Entanglement of localized states

We derive exact expressions for the mean value of Meyer-Wallach entanglement $Q$ for localized random vectors drawn from various ensembles corresponding to different physical situations. For vectors localized on a randomly chosen subset of the basis, $⟨Q⟩$ tends for large system sizes to a constant which depends on the participation ratio, whereas for vectors localized on adjacent basis states it goes to zero as a constant over the number of qubits. Applications to many-body systems and Anderson localization are discussed.

Figure 1

(Color online) Scaled entanglement with respect to the reduced localization length ${⟨N_b∕\xi ⟩}^{-1}$ with $N_b=\left(\genfrac{}{}{0ex}{}{n}{n_b}\right)$. The blue thick solid curves correspond to the second band (for $n=11-20$), the red thin dashed curves to the third band (for $n=11-20$), the green thin dotted curves to the fourth band (for $n=13,15,17$), the black thin solid curves to the fifth band (for $n=10,11,13,15$), the yellow thick dashed curve to the sixth band (for $n=12$), and the violet thick dotted curve to the seventh band (for $n=14$).

Figure 2

(Color online) Scaled mean MWE $⟨Q⟩(N-2)∕N$ of Eq. (9) vs IPR for $\delta ={\Delta }_0$, $n=10$ (blue circles) and $n=11$ (green squares). Average is over $N∕16$ central eigenstates and 100–200 disorder realizations. Red solid line is the theory, and stars are the data for $n=10$ with random shuffling of components. Inset: scaled correlator $(N∕2)(N∕2-1)⟨C_{xx}⟩$ with same parameters; red line is the result when $⟨C_{xx}⟩=⟨C_{xy}⟩$.

Figure 3

(Color online) Mean MWE vs the number of qubits for a 1D Anderson model with disorder from top to bottom $w=0.2$ (blue), 0.5 (red), 1.0 (green), 1.5 (magenta), 2.0 (cyan), and 2.5 (orange). Average is over 10 central eigenstates for 1000 disorder realizations. Solid lines are the $C∕n$ fits of the tails. Inset: value of $C={\lim }_{n\to \infty }n⟨Q⟩$ as a function of IPR $\xi$ (green dots) for the values of $w$ above and $w=0.4$, together with analytical result from Eq. (7) (red line, top) and from Eq. (10) (blue line, bottom) both for $M=2\xi$. Stars are the $C$ values resulting from a $C∕n$ fit of the numerical data for CUE vectors of size $N$ with exponential envelope $\exp (-x∕l)$.

Figure 4

(Color online) Mean MWE (left) and IPR (right) vs number of qubits for quantum small-world networks with $w=1$ and $p=0.001$ (blue +), 0.005 (red ×), 0.01 (green 엯), 0.03 (magenta ◻), and 0.06 (orange ⋆). Logarithm is decimal.