Propagation of nonclassical correlations across a quantum spin chain

We study the transport of quantum correlations across a chain of interacting spin-$1/2$ particles. As a quantitative figure of merit, we choose a symmetric version of quantum discord and compare it with the transported entanglement, addressing various operating regimes of the spin medium. Discord turns out to be better transported for a wide range of working points and initial conditions of the system. We relate this behavior to the efficiency of propagation of a single excitation across the spin chain. Moreover, we point out the role played by a magnetic field in the dynamics of discord in the effective channel embodied by the chain. Our analysis can be interestingly extended to transport processes in more complex networks and the study of nonclassical correlations under general quantum channels.

Figure 1

We consider a chain of $N$ interacting spin-$1/2$ particles coupled through the Hamiltonian model $\widehat{\mathcal{H}}$ [cf. Eq. (1)]. A further spin-$1/2$ particle, labeled 0 and completely isolated from the chain, is prepared in a joint mixed state ${\rho }^{(1,0)}\left(0\right)$ with particle 1. We study how the general quantum correlations of such a state propagate across the chain.

Figure 2

Behavior of quantum correlations between spins 0 and 3 against dimensionless time $Jt$ in a system ruled by $\widehat{\mathcal{H}}$ with uniform interaction strengths and arbitrary $h$. (a)–(d) $C_{(1,0)}=0.1,0.4,0.7,\text{and}1$, respectively, with the initial EOF shown as a straight line. The solid (red) curve shows the EOF, while the dashed (gray) one is for the shared two-way QD.

Figure 3

Behavior of quantum correlations between spin 0 and 50 against dimensionless time $Jt$ for a system ruled by ${\widehat{\mathcal{H}}}_T$ with arbitrary $h$. (a)–(d) $C_{(1,0)}=0.1,0.4,0.7,\text{and}1$, respectively, with the initial EOF shown as a straight line. The solid (red) curve shows the EOF $\mathcal{E}$, while the dashed (gray) one is for the shared two-way QD $\mathcal{D}$.

Figure 4

Behavior of QD against entanglement and dimensionless propagation time $Jt$ for a chain of 15 spins, homogeneous intrachain couplings, and arbitrary $h$. The yellow plane at $\mathcal{D}=\mathcal{E}$ is used as a guide to the eye for discerning whether or not $\mathcal{D}⩾\mathcal{E}$. Each panel is for a different initial value of concurrence in the pair $(1,0)$. (a)–(c) $C_{(1,0)}=0.1,0.6$, and 1, respectively. As the initial degree of entanglement grows, the efficiency of transport of $\mathcal{D}$ gets very close to that for $\mathcal{E}$.

Figure 5

Comparison between the rescaled quantities $\tilde{\mathcal{D}}$ (gray dashed line) and $\tilde{\mathcal{E}}$ (red solid one) propagated across our chain of uniform couplings and the arbitrary magnetic field. The spin pair $(1,0)$ is prepared in a Werner state with $a=0.4,0.7$, and 1 [(a)–(c), respectively]. The curves are plotted against the transition amplitude $\left|f\right|$.

Figure 6

Comparison between the rescaled quantities $\tilde{\mathcal{D}}$ (gray dashed line) and $\tilde{\mathcal{E}}$ (red solid one) propagated across our chain of uniform couplings and the arbitrary magnetic field. The spin pair $(1,0)$ is prepared in state ${\rho }_{(1,0)}^P$ with $b=0$ and two values of $a$. The curves are plotted against the transition amplitude $\left|f\right|$.

Figure 7

Comparison between the rescaled quantities $\tilde{\mathcal{D}}$ (gray dashed line) and $\tilde{\mathcal{E}}$ (red solid one) propagated across our chain of uniform couplings and no magnetic field. The spin pair $(1,0)$ is prepared in state ${\rho }_{(1,0)}^R$ with (a) $(a,g)=(0.0150,0.9747)$ and (b) $(a,g)=(0.1625,0.7649)$. The curves are plotted against the transition amplitude $\left|f\right|$.

Figure 8

QD transported across a chain of 50 spins whose $(1,0)$ pair is initialized in an $X$-type state with maximally mixed marginals having $c_x\simeq 0.53$, $c_y\simeq 0.340$, and $c_z\simeq 0.035$. The associated value of QD is ${\mathcal{D}}_{\left(01\right)}\left(0\right)=0.210$. We have taken (a) $h/J=0$, (b) $h/J=0.5$, (c) $h/J=1$, and (d) $h/J=5$.

Figure 9

QD between spin 0 and 3, for a chain of N = 3 spins whose pair $(1,0)$ is initialized in an $X$-type state with maximally mixed marginals having $c_x=1$, $c_y=c_z=0$, and thus $\mathcal{D}\left(0\right)=0$. We have taken $h/J=0$ (blue dashed line) and $h/J=2$ (full red line). The dot-dashed line is the transition amplitude $|f_3\left(t\right)|$.