Order Statistics in Haar-Random Quantum States

• The paper presents an exact analysis using Laplace methods to derive cumulative and differential record statistics for Haar-random states.
• It demonstrates that record occurrence follows a universal Bernoulli process with a 1/j success rate, leading to harmonic scaling of record counts.
• The study reveals that finite-size effects and dynamical correlations induce deviations from the Gumbel law, offering diagnostic tools for quantum chaos.

Order statistics of Haar-random quantum states concern the extremal and ranked values of observables derived from quantum pure states sampled uniformly (via the Haar measure) in high-dimensional Hilbert spaces. These statistics encapsulate the universal behavior of maxima, record-setting intensities, and their distributions in the computational basis, revealing the interplay between global constraints (normalization, invariance) and local properties (ordering, correlations). They are directly relevant for quantum chaos, benchmarking, and the statistical theory of quantum measurements.

1. Exact Record Statistics in Haar-Random States

For a complex normalized pure state $|\psi\rangle$ expanded in an orthonormal basis as $z_n = \langle n|\psi\rangle$, the intensities $x_n = |z_n|^2$ are constrained by normalization $\sum_n x_n = 1$. The joint probability density function (jpdf) of the $x_n$ for Haar-random states is

$$P(x_1, ..., x_N; u) = \Gamma(N) \delta\left(\sum_{n=1}^N x_n - u\right)$$

with $u=1$. The cumulative probability that the record up to time $t$ is less than $R$,

$$Q(R, t) = \int_0^R dx_1 \ldots dx_t\, P_t(x_1, ..., x_t)$$

admits an exact solution via Laplace methods:

$$Q(R, t) = \sum_{m=0}^t (-1)^m {t \choose m} (1 - mR)^{N-1} \Theta(1 - mR)$$

Differentiation yields the record probability density

$$P(R, t) = \sum_{m=1}^t (-1)^{m+1} {t \choose m} m (N-1)(1-mR)^{N-2} \Theta(1-mR)$$

These results precisely characterize the order statistics of intensities (such as maxima) in Haar-random vectors.

2. Universal Record Occurrence: Bernoulli Process Nature

Despite the $\delta$-correlations among $x_n$ induced by normalization, the probability that element $j$ is a new record is exactly $1/j$, replicating the result for i.i.d. random variables:

$p(j) = \frac{1}{j}$

This establishes that the sequence of record-setting events behaves as a Bernoulli process with success probability $1/j$ at each step, regardless of the underlying intensity distribution. The universality of this process underpins statistical properties of orderings—such as the mean number of records, which equals the $N$th harmonic number $H_N \sim \log N + \gamma$, and is independent of correlations in the underlying distributions at finite $N$.

3. Correlation-Induced Non-Universality and Extreme Value Distributions

Normalization imposes $\delta$-correlations on the $x_n$, affecting the detailed statistics of the record (maximal) values themselves. For large $N$ and $t$,

$$Q(R, t) \approx (1 - e^{-N R})^t \approx \exp\left(-t e^{-N R}\right)$$

which is the cumulative distribution function of a Gumbel law for the largest intensity.

However, at finite $N$, the distribution of record values departs from the Gumbel form—non-universal corrections persist due to the normalization constraint, only vanishing as $N \to \infty$. The scaling parameters for the Gumbel regime are set by $t$ and $N$. Thus, while the ordering statistics (record positions) are universal, the record intensities remain non-universal at finite sizes.

4. Dynamical Correlations and the Quantum Standard Map

To examine the impact of correlations beyond normalization, the quantum standard map—whose Floquet operator is

$$U_{nn'} = \frac{1}{N} \sum_{m=0}^{N-1} \exp\left[ -i\pi\frac{(m+\beta)^2}{N} + 2\pi i \frac{(m+\beta)(n-n')}{N} \right] \exp\left[i \frac{K N}{2\pi} \cos\left(\frac{2\pi (n + \alpha)}{N}\right) \right]$$

—is studied as a testbed for quantum chaotic and mixed regimes. In the chaotic regime ($K\gg5$), the eigenstates are distributed like Haar-random vectors and reproduce the universal order statistics. In the mixed (non-ergodic) regime (intermediate $K$), additional correlations emerge from phase-space localization and non-chaotic components.

5. Scaling Laws, Phase Space, and Quantum Chaos Diagnostics

The mean number of records, $\langle N_R \rangle$, and the scaling of records as a function of Hilbert space dimension $N$ serve as sensitive probes of emergent correlations:

• In the chaotic regime: $\langle N_R \rangle \sim \log N$ (the harmonic number, as for i.i.d. Haar-random intensities).
• In the mixed/critical regime: $\langle N_R \rangle \sim N^{\delta}$ with $\delta \approx 0.5$ at $K \simeq 0.98$ (indicative of a square-root law).

This is in stark contrast to the random (logarithmic) scaling and signals the presence of nontrivial structure similar to record statistics in random walks. The location of the maximal intensity (i.e., final record) in the basis is also sensitive to remnants of classical dynamics, with increased likelihood of occurrence near classical turning points in regular regimes. Such deviations in order-statistics scaling can serve as diagnostics for the ergodicity and chaos properties in quantum systems.

6. Broader Implications, Universality, and Applications

The formalism and results have key implications:

• For Haar-random quantum states, extremes and orderings—e.g., distribution of maxima (Gumbel) and the log-scaling of record counts—are universal features that transcend the details of the system, provided quantum states are sampled from the Haar measure.
• The Bernoulli process nature of record occurrence, and the harmonic mean scaling of record numbers, are robust even when normalization-induced correlations are present.
• In systems exhibiting additional dynamical correlations, deviations in order-statistical patterns—such as the emergence of square-root scaling—provide a tool for detecting non-ergodic physics, localization, or the breakdown of quantum chaos.
• These methods are applicable in quantum ergodicity studies, diagnostics of many-body chaos, and as a statistical test for deviations from Haar randomness in experimental quantum simulators or quantum information processing.

7. Summary Table: Order-Statistics Properties for Haar-Random Vectors and Quantum Chaos Models

Property	Haar-random vectors	Mixed/standard map (critical)
Mean number of records $\langle N_R \rangle$	$\log N + \gamma$	$N^{1/2}$
Record occurrence probability $p(j)$	$1/j$	$1/j$
Maximal intensity distribution	Gumbel (asymptotically)	Deviates from Gumbel
Scaling of record values (finite $N$)	Non-universal/Gumbel for large $N$	Non-universal, strongly affected
Sensitivity to dynamical correlations	Low	High (diagnostic)

This provides a concise summary of how the universal features of order statistics from Haar-random quantum states are modified in dynamical systems with correlated structures, with explicit identification of universal and non-universal aspects.

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In summary, order statistics of Haar-random quantum states reveal deep universalities: record occurrences conform to a Bernoulli process, maxima follow extreme value laws for large dimension, and deviations from these patterns under dynamical correlations in quantum models provide diagnostic power for studying quantum chaos and ergodicity (Srivastava et al., 2012).