Entanglement between two subsystems, the Wigner semicircle and extreme value statistics

TL;DR

This paper investigates the entanglement properties of random pure states through the spectral analysis of the partial transpose of their density matrices, revealing connections to random matrix theory, extreme value statistics, and phase transitions.

Contribution

It introduces a simple random matrix model for the partial transpose, derives analytic formulas for log-negativity, skewness, and extreme eigenvalue distributions, and links these to entanglement transitions in quantum systems.

Findings

Density of states follows the semicircle law for large subsystems.

Log-negativity can be accurately predicted by the model.

Smallest eigenvalue follows Tracy-Widom distribution, aiding in entanglement detection.

Abstract

The entanglement between two arbitrary subsystems of random pure states is studied via properties of the density matrix's partial transpose, $\rho_{12}^{T_2}$. The density of states of $\rho_{12}^{T_2}$ is close to the semicircle law when both subsystems have dimensions which are not too small and are of the same order. A simple random matrix model for the partial transpose is found to capture the entanglement properties well, including a transition across a critical dimension. Log-negativity is used to quantify entanglement between subsystems and analytic formulas for this are derived based on the simple model. The skewness of the eigenvalue density of $\rho_{12}^{T_2}$ is derived analytically, using the average of the third moment over the ensemble of random pure states. The third moment after partial transpose is also shown to be related to a generalization of the Kempe invariant. The smallest eigenvalue after partial transpose is found to follow the extreme value statistics of random matrices, namely the Tracy-Widom distribution. This distribution, with relevant parameters obtained from the model, is found to be useful in calculating the fraction of entangled states at critical dimensions. These results are tested in a quantum dynamical system of three coupled standard maps, where one finds that if the parameters represent a strongly chaotic system, the results are close to those of random states, although there are some systematic deviations at critical dimensions.