Approximate Decoherence, Recoherence and Records in Isolated Quantum Systems

TL;DR

The paper investigates how approximate decoherence manifests in isolated quantum systems, separating the problem into (i) counting histories that are nearly decoherent versus those that can leave detectable records, and (ii) analyzing the structure of decoherence for long histories via a random-matrix model. By combining analytic results for random histories (Haar and 2-designs) with a numerically exact long-history study, it shows that many more histories can be approximately decoherent than can be reliably distinguished by records, highlighting a branch-selection problem within the Many-Worlds framework. The long-history analysis reveals a clear decoherence structure where some histories recohere while others remain decoherent, and correlates recoherence with localization, high Petz-purity, small Hamming distance, and atypical Born-rule frequencies. The work connects decoherence, quantum state discrimination, and Born’s rule, providing quantitative bounds and numerical evidence for how classical-like records emerge and under what conditions the Born rule and theory confirmation can arise. Overall, the results illuminate fundamental limits on information extraction from histories, the emergence of classical records, and potential implications for the interpretation of quantum mechanics and the nature of the quantum-to-classical transition.

Abstract

Using the framework of decoherent histories, we study which past events leave detectable records in isolated quantum systems under the realistic assumption that decoherence is approximate and not perfect. In the first part we establish -- asymptotically for a large class of (pseudo-)random histories -- that the number of reliable records can be much smaller than the number of possible events, depending on the degree of decoherence. In the second part we reveal a clear decoherence structure for long histories based on a numerically exact solution of a random matrix model that, as we argue, captures generic aspects of decoherence. We observe recoherence between histories with a small Hamming distance, for localized histories admitting a high purity Petz recovery state, and for maverick histories that are statistical outliers with respect to Born's rule. From the perspective of the Many Worlds Interpretation, the first part -- which views the self-location problem as a coherent version of quantum state discrimination -- reveals a "branch selection problem", and the second part sheds light on the emergence of Born's rule and the theory confirmation problem.

Figures (25)

• Figure 1: We plot $\sqrt[N]{\det G}$ for $d\in\{20,400\}$ (filled, empty magenta circles) and $\sqrt[N]{\det W}$ for $d\in\{20,400\}$ (filled, empty blue diamonds) for a single realization of the Haar random or Gaussian vectors as a function of $\gamma=N/d$. The solid black line displays the asymptotic, analytical result of eqn (\ref{['eq analytical Andreas']}). Discrepancies between $G$, $W$ and the asymptotic result are almost invisible for $d=400$.
• Figure 2: Plot of the success probability for the sqrt measurement as a function of $\gamma$ for $d=20$ (green diamonds), $d=100$ (red squares), $d=500$ (orange circles) and $d=2500$ (blue crosses), numerically evaluated for a single realization of the Gram matrix $G_{jk} = {\langle {\psi_j|\psi_k}\rangle}$ and for weights $q_j=1/N$. The black solid line is the analytical result of eqn (\ref{['eq p success result']}). Note that in our setting the plot makes only sense for $\gamma\le1$, but we found it illuminating to show that eqn (\ref{['eq p success result']}) remains valid also for $\gamma>1$. Inset: Empirical standard deviation $\sigma$ of the success probability as a function of $d$, averaged over all values of $\gamma$ and obtained from ten realizations of the Gram matrix. We see the expected suppression $\sigma\sim d^{-1} \sim N^{-1}$.
• Figure 3: Plot of various central quantities as a function of the ratio $\gamma = N/d$. We evaluate numerically for $d=40$ (a) and $d=400$ (b) the average success probability of QSD (yellow circles) and SLP (green diamonds) for a single random realization of the Gram matrix $G$. The red squares show what happens to $\overline Q_S$ if one replaces $G$ by $W$. The blue crosses show the bound of eqn (\ref{['eq QSD SLP bound']}). Furthermore, we show the asymptotic analytical result for $\overline P_S$ [eqn (\ref{['eq p success result']}), black solid line] and the bound $\mu_\text{sqrt}(\gamma)\ge\overline P_S^\text{opt}$ (black dashed line).
• Figure 4: Plot of the normalized mutual information $I/\ln(N)$ for $d=60$ (in magenta) and $d=2000$ (in blue) as a function of $\gamma$. $I(G)$ is the exact mutual information for a single realization of the Gram matrix $G$ (crosses), the dashed line displays the mean field approximation [eqn (\ref{['eq MI mean field']})] and the solid line takes off-diagonal fluctuations into account [eqn (\ref{['eq MI fluct']})].
• Figure 5: Plot of the average success probability $\overline P_S$ as a function of the ratio $\gamma$ for three different classes of states (a), (b) and (c) as described in the main text. Plots are done for a single realization of the Gram matrix $G$ for different Hilbert space dimensions $d$. As for the Haar random case, we observed measure concentration for large $d$ (not shown here for brevity). The black solid line denotes the asymptotic analytical expression $\mu_\text{sqrt}(\gamma)^2$.

Theorems & Definitions (10)

• Lemma 1
• Lemma 2: Theorem 4.2 in Ref. LytovaPasturAP2009
• Lemma 3
• Lemma 4
• proof
• Remark 6.1
• Lemma 5
• proof
• Lemma 6
• proof