Quantum-information processing in disordered and complex quantum systems

We study quantum information processing in complex disordered many body systems that can be implemented by using lattices of ultracold atomic gases and trapped ions. We demonstrate, first in the short range case, the generation of entanglement and the local realization of quantum gates in a disordered magnetic model describing a quantum spin glass. We show that in this case it is possible to achieve fidelities of quantum gates higher than in the classical case. Complex systems with long range interactions, such as ions chains or dipolar atomic gases, can be used to model neural network Hamiltonians. For such systems, where both long range interactions and disorder appear, it is possible to generate long range bipartite entanglement. We provide an efficient analytical method to calculate the time evolution of a given initial state, which in turn allows us to calculate its quantum correlations.

Figure 1

(Color online) Temporal behavior of NN averaged entanglement in a 2D SG model, starting from ${\prod }_i{\mid +⟩}_i$. $E_{\mathit{LN}}\left(t\right)$ denotes the NN averaged logarithmic negativity and $t$ denotes the time. (Note that $t$ is implicitly multiplied by $c∕\hslash$, so that the horizontal axis is actually of $ct∕\hslash$.) For a model with frustration $(J=0)$, $E_{\mathit{LN}}\left(t\right)$ converges quickly to a constant value (red dotted curve). For a nonfrustrated case $(J=+5)$, $E_{\mathit{LN}}\left(t\right)$ exhibits damped oscillations (blue curve), converging to the same value $\approx 0.0154$, as reached in the frustrated case. Standard deviation for $t\to \infty$ is $\approx 0.0704$. It is interesting to note that the dynamical behavior of $E_{\mathit{LN}}$ depends on $J$, although at large times, they all converge to the same value (see inset). The same behavior is encountered in the 1D case, even though there is no frustration in that case.

Figure 2

(Color online) Probability distribution of NN entanglement for given times $t$ in the evolved state ${\varrho }_{12}(t,\left\{J_{ij}\right\})$ of the Edwards-Anderson spin glass model [27]. The value of entanglement for a large number of sample $J_{ij}$ points (and hence a large probability) is concentrated in the “zero” entanglement zone [0,0.01]. This probability column is not shown in both of the histograms for easy viewing of the histograms. This large zero entanglement zone probability leads to the low average value of entanglement. However, the histograms show that there is a substantial probability of obtaining entanglement that is one order of magnitude more than the average value (see text). Note that including the zero entanglement probability, the histograms are in fact correctly normalized.

Figure 3

(Color online) Plot of probability vs NN entanglement for time $t=1.25$ (and $t=20$ in inset) of ${\varrho }_{12}(t,\left\{J_{ij}\right\})$ [27]. The probability for the entanglement range $[0.01i,0.01(i+1)]$ is plotted against an entanglement of $0.01i$, for $i=1$ to 100. The dashed (blue) lines correspond to the actual values, obtained by sampling a large number of $J_{ij}$’s. The undashed (red) lines are exponential fits. The exponential fit in the inset (for $t=20$) is almost exact, while the one for $t=1.25$ is seen to not fit very well. In both cases, the exponential fit is $\text{probability}=a\exp (-\mathrm{entanglement}∕b)$. For $t=1.25$, we have $a=0.01837$, $b=0.21543$, while for $t=20$, we have $a=0.00577$, $b=0.15053$. Just like in Fig. 2, the “zero” entanglement zone is not shown in the figure (see caption of Fig. 2 and text). Also, including the zero entanglement probability, the curves are in fact correctly normalized.

Figure 4

(Color online) Generation of entanglement [$E_{\mathit{LN}}\left(t\right)$, on the vertical axis] of bipartite states ${\varrho }_{12}^{\mathit{lro}}\left(t\right)$ with respect to time $t$ and number of spins $N$ [27]. Collapses and revivals of the entanglement are clearly depicted.