Testing Statistical Bounds on Entanglement Using Quantum Chaos

Previous results indicate that while chaos can lead to substantial entropy production, thereby maximizing dynamical entanglement, this still falls short of maximality. Random matrix theory modeling of composite quantum systems, investigated recently, entails a universal distribution of the eigenvalues of the reduced density matrices. We demonstrate that these distributions are realized in quantized chaotic systems by using a model of two coupled and kicked tops. We derive an explicit statistical universal bound on entanglement, which is also valid for the case of unequal dimensionality of the Hilbert spaces involved, and show that this describes well the bounds observed using composite quantized chaotic systems such as coupled tops.

Figure 1

Entanglement saturation of a completely unentangled initial state (solid line) and a maximally entangled initial state (dotted line) under time evolution operator $U_T$. Here $k=3,ϵ=0.1$ and the phases ${\alpha }_1={\alpha }_2=0.47$. The inset shows the similar behavior of linear entropy.

Figure 2

The spectral average of the entanglement present in eigenstates of $U_T$ ($k=9,ϵ=10$) as a function of $Q=M/N$, where $N=2j_1+1=33$. Solid triangles are kicked top results with parity symmetry (${\alpha }_1={\alpha }_2=0$) and solid circles are the corresponding results without symmetry (${\alpha }_1={\alpha }_2=0.47$). Solid squares are the result of corresponding RMT Monte Carlo simulations and the solid line is the theoretical curve [Eq. (8)]. The horizontal line is the maximum possible entanglement [$\ln (N)$]. The inset shows the behavior of the linear entropy.

Figure 3

Distribution of the eigenvalues of the RDMs of coupled kicked tops, averaged over all the eigenstates ($N=2j_1+1=33$). Solid curves correspond to the theoretical distribution function [Eq. (7)].