Nature is stingy: Universality of Scrooge ensembles in quantum many-body systems

TL;DR

The paper addresses how quantum many-body systems exhibit universal randomness in state ensembles under physical constraints, extending beyond Haar randomness to Scrooge ensembles that maximize entropy subject to constraints. It develops a rigorous framework of Scrooge $k$-designs and proves three main theorems: global Scrooge from chaotic dynamics, emergent Scrooge $k$-designs from generators drawn from a Scrooge$2k$-design, and local Scrooge designs from scrambling the measurement basis. The results are complemented by numerical studies across commuting circuits, doped Clifford circuits, and ground-state Hamiltonians, identifying coherence, magic, and scrambling as essential resources for Scrooge behavior. The findings unify deep thermalization with Hilbert-space ergodicity under constrained randomness and offer practical designs for realizing Scrooge randomness in finite systems, with implications for benchmarking and quantum information tasks at finite temperature or under conservation laws.

Abstract

Recent advances in quantum simulators allow direct experimental access to the ensemble of pure states generated by measuring part of an isolated quantum many-body system. These projected ensembles encode fine-grained information beyond thermal expectation values and provide a new window into quantum thermalization. In chaotic dynamics, projected ensembles exhibit universal statistics, a phenomenon known as deep thermalization. While infinite-temperature systems generate Haar-random ensembles, realistic physical constraints such as finite temperature or conservation laws require a more general framework. It has been proposed that deep thermalization is governed in general by the emergence of Scrooge ensembles, maximally entropic distributions of pure states consistent with the underlying constraints. Here we provide rigorous arguments supporting this proposal. To characterize this universal behavior, we invoke Scrooge $k$-designs, which approximate Scrooge ensembles, and identify three physically distinct mechanisms for their emergence. First, global Scrooge designs can arise from long-time chaotic unitary dynamics alone, without the need for measurements. Second, if the global state is highly scrambled, a local Scrooge design is induced when the complementary subsystem is measured. Third, a local Scrooge ensemble arises from an arbitrary entangled state when the complementary system is measured in a highly scrambled basis. Numerical simulations across a range of many-body systems identify coherence, entanglement, non-stabilizerness, and information scrambling as essential resources for the emergence of Scrooge-like behavior. Taken together, our results establish a unified theoretical framework for the emergence of maximally entropic, information-stingy randomness in quantum many-body systems.

Figures (15)

• Figure 1: Scrooge $k$-designs in temporal and projected ensembles. (1) Temporal ensemble $\{e^{-iHt} {|{0}\rangle}\}_{t > 0}$ obtained by evolving an initial reference state ${|{0}\rangle}$ with a Hamiltonian $H$ satisfying the $k$th no-resonance condition, for late times $t$. (2) Projected ensemble generated by a state ${|{\Psi}\rangle}_{AB}$ drawn from a global Scrooge $2k$-design, with subsystem $B$ measured in an arbitrary fixed basis. (3) Projected ensemble generated by an arbitrary bipartite entangled state ${|{\Psi_0}\rangle}_{AB}$, with subsystem $B$ scrambled by a unitary $U_B$ drawn from a $2k$-design, prior to measurements in an arbitrary fixed basis. We prove rigorously that the temporal ensemble (1) forms a global Scrooge $k$-design [Theorem \ref{['thm:global_scrooge']}], the projected ensemble in (2) forms a probabilistic mixture of local Scrooge $k$-designs [Theorem \ref{['thm:2kgenerator']}], and the projected ensemble in (3) forms a local Scrooge $k$-design [Theorem \ref{['thm:ScroogeByMeasBasis']}]. Scrooge designs are depicted by the purple Bloch vectors. The Scrooge ensemble is uniquely defined by its density matrix $\sigma$, whose explicit form depends on the setting in consideration. As the von Neumann entropy of the density matrix $S(\sigma)$ increases, the Scrooge ensemble approaches the Haar ensemble, converging to it in the infinite-temperature limit where $\sigma$ is maximally mixed.
• Figure 2: Essential physical ingredients for emergent Scrooge designs. The emergence of Scrooge designs in projected ensembles requires the generator state ${|{\Phi}\rangle}_{AB}$ to exhibit magic (or nonstabilizerness), quantum information scrambling (via nonlocal entanglement), and coherence. When any of these ingredients is absent or insufficient, obstructions to Scrooge behavior can arise. Representative examples include stabilizer states, which lack magic; diagonal-scrambled stabilizer states (where a unitary diagonal in the computational basis is applied to subsystem $B$ of a stabilizer state), which only weakly scrambles quantum information between subsystems $A$ and $B$; and subset phase states, which may exhibit low coherence density depending on subset size.
• Figure 3: Emergent $2$-designs from commuting quantum circuit evolution.a) Circuit to generate projected ensemble from commuting quantum circuit dynamics, Eq. \ref{['eq:randphase']}. We apply a random diagonal unitary $U_{AB}^{\text{diag}}$ on ${|{+}\rangle}^{\otimes N}$, rotate $B$ by angle $\theta$ around $y$-axis with single-qubit unitaries $R_y(\theta) = \exp(-i\theta Y/2)$, and measure subsystem $B$ in the computational basis. b) Trace distance to Haar $2$-design $\Delta^{(2)}$ against $N_\text{A}$ for $\theta=0$. For finite $N_\text{B}$, the moment operator of the projected ensemble is computed exactly (or using $3\times10^6$ measurement samples for $N_\text{B}=22$), and $\Delta^{(2)}$ itself is averaged over up to $20$ random instances of the phase state. $N_\text{B}=\infty$ is computed from the ensemble of uniform random phase states over $N_\text{A}$ qubits, where the moment operator is averaged over up to $10^7$ random instances. c)$\Delta^{(2)}$ against rotation angle $\theta$ for different $N_\text{B}$. The inset shows rescaled $\theta N_\text{B}^{1/\nu}$, where $\nu=2$. Here, $N_\text{A}=2$, $N=N_\text{A}+N_\text{B}$, and $\Delta^{(2)}$ is averaged over $10$ random instances.
• Figure 4: Emergent Scrooge $2$-designs from doped Clifford circuit evolution.a) We consider projected ensembles from the generator state Eq. \ref{['eq:BellGen']}, where we measure in the computational basis on $B=B_1\cup B_2$ after applying there a quantum circuit composed of $d$ layers of local Clifford gates (random single-qubit Clifford gates together with fixed CNOT gates) doped with $N_\text{T}$ T-gates in total, randomly placed within the circuit. b) Heat map of the trace distance to Scrooge $2$-design $\Delta^{(2)}$ in the $d$ versus $N_\text{T}$ plane, for $\chi = \pi/6$. Here we fixed $N_A=1$, $N_B=N-N_\text{A}$, and $N=20$. c)$\Delta^{(2)}$ against $N_\text{T}$ for different total qubit numbers $N$ and fixed $d=30$. We choose $N_A=1$ and $\Delta^{(2)}$ is averaged over 500 random realizations of the circuit. Similar behavior is observed for higher moments $k > 2$, see Appendix \ref{['sec:Cliffordmagicdepth']}.
• Figure 5: Emergent Scrooge 2-designs from 1D integrable ground states. We study projected ensembles generated from the ground state of the transverse-field Ising model \ref{['eq:ising']}, where we measure $B$ in various bases which arise from rotations of the computational basis by various unitaries $U_\text{B}$ applied on $B$ only. a)$\Delta^{(2)}$ against field $h$ for $N_A = 1$ qubits, with $U_\text{B}$ chosen from the identity, single-qubit Haar random unitaries, random Clifford unitaries, or (global) unitaries drawn from the Haar measure on $N_\text{B}=19$ spins. b)$\Delta^{(2)}$ against $N_\text{B}$ for different $U_\text{B}$ and $h=1$. Dashed lines show the fit with $\Delta^{(2)}\sim 2^{-\alpha N_\text{B}}$, where we find $\alpha_\text{Clifford}\approx0.23$ and $\alpha_\text{Haar}\approx0.5$. c)$\Delta^{(2)}$ against $h$ for $U_\text{B}$ being random Clifford unitaries and different $N_\text{B}$. Inset shows same data rescaled by defining $y=\log_2(\Delta^{(2)})/N_\text{B}$ and subtracting the field $h_0$ with minimal $y_0$ around the critical field $h_\text{c}=1$. When rescaling the field with $N_\text{B}^{1/\nu}$ where $\nu=1$ from the Ising universality class, the data for different $N_\text{B}$ nearly collapse to a single curve, a hallmark of universality of the critical field $h_\text{c}=1$ (see Appendix \ref{['sec:ising']}). We fit the collapsed data with a third-order polynomial as dashed line. Similar behavior is observed for higher moments $k > 2$, see Appendix \ref{['sec:ising']}.

Theorems & Definitions (41)

• Definition 1: Scrooge ensemble josza1994lower
• Definition 2: Approximate Scrooge $k$-designs
• Lemma 1: Scrooge approximation
• Theorem 1: Scrooge designs from late-time chaotic dynamics
• Theorem 2: Emergent Scrooge $k$-design from Scrooge $2k$-design generator
• Corollary 1: Emergent $k$-design from $2k$-design generator
• Proposition 1: Emergent Scrooge $k$-design from late-time chaotic dynamics, informal
• Theorem 3: Emergent Scrooge $k$-design from $2k$-design measurement basis
• Definition 3: Approximate Scrooge $k$-designs
• Lemma 2