Entropy of entanglement and multifractal exponents for random states

We relate the entropy of entanglement of ensembles of random vectors to their generalized fractal dimensions. Expanding the von Neumann entropy around its maximum we show that the first order only depends on the participation ratio, while higher orders involve other multifractal exponents. These results can be applied to entanglement behavior near the Anderson transition.

Figure 1

(Color online) Mean entropy of entanglement as a function of the mean IPR. Left: eigenvectors of Eq. (6) with $\gamma =1/3$; the average is taken over ${10}^6$ eigenvectors. Right: eigenvectors of the Hamiltonian $H$ (see text) with $\delta ={\Delta }_0$ and $J/\delta =1.5$; average over $N/16$ central eigenstates, with a total number of vectors $\approx 3\times {10}^5$. Triangles correspond to $\nu =1$, squares to $\nu =2$ and circles to $\nu =n_r/2$, with $n_r=4–10$ (bipartition of the $\nu$ first qubits with the $n_r-\nu$ others). Black symbols are the theoretical predictions for $⟨S⟩$ at first order [Eq. (5)] and green (gray) symbols are the computed mean values of the exact $⟨S⟩$.

Figure 2

(Color online) Relative difference of the entropy of entanglement (7) and its successive approximations $S_m$ $\left(m=1,2\right)$ with respect to the number of qubits for eigenvectors of Eq. (6) for (left) $\gamma =1/3$ and (right) $\gamma =1/7$. The average is taken over ${10}^7$ eigenvectors, yielding an accuracy $\lesssim {10}^{-6}$ on the computed mean values. Green triangles correspond to the first-order expansion $S_1$, blue squares and red circles to the second-order expansion $S_2$. The difference between the latter two is that for blue squares $⟨p_2^2⟩$ appearing in Eq. (12) has been replaced by ${⟨p_2⟩}^2$ yielding a less accurate approximation. Dashed line is a linear fit yielding $1-⟨S_1⟩/⟨S⟩\sim N^{-0.84}$ for $\gamma =1/3$ and $N^{-1.58}$ for $\gamma =1/7$.