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

Abstract

The entanglement between two arbitrary subsystems of random pure states is studied via properties of the density matrix's partial transpose, ρ T 2 12 . The density of states of ρ T 2 12 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 ρ T 2 12 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.

Bures–Hall ensemble: spectral densities and average entropies

“…Expressing the (complete) beta function in terms of gamma functions and simplifying gives us (40). Similarly, when n is odd, using (29), (30) we also have…”

Local entanglement structure across a many-body localization transition

“…1 for an illustration). Exactly how large the subsystems must be before they become entangled has been explored in the case of random states 51 . In the extreme case, the entanglement between two spins in a spin chain whose state obeys the volume law is typically zero.…”

Disentangling quantum matter with measurements

“…The intuition is that nothing dramatic happens if one integrates out a very small region, so negativity will continue to be volume law until the subsystem looks thermal. This intuition is verified numerically in ergodic spin chains [19] and proved analytically for Haar random states [20].…”

“…Such a thermal state is obtained by tracing out more than half the degrees of freedom in a purely ergodic wavefunction. In contrast, for an ergodic wavefunction, integrating out less than half the degrees of freedom leads to a volume law negativity [19,20]. The intuition is that nothing dramatic happens if one integrates out a very small region, so negativity will continue to be volume law until the subsystem looks thermal.…”