$(41)where $|\gamma ⟩$ are members of an overcomplete set $\{\mathit{\gamma }\}$ of coherent states in phase space.

This argumentation can generalized to the phase-space distribution in terms of the generalized ${\mathrm{\Gamma }}_{\gamma }$: In the quantum evolution of a classically chaotic Hamiltonian system, the phase-space wave packet never again, even in the infinite future, spontaneously collects into another compact form with a small value of ${\mathrm{\Gamma }}_{\gamma }$, for if it did due to symmetry or focus points, that form would also leave its own pattern.

Even for integrable systems, this time can be tremendous, depending on the size of the Hilbert space, and it is even longer in chaotic ones. For both the soft- and hard-wall billiards considered here, this timescale lies far beyond our simulation window and thus does not affect our analysis of QBs.