Entanglement transitions induced by large deviations

The probability of large deviations of the smallest Schmidt eigenvalue for random pure states of bipartite systems, denoted as $A$ and $B$, is computed analytically using a Coulomb gas method. It is shown that this probability, for large $N$, goes as $\exp[-\beta N^2\Phi(\zeta)]$, where the parameter $\beta$ is the Dyson index of the ensemble, $\zeta$ is the large deviation parameter while the rate function $\Phi(\zeta)$ is calculated exactly. Corresponding equilibrium Coulomb charge density is derived for its large deviations. Effects of the large deviations of the extreme (largest and smallest) Schmidt eigenvalues on the bipartite entanglement are studied using the von Neumann entropy. Effect of these deviations is also studied on the entanglement between subsystems $1$ and $2$, obtained by further partitioning the subsystem $A$, using the properties of the density matrix's partial transpose $\rho_{12}^\Gamma$. The density of states of $\rho_{12}^\Gamma$ is found to be close to the Wigner's semicircle law with these large deviations. The entanglement properties are captured very well by a simple random matrix model for the partial transpose. The model predicts the entanglement transition across a critical large deviation parameter $\zeta$. Log negativity is used to quantify the entanglement between subsystems $1$ and $2$. Analytical formulas for it are derived using the simple model. Numerical simulations are in excellent agreement with the analytical results.