Disorder overtakes order in information concentration over quantum networks

We consider different classes of quenched disordered quantum $XY$ spin chains, including a quantum $XY$ spin glass and a quantum $XY$ model with a random transverse field, and investigate the behavior of genuine multiparty entanglement in the ground states of these models. We find that there are distinct ranges of the disorder parameter that give rise to a higher genuine multiparty entanglement than in the corresponding systems without disorder: an order-from-disorder phenomenon in genuine multiparty entanglement. Moreover, we show that such a disorder-induced advantage in the genuine multiparty entanglement is useful: It is almost certainly accompanied by a order-from-disorder phenomenon for a multiport quantum dense coding capacity with the same ground state used as a multiport quantum network.

Figure 1

Order-from-disorder phenomenon for a genuine multiparty entanglement in quantum $XY$ spin glass. The upper plot is for seven spins, while the lower one is for eight. In both the plots, GGMs are plotted on the vertical axes. The curves with black circles are for the ordered systems (with the Hamiltonian $H$), and in these cases, the GGMs are plotted against $\lambda =J/h$ on the horizontal axes. The curves with red squares are for the disordered systems (with the Hamiltonian $H_{\text{SG}}$), and in these cases, the GGM are plotted against the mean values $\lambda =J/h$ of the Gaussian distributed $J_i/h$ (with mean $J/h$ and unit standard deviation) on the horizontal axes. The quenched averaging is performed over $5\times {10}^3$ disorder realizations. Both axes represent dimensionless quantities in both the plots. The whole range of parameters considered offers disorder-induced enhancement of GGM for the case of an odd number of spins. For an even number of spins, there is a crossover $\lambda$, before which the disorder-induced enhancement occurs. See [63] for the strategy assumed for the very small number of degenerate ground states that appear.

Figure 2

Order-from-disorder phenomenon for a multiport classical capacity of a quantum channel in quantum $XY$ spin glass. The upper and lower plots are for seven and eight spins respectively. In both plots, the normalized capacities (dimensionless) are plotted on the vertical axes. All other notations are the same as in Fig. 1, and the quenched averaging is performed over $5\times {10}^3$ disorder realizations. Again, the whole range of parameters considered offers disorder-induced enhancement for the capacity for an odd number of spins, while there is a crossover $\lambda$ before which the disorder-induced enhancement occurs for an even number of spins. The crossover $\lambda$ for the capacity is lower than that for the GGM. The horizontal lines (with blue diamonds) correspond to the classical capacities that can be attained if there are no previously shared quantum states between the senders and the receivers.

Figure 3

Order-from-disorder phenomenon for GGM in a quantum $XY$ system with random transverse field. The upper plot is for $N=7$, while the lower one is for $N=8$. The vertical axes in both plots represent the GGM. The curves with black circles are for the systems without disorder (represented by the Hamiltonian $H$), in which case the GGM are plotted against $\mu =h/J$ on the horizontal axes. The curves with red squares are for the systems with disorder and are described by the Hamiltonian $H^{\text{RF}}$. In these cases, the horizontal axes represent the mean values $\mu =h/J$ of the Gaussian distributed random variables $h_i/J$ (with mean $h/J$ and unit standard deviation). The quenched average is taken over $5\times {10}^3$ disorder realizations. All the axes represent dimensionless quantities. In contrast to the situation in Fig. 1, the crossovers now appear for both odd and even $N$.

Figure 4

Order-from-disorder phenomenon for the multiport capacity in quantum $XY$ system with a random transverse field. The vertical axes in both the plots represent the normalized capacities (dimensionless). All other notations remain the same as in Fig. 3, and $5\times {10}^3$ disorder realizations are used for the quenched averaging. There are crossovers again, and the crossover $\mu$'s for the capacity are higher than those for GGM. The horizontal lines (with blue diamonds) have the same meaning as in Fig. 2.

Figure 5

Order-from-disorder phenomenon for genuine multiparty entanglement in quantum $XY$ spin glass with random transverse field. The upper plot is for $N=7$, and the lower one is for $N=8$. The vertical axes represent ${\lambda }_1=J/\kappa$, while the horizontal axes represent ${\mu }_1=h/\kappa$, with $\kappa >0$. The plotted quantity is the difference (disorder case $-$ order case) between the quenched averaged GGM for the system described by $H_{\text{SG}}^{\text{RF}}$ [with i.i.d. random variables $J_i$ and $h_i$, where the $J_i/\kappa$ are distributed as $N(J/\kappa ,1)$ and the $h_i/\kappa$ are distributed as $N(h/\kappa ,1)$] and the GGM for the system described by $H$. Again, the quenched averaging is performed over $5\times {10}^3$ disorder realizations. See the text for further details. The plotted quantity as well as the quantities represented by the axes are dimensionless.

Figure 6

Order-from-disorder phenomenon for the multiport capacity in quantum $XY$ spin glass with random transverse field. All notations are the same as those in Fig. 5, except that the plotted quantity is the difference between the multiport capacities in the two situations considered. Again, the quenched averaging is performed over $5\times {10}^3$ disorder realizations. See the text for further details.

Figure 7

Behavior of quenched averaged GGM for different anisotropy parameters in the spin glass system. We consider a system of eight spins. The vertical axis represents the quenched averaged GGM. The different curves are for different values of the anisotropic parameter $\gamma$. From bottom to top, they are respectively for $\gamma =0.1$ (black), 0.3 (red), 0.5 (blue), 0.7 (green), and 1.0 (magenta). The GGM is plotted for the ground state of the spin glass system described by the Hamiltonian $H_{\text{SG}}$ and is plotted against the mean values $\lambda =J/h$ of the Gaussian quenched random variables $J_i/h$. The standard deviations of the random variables are all unity. The quenched averaging of GGM is taken over $5\times {10}^3$ disorder configurations. Both the axes represent dimensionless quantities.

Figure 8

Order-from-disorder phenomenon for the genuine multiparty entanglement and capacity in quantum $XY$ spin glass with transverse field. Here we consider $N=10$ spins. All notations are the same as in Fig. 2. The quenched averaging is performed over $5\times {10}^3$ disorder realizations.