Non-Gaussianity of random quantum states

TL;DR

This paper analyzes the typical fermionic non-Gaussianity in random quantum states, revealing how it scales with subsystem size and symmetry, using analytical methods based on Weingarten calculus.

Contribution

It provides the first analytical predictions for the non-Gaussianity of Haar random states, including the effects of global $U(1)$ symmetry, and identifies different regimes based on subsystem size.

Findings

Non-Gaussianity vanishes for small subsystems without symmetry.

With $U(1)$ symmetry, non-Gaussianity remains small but finite.

Non-Gaussianity becomes extensive when the subsystem exceeds half the system size.

Abstract

We study the fermionic non-Gaussianity in typical quantum states, focusing on Haar random states of qubits with or without a global $U(1)$ symmetry. Using the Weingarten calculus, we derive analytical predictions for the non-Gaussianity, defined as the relative entropy between the reduced density matrix and its Gaussianized counterpart. We identify two regimes controlled by the ratio between the subsystem and the system size, $\ell/L$. For $\ell/L < 1/2$, the non-Gaussianity vanishes in the absence of symmetries, because typical reduced density matrices are exponentially close to the maximally mixed state. In the presence of a global $U(1)$ symmetry, instead, it remains small but finite. By contrast, in the regime $\ell/L > 1/2$, the non-Gaussianity becomes extensive. These results establish the typical scaling of fermionic non-Gaussianity in random states and analyze how this is modified by the presence of global symmetries.