Scrooge Ensembles in Quantum Information

• Scrooge ensembles are canonical pure state distributions defined by a fixed average density matrix and minimal accessible information, equating to subentropy.
• They are constructed by distorting the Haar measure via a ρ-transformation, with moment calculations revealing insights into random state statistics and complexity growth.
• Their applications range from benchmarking quantum devices to probing randomness and entanglement in quantum statistical mechanics and many-body systems.

The Scrooge ensemble or Scrooge distribution is a canonical ensemble of pure quantum states, defined by extremal information-theoretic properties. For a given density matrix $\rho$ on a finite-dimensional Hilbert space, the Scrooge ensemble is the unique (continuous) ensemble of pure states having average state $\rho$ and minimal accessible information—i.e., it is the maximally entropic ("stingy") distribution consistent with that constraint. The structure of Scrooge ensembles underlies key phenomena in quantum statistical mechanics, quantum information, and the theory of random quantum states, notably in settings where physical or operational constraints render the Haar distribution inappropriate. Scrooge ensembles generalize the uniform Haar measure, and their deep connections to subentropy, complexity, and physical randomness have become central in recent work on quantum many-body systems (Wootters, 2018, McGinley et al., 21 Nov 2025, Mok et al., 1 Jan 2026).

1. Definition and Mathematical Construction

Given an $n$-dimensional density operator $$\rho = \sum_{j=1}^n \lambda_j |e_j\rangle\langle e_j|$$, the Scrooge ensemble is the unique continuous probability distribution $\sigma(x)$ over pure states $|x\rangle$ such that:

• The average state is $\rho$:

$\int \sigma(x) |x\rangle\langle x|\, dx = \rho$

• The accessible information is minimized among all pure-state decompositions of $\rho$.

The explicit construction is by "rho-distorting" the Haar-uniform ensemble:

1. Draw $|y\rangle$ Haar-uniform.
2. Form $|\tilde\phi\rangle = \sqrt{n\rho}\,|y\rangle$, normalize to $|x\rangle = |\tilde\phi\rangle/\|\tilde\phi\|$.
3. The resulting weight is

$$\sigma(x)\,dx = n\,\tau(y)\,\langle y|\rho|y\rangle\,|\det(\partial y/\partial x)|\,dy$$

evaluated at $y = f^{-1}(x)$ (Wootters, 2018).

Alternatively, in terms of expectation over Haar-random $|\phi\rangle$:

$$d\mu_{\mathrm{Scrooge}}(\psi) = p_\rho(\phi)\,d\mu_{\rm Haar}(\phi)\,\delta\bigl(\psi-\psi_\phi\bigr),$$

where $p_\rho(\phi)=\langle\phi|\rho|\phi\rangle$ and $$|\psi_\phi\rangle=\frac{\sqrt{\rho}|\phi\rangle}{\sqrt{p_\rho(\phi)}}$$ (McGinley et al., 21 Nov 2025, Mok et al., 1 Jan 2026).

For the complex-amplitude case, the distribution over squared amplitudes $x_j = |\langle e_j|\psi\rangle|^2$ is:

$$\sigma(x) = \frac{n!}{\lambda_1\cdots\lambda_n} \frac{1}{\bigl(\sum_j x_j/\lambda_j\bigr)^{n+1}}$$

normalized over the probability simplex (Wootters, 2018).

2. Extremal Information-Theoretic Properties

The Scrooge ensemble is defined by its minimal accessible information among all pure-state ensembles with average $\rho$:

• Accessible information is the supremum, over quantum measurements, of the Shannon mutual information between the pure state label and the measurement outcome.
• For the Scrooge ensemble, mutual information is independent of the measurement basis and equals the subentropy $Q(\rho)$:

$$Q(\rho) = -\sum_{k=1}^n\left[\prod_{l\neq k}\frac{\lambda_k}{\lambda_k-\lambda_l}\right]\lambda_k\ln\lambda_k$$

• For any pure-state $\rho$-ensemble, the accessible information cannot be less than $Q(\rho)$. The Scrooge ensemble achieves this lower bound, establishing it as the global minimizer (Wootters, 2018).

This stingy property motivates the terminology: Scrooge ensembles are maximally entropic, “giving up the least information” about the underlying pure state upon measurement.

3. Classical and Operational Interpretations

In the real-amplitude Hilbert space, the Scrooge density $\sigma_r(x)$ admits a natural classical model:

• Consider $n$-sided dice, with face $j$ appearing $N_j$ times, Poisson-distributed with mean $M_j$. Impose upper bounds $\bar M_j$ and maximize signal packings under Fisher-information distinguishability.
• The induced distribution over $x_j = M_j/\sum_k M_k$ is, up to normalization:

$$\sigma_r(x) = \frac{n\,\Gamma(n/2)}{\pi^{n/2}\sqrt{\lambda_1\cdots\lambda_n}\sqrt{x_1\cdots x_n}} \frac{1}{\bigl(\sum_j x_j/\lambda_j\bigr)^{n/2+1}}$$

with $\lambda_j = \bar M_j/\sum_k \bar M_k$ (Wootters, 2018).

To obtain the complex-amplitude Scrooge density, one "doubles" the faces and imposes constraints on summed pairs, then integrates out the internal structure—a step with no direct classical analog. This demonstrates a bridge (albeit partial) between classical Fisher geometry and the quantum Scrooge measure.

4. Moment Calculus and Statistical Properties

The $k$-th moment of the Scrooge ensemble is:

$$\chi^{(k)}_{\mathcal S_\rho} = \mathbb{E}_{\psi\sim\mathcal S_\rho}(\ket\psi\bra\psi)^{\otimes k}$$

and can be computed as:

$$\int d\mu_{\rm Haar}(\phi)\,p_\rho(\phi)\,(\ket{\psi_\phi}\bra{\psi_\phi})^{\otimes k}$$

(McGinley et al., 21 Nov 2025). For $k \leq 2^{S_\infty(\rho)/3}$, with $S_\infty(\rho) = -\log \|\rho\|_\infty$, the $k$-th moment is approximately:

$$\chi^{(k)}_{\mathcal S_\rho} \approx \rho^{\otimes k}\,\sum_{\pi\in S_k}\pi$$

up to small corrections $$\delta_{\rho,k} = \mathcal O(k^2\,2^{-S_\infty(\rho)/2})$$. This yields exponential concentration of reduced (local) moments and output probabilities, manifesting as rescaled Porter-Thomas and Wishart statistics.

Table: Key Statistical Properties

Property Type	Characteristic Law	Scaling/Error
Output probability $p^\psi(x)$	$k!\langle x|\rho|x \rangle^k$ (moments); rescaled Porter-Thomas distribution	$\mathcal O(k^2 2^{-S_\infty/2})$
Joint $p^\psi(x),p^\psi(x')$	Bivariate gamma (Kibble law), depends on coherences $\langle x|\rho|x'\rangle$	-
Local reduced density (on $A$)	Approaches $\operatorname{Tr}_B \rho^{\otimes k}$ exponentially in $|B|$	$\mathcal O(k^2 2^{-\hat S_\infty(B|A)})$
CMI over partitions $A:B:C$	Order-$1$ for all non-overlapping $A,C$; equals $Q(\rho^A)+Q(\rho^C)-Q(\rho^{AC})$ for product $\rho$ (McGinley et al., 21 Nov 2025)	$\approx 0.61$ bits (large entropy)

These properties reflect the universality and concentration phenomena of Scrooge ensembles in large systems.

5. Mechanisms and Emergence in Many-Body Quantum Systems

Three principal mechanisms produce (approximate) Scrooge $k$-designs in quantum many-body systems (Mok et al., 1 Jan 2026):

1. Global Scrooge from Chaotic Dynamics: For a closed system under generic Hamiltonian evolution, the long-time random-phase ensemble yields a Scrooge ensemble with first moment given by the diagonal population $\sigma_{\mathrm{diag}}$. Additive errors in the $k$-th moment are controlled as $$\mathcal{O}[(D\|\sigma_{\mathrm{diag}}\|_\infty)^k k^2/D + k \operatorname{Tr}\sigma_{\mathrm{diag}}^2]$$.
2. Local Scrooge via Measurement on a Global Scrooge State: Given $|\Psi\rangle_{AB}$ from a Scrooge $2k$-design, measurement of $B$ in a fixed basis projects $A$ onto a mixture of local Scrooge ensembles with rigorous error bounds.
3. Local Scrooge from Scrambled Measurements: For arbitrary entangled $|\Psi\rangle_{AB}$, applying a Haar-random (or 2$k$-design) unitary to $B$ before measurement ensures the projected state on $A$ is an approximate Scrooge $(\sigma_A)$ $k$-design, with additive errors governed by $(D_A \operatorname{Tr}\sigma_A^2)^k + 1/\sqrt{D_B}$.

Numerical simulations confirm that coherence, entanglement entropy, non-stabilizerness (i.e., “magic”), and scrambling power are all essential for Scrooge ensemble formation. Removing any of these resources precludes the emergence of deep Scrooge statistics (Mok et al., 1 Jan 2026).

6. Complexity, Randomness, and Physical Implications

Scrooge ensembles are far from algorithmically trivial: any $2^m$-state ensemble can only match Scrooge moments up to $k$ and error $\epsilon$ if

$$m \geq k\left(S_\infty(\rho) - \log k\right) - \log(1-\epsilon-2\delta_{\rho,k})$$

For full-rank $\rho$ of entropy density $\Theta(n)$, $m = \Omega(kn)$ bits are required (McGinley et al., 21 Nov 2025). Conventional temporal evolution under Hamiltonians fails to generate high $k$-designs quickly: exponential time in $k$ and $n$ is required. Furthermore, any circuit of fewer than $\Omega(k S_\infty(\rho)/\log(k S_\infty(\rho)))$ two-qubit gates cannot approximate a Scrooge $k$-design to constant error.

Scrooge ensembles provide nontrivial benchmarks and diagnostics in physical contexts:

• Device Benchmarking & Cross-Entropy Tests: Porter-Thomas and Wishart statistics directly reveal coherent errors; even small amounts of local noise collapse the Scrooge distribution to the classical (diagonal) law, mirroring Haar-random circuits (McGinley et al., 21 Nov 2025).
• Quantum Complexity Growth: The approach to a Scrooge $k$-design in circuit evolution (including symmetry-constrained random circuits) correlates to circuit complexity growth rates.
• Sampling and Universality: The persistence of significant classical conditional mutual information (CMI) and output probability fluctuations implies classical patching/tensor-network sampling fails for generic Scrooge ensembles. Trivial sampling arises only with added noise.
• Physical Origin in Many-Body Systems: Microcanonical time evolution, projected or measured subsystems in scrambled states, and symmetric random circuits all yield (approximate) Scrooge designs. This places universality in projected ensemble statistics ("deep thermalization") on firm theoretical footing (Mok et al., 1 Jan 2026).

7. Applications and Open Problems

Scrooge ensembles unify the maximum-entropy approach to quantum state statistics with operational constraints from quantum thermodynamics, information scrambling, and circuit complexity theory. They serve as accurate models for universal properties in realistic finite-temperature and symmetry-constrained quantum many-body systems.

Open directions include:

• Rigorous proofs of short-depth circuit emergence of Scrooge designs for non-Gibbsian $\rho$.
• Extensions to classical shadow tomography and learning scenarios.
• Quantification of necessary physical resources for deep thermalization in experimental quantum simulators.

The study of Scrooge ensembles continues to facilitate a deeper theoretical understanding of randomness, information, and complexity in quantum statistical physics and quantum information theory (Wootters, 2018, McGinley et al., 21 Nov 2025, Mok et al., 1 Jan 2026).