Subsystem Statistics and Conditional Self-Similarity of Random Quantum States

Abstract

We analytically derive the bit-string probability distributions of subsystems of random pure states and depolarized random states using the Dirichlet distribution. We identify the exact Beta distribution as the universal statistical law of random quantum states, providing a unified finite-size description of full-system, subsystem, and conditional statistics. In the presence of depolarizing noise, these distributions are scaled and shifted by the noise strength, producing a noise-induced gap in their support. Remarkably, we prove that random states exhibit exact conditional self-similarity: the distribution of subsystem bit-string probabilities conditioned on specific outcomes of the complementary subsystem is identical to that of the full system. This hidden scale invariance enables the exact restoration of the full-system statistics from the marginalized Beta distribution via post-selection, and persists under depolarizing noise. Our results uncover a fundamental symmetry of Hilbert space and provide a scalable, rigorous framework for validating random circuit sampling via subsystem or conditional cross-entropy benchmarking.

Figures (3)

• Figure 1: Plot of Eq. (\ref{['Eq:Sub_Beta_x']}), the Beta distributions of subsystem bit-string probabilities in a random pure state as a function of $x = Mp_A$ for various subsystem sizes. Here the total number of qubits is $n=30$.
• Figure 2: The empirical distribution of bit-string probabilities for Google Sycamore with $n=12$ qubit (histogram), the ideal exponential distribution (dotted black line), the scaled-shifted exponential distribution, Eq. (\ref{['Eq:scaled_shifted_exponential']}) (blue solid line), and the exponentially modified distribution (red solid line) are plotted. The strengths of depolarizing noise is $\lambda= 0.52$.
• Figure 3: Using Google Sycamore data for $n=12$ qubits, the empirical distributions are plotted for conditional probabilities (a) $p(y|a=0)$, (b) $p(y|a=01)$, and (c) the marginal probability $p(y)$ after tracing out last two bits.