Statistics of a quantum-state-transfer Hamiltonian in the presence of disorder

We present a statistical analysis on the performance of a protocol for the faithful transfer of a quantum state in finite qubit or spin chains, in the presence of diagonal and off-diagonal disorder. It is shown that the average-state fidelity, typically employed in the literature for the quantification of the transfer, may overestimate considerably the performance of the protocol in a single realization, leading to faulty conclusions about the success of the transfer.

Figure 1

Probability distribution of the fidelity for $N=12$ and 1000 independent realizations. The histograms (a)–(e) correspond to $F_{\psi }{\left(\right|\beta |}^2,p,\Delta \phi )$ for ${\left|\beta \right|}^2=1.0,0.8,0.6,0.5,$ and $0.4$, respectively. The histogram (f) corresponds to $\overline{F}(p,\Delta \phi )$. Other parameters: ${\sigma }_{\eta }={\sigma }_{\xi }=0.1$.

Figure 2

The average fidelities $\langle F_{\psi }{\left(\right|\beta |}^2,p,\Delta \phi )\rangle$ (solid curves) and $\langle \overline{F}(p,\Delta \phi )\rangle$ (horizontal solid lines) as functions of ${\left|\beta \right|}^2$, for different values of $N$: (a) $N=12$; (b) $N=18$; (c) $N=25$; and (d) $N=31$. The averages have been obtained on 1000 independent realizations and the error bars show the standard deviations as obtained from the corresponding distributions of $\overline{F}(p,\Delta \phi )$ and $F_{\psi }{\left(\right|\beta |}^2,p,\Delta \phi )$ for various values of ${\left|\beta \right|}^2$. The standard deviation of $\overline{F}(p,\Delta \phi )$ is independent of ${\left|\beta \right|}^2$. The dashed horizontal line shows the classical threshold $F_{\text{cl}}=2/3$. Other parameters as in Fig. 1.

Figure 3

(a) The average fidelities $\langle F_{\psi }{\left(\right|\beta |}^2,p,\Delta \phi )\rangle$ for various values of ${\left|\beta \right|}^2$ (solid curves with symbols) and $\langle \overline{F}(p,\Delta \phi )\rangle$ (dashed curve with circles) are plotted as functions of $N$. (b) As in (a) for the standard deviations. (c) The intervals ${\Delta }_j$ for which the average fidelity $\langle F_{\psi }{\left(\right|\beta |}^2,p,\Delta \phi )\rangle$ is below $\langle \overline{F}(p,\Delta \phi )\rangle$ (dot-dashed curve with triangles) and the classical limit (solid curve with squares), as functions of $N$. (d) The probability of failure for various values of ${\left|\beta \right|}^2$, as a function of $N$. The vertical arrows mark the values of $N$ for which $\langle \overline{F}(p,\Delta \phi )\rangle =F_{\text{cl}}$. The inset of Fig. 3b indicates the values of ${\left|\beta \right|}^2$ that correspond to the depicted curves. Other parameters as in Fig. 2.

Figure 4

The average fidelities $\langle \overline{F}(p,\Delta \phi )\rangle$ (a) and $\langle F_{\psi }{\left(\right|\beta |}^2,p,\Delta \phi )\rangle$ for ${\left|\beta \right|}^2=0.8$ (b), ${\left|\beta \right|}^2=0.6$ (c), and ${\left|\beta \right|}^2=0.4$ (d) as functions of disorder for $N=15$. The averages have been obtained on 1000 independent realizations.

Figure 5

The probability of failure as estimated by $\langle \overline{F}(p,\Delta \phi )\rangle$ (a), and $\langle F_{\psi }{\left(\right|\beta |}^2,p,\Delta \phi )\rangle$ for ${\left|\beta \right|}^2=0.8$ (b), ${\left|\beta \right|}^2=0.6$ (c), and ${\left|\beta \right|}^2=0.4$ (d), as a function of disorder for $N=15$. The averages have been obtained on 1000 independent realizations. Successive contours differ by 0.1 and the color scale is as in Fig. 4.

Figure 6

The quantities ${\Delta }_0$ (a) and ${\Delta }_1$ (b) as functions of disorder for $N=15$. The averages have been obtained on 1000 independent realizations.

Figure 7

The values of ${\left|\beta \right|}^2$ that minimize $F_{\psi }{\left(\right|\beta |}^2,p,\Delta \phi )$ (a) as well as the fidelity differences $F_{\min }(p,\Delta \phi )-F(1,p,\Delta \phi )$ (b) and $F_{\min }(p,\Delta \phi )-\overline{F}(p,\Delta \phi )$ (c), for various combinations of $p$ and $\Delta \phi$. The plots are mirror symmetric with respect to $\Delta \phi =0$, and can be extended to $\Delta \phi <0$ by the transformation $\Delta \phi \to -\Delta \phi$.

Figure 8

Density plot of probability (18) as a function of the diagonal (${\sigma }_{\eta }$) and off-diagonal (${\sigma }_{\xi }$) disorder, for a chain of length (a) $N=12$ and (b) $N=21$. The probabilities have been estimated for $\epsilon ={10}^{-2}$ on 1000 realizations.