Quantum state transfer in spin-1 chains

We study the transfer of a quantum state through a Heisenberg spin-1 chain prepared in its ground state. We characterize the efficiency of the transfer via the fidelity of retrieving an arbitrarily prepared state and also via the transfer of quantum entanglement. The Heisenberg spin-1 chain has a very rich quantum phase diagram. We show that the boundaries of the different quantum phases are reflected in sharp variations of the transfer efficiency. In the vicinity of the border between the dimer and the ferromagnetic phase (in the conjectured spin-nematic region), we find strong indications for a qualitative change of the excitation spectrum. Moreover, we identify two regions of the phase diagram that give rise to particularly high transfer efficiency; the channel might be nonclassical even for chains of arbitrary length, in contrast to spin-$1∕2$ chains.

Figure 1

(a) Quantum phases of the spin-1 chain. (b) Scheme for state transfer with an underlying ferromagnetic state and (c) with the channel initialized to the ground state of $\widehat{H}\left(\theta \right)$, and (d) transfer of quantum entanglement.

Figure 2

Channel fidelity $F$ (open circles) and entanglement measured via logarithmic negativity LN (filled circles) for the transfer between the end points of a chain with 25 sites (obtained with the MPS methods, using $D=20–25$). The dashed horizontal line denotes the maximal fidelity $F_{\text{class}}=1∕2$ that can be obtained through classical communication. Vertical dashed lines indicate the quantum phase boundaries. (See Fig. 4 for a zoom into the region $-0.76\pi ⩽\theta ⩽-0.7\pi$.)

Figure 3

Fidelity $F$ (left column) and logarithmic negativity LN (right column) versus chain length $N$ for (a), (b) the spin-1 chain at $\theta =\pi ∕4$; (c), (d) the spin-1 chain at $\theta =-\pi ∕2$; and (e), (f) the spin-$1∕2$ Heisenberg chain. Dark circles correspond to the channel prepared in the ground state; open circles in (a), (b) to a ferromagnetic initial state. $F$ is fitted through an exponentially decaying function. Dashed lines indicate the classical communication limit $F_{\text{class}}=1∕2$ $(2∕3)$ for spin-1 $(-1∕2)$. [Results obtained with MPS simulations using up to $D=25$ (24) for $\theta =\pi ∕4$ $(-\pi ∕2)$.]

Figure 4

(a) $F$ (open circles) and LN (filled circles) around the region for which a spin-nematic phase has been conjectured (for $N=25$). The transfer efficiency is strongly reduced for $-0.75\pi <\theta \lesssim 0.72\pi$, as most clearly visible from $\mathrm{LN}=0$. To characterize the ground state we plot (b) the dimerization $D_{(N-1)∕2}$ $(D_i=⟨{\widehat{H}}_{i,i+1}⟩-⟨{\widehat{H}}_{i+1,i+2}⟩)$, (c) the nematic order parameter $Q={\max }_{\Omega }{\sum }_{i=2}^N[⟨{({\vec{n}}_{\Omega }\bullet {\vec{S}}_i)}^2⟩-2∕3]∕(N-1)$ (${\vec{n}}_{\Omega }$ is the unit vector pointing in direction $\Omega$), and (d) the magnetization $M={\max }_{\Omega }{\sum }_{i=2}^N⟨{\vec{n}}_{\Omega }\bullet {\vec{S}}_i⟩∕(N-1)$.