Quantum microstate counting from Brownian motion: from many-body systems to black holes

TL;DR

The paper develops a universal framework to count the dimension of Hilbert spaces spanned by families of Brownian-driven quantum states, applicable to finite- and infinite-temperature ensembles in many-body systems and to black holes. By analyzing overlap moments Z_n(t) through Gram matrices and replica techniques, it derives exact finite-N results for Brownian GUE, spin clusters, and SYK models, and shows that the late-time dimension saturates at the square of the relevant density-matrix rank, with a gap/scrambling-time scaling. In gravity, the authors construct replica wormholes (caterpillars) to compute the same moments microscopically and demonstrate that black hole entropy is universally reproduced via microcanonical projections, with extensions to higher dimensions and universal structures. The approach addresses several caveats of previous gravitational state-counting programs and provides a robust, broadly applicable method for understanding microstate counting in quantum gravity and chaotic many-body systems. The results point to a deep, universal connection between replica-wormhole dynamics, density-matrix rank, and Hilbert-space dimensionality across disciplines.

Abstract

We introduce a new way to produce infinite families of bases of a quantum system's Hilbert space, as well as methods to find its dimension. These families are constructed via Brownian motions in the Hilbert space, defined using disordered, time-dependent couplings. The dimension spanned by them has zero variance over the ensemble of disordered couplings, and it is determined by certain replica partition functions. We apply these methods to finite dimensional $q$-local systems (spin clusters and SYK) and black holes. For $q$-local systems we find exact expressions at finite $q$, $N$, and $t$, for replica partition functions, together with universal behavior at large times and a semiclassical analysis at large-$N$ in appropriate master field variables. The right Hilbert space dimension is obtained for any time $t>0$, $q>1$ and $N>0$. For black holes, the Brownian motions prepare quantitatively generic ensembles of black hole microstates, and the Bekenstein-Hawking entropy is universally reproduced by counting those. There the $n$-replica partition functions are constructed by gluing $n$-traversable womrholes at their future and past. This method demonstrates the robustness of black hole microstate counting methods, avoiding several limitations of previous constructions, including the non-genericity of the microstates and associated interiors, implicit statistics of heavy operators, limited microscopic control over overlaps, and the need for specific limits in the calculation.

Figures (24)

• Figure 1: We will count families of caterpillars, corresponding to states of two black holes prepared for a fixed Euclidean time $t$, with semiclassical ER bridge of length of order $t$ supported by large numbers of erratic matter inhomogeneities. Using the gravitational path integral, we will show that, for any time (or wormhole length) larger than the scrambling time $t\gg t_\ast$, families of caterpillars span a finite-dimensional Hilbert space of dimension $e^{2S_{\rm BH}}$.
• Figure 2: On the left, the quantity $|\mathrm{Tr}(U_1(t)^\dagger U_2(t))|^2$ is proportional to the square of the overlap between the two states. On the right, averaging over the Brownian couplings generates local-in-time effective interactions between the two replicas, resulting in Euclidean evolution governed by a time-independent Hamiltonian $H_{\mathrm{eff}}^{1\bar{1}}$. The average overlap is the thermal partition function of this effective Hamiltonian at inverse temperature $2t$.
• Figure 3: On the left, the quantity $\mathrm{Tr}(U_1(t)^\dagger U_2(t))\,\mathrm{Tr}(U_2(t)^\dagger U_3(t))\,\mathrm{Tr}(U_3(t)^\dagger U_1(t))$ is represented so that the time-ordering is aligned across replicas. On the right, averaging over the Brownian couplings generates local-in-time effective interactions between each pair of replicas, resulting in Euclidean evolution governed by time-independent Hamiltonians $H_{\mathrm{eff}}^{a\bar{a}}$ for $a = 1,2,3$. Following the replica structure of the original quantity, the resulting object $Z_3(t)$ corresponds to the survival amplitude under Euclidean evolution by time $t$ in the cyclic state $\ket{\eta}$ in the six-replica Hilbert space.
• Figure 4: A random quantum circuit preparing a maximally entangled state $\ket{U(t)}$.
• Figure 5: On the left, the "wormhole" saddle point dominates $Z_2(t)$ at $O(1)$ times and reproduces the contribution from the tower of quasi-particle excitation with mass $E_{\text{gap}} = 2 q \mathcal{J}$. On the right, the corresponding replica wormhole providing the analogous contribution to $Z_4(t)$.

Theorems & Definitions (12)

• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• proof : Proof of Lemma \ref{['lemma:densitymatrix']}
• proof : Proof of Lemma \ref{['lemma:bspin1']}
• proof : Proof of Lemma \ref{['lemma:bspin2']}
• proof : Proof of Lemma \ref{['lemma:bSYK1']}
• proof : Proof of Lemma \ref{['lemma:syk2']}