Non-Gaussianity of random quantum states

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.