Harnessing non-Hermiticity for efficient quantum state transfer

TL;DR

This work investigates how non-Hermitian dynamics, arising from continuous monitoring of a quantum system’s environment, can enhance quantum state transfer (QST) along spin chains. It derives a general fidelity expression for U(1)-symmetric non-Hermitian evolutions and applies it to PT-symmetric XX and SSH models and RT-symmetric iXY models, identifying parameter regimes where non-Hermiticity beats the classical limit and, in some cases, its Hermitian counterpart. A key finding is near-unit fidelity in the SSH model when inter-cell coupling dominates, facilitated by the broken PT-symmetric phase, along with an ellipse–hyperbola correspondence linking non-Hermitian and Hermitian parameter regions. The results demonstrate a constructive role for non-Hermiticity in QST under decoherence, with entanglement dynamics closely tied to fidelity and favorable scaling for moderate system sizes, suggesting routes for robust quantum links in realistic devices.

Abstract

The non-Hermitian Hamiltonian describes the effective dynamics of a system coupled to a continuously measured bath, and can exhibit anti-unitary symmetries that give rise to exceptional points and broken phases with complex eigenvalues, features unique to non-Hermitian systems. Going beyond conventional Hermitian physics, we analyze the impact of non-Hermiticity in the quantum state transmission by employing a non-Hermitian spin chain that functions as a quantum data bus. By deriving a general expression for the fidelity of quantum state transfer for a U(1)-symmetric non-Hermitian Hamiltonian, we analyze PT-symmetric XX and SSH models, complemented by a numerical study of the RT-symmetric iXY model. We demonstrate that, in several parameter regimes, the transfer fidelity in the non-Hermitian setting exceeds the classical threshold and can even exceed the performance of the corresponding Hermitian models. In particular, for the SSH model with dominant inter-cell coupling, the broken phase supports near-unit-fidelity quantum state transfer, a level of performance that the corresponding Hermitian model fails to attain. Moreover, we establish a correspondence between the non-Hermitian and Hermitian descriptions by identifying related parameter regions in which the fidelity fails to surpass the classical bound.

Figures (9)

• Figure 1: Schematic of the QST protocol through a spin chain. Each spin is coupled to a local bath $B_k$ ($k=1, 2, \ldots N$). Owing to the presence of these baths and the suitable continuous measurements followed by post-selection, the state-transfer dynamics is governed by an effective $\hat{\mathcal{P}}\hat{\mathcal{T}}$-symmetric non-Hermitian Hamiltonian. In the protocol, $|\psi\rangle_1$ denotes the arbitrary quantum state to be transferred, ($S_2,S_3,...,S_{N-1}$) represent the intermediate spins of the chain and $\rho_N(t)$ is the final state received at Bob’s site after time evolution. By appropriately selecting the initial state, the governing Hamiltonian, and its parameters, our goal is to maximize the fidelity, which serves as a measure of the protocol’s success.
• Figure 2: Comparison between fidelities $\mathcal{F}_N(t)$ of QST for the non-Hermitian and Hermitian $XX$ models. (a) and (b) $\mathcal{F}_N(t)$ (ordinate) against $t$ (abscissa) for specific choices of $(h_1,h_2)$, with advantage in the nH case, i.e., $\mathcal{F}_{N, nH}^{(1)}>\mathcal{F}_{N, H}^{(1)}$. Here, solid (blue) lines indicate non-Hermitian case while dashed-dot (red) lines show the corresponding Hermitian behavior, with dashed (black) line denoting the classical limit $\mathcal{F}_c$. The fidelity corresponding to non-Hermitian evolution beats the classical limit of $2/3$ earlier, and the first maxima rises above the corresponding Hermitian evolution, where $h_1$ and $h_2$ belong to the unbroken regime of the non-Hermitian Hamiltonian in both cases. The map plot of first maximum fidelity $\mathcal{F}_N^{(1)}$ in the $(h_2,h_1)$-plane for (c) non-Hermitian, and (d) Hermitian $XX$ models. The points where fidelity rises above $0.9$ are shown in pink. The red dotted line in (c) distinguishes the broken (B) region ($h_2 \gtrsim 0.18$) from the unbroken (U) one. The parameters in the unbroken region of (c) and (d) represent an ellipse and a hyperbola, respectively, of the points where the fidelity could not reach the classical threshold. The corresponding parameters of these curves in Eq. (\ref{['eq:ellipse']}) and (\ref{['eq:hyperbola']}) are ($a=0.188, b=0.186$). Here $N=16$ and all the axes are dimensionless.
• Figure 3: Connecting parameter space of the pseudo non-Hermitian model with the Hermitian ones. The lines represent the relation between $y\equiv h_1$ ($h$) and $x \equiv h_2$ ($\gamma$) (see Eqs. (\ref{['eq:ellipse']}) and (\ref{['eq:hyperbola']})) where the fidelity $\mathcal{F}_N(t)$ could not reach the classical threshold of $2/3$ for the $XX$, and SSH ($iXY$) models. The gradient of colors, from dark to light (pink), represent parameters of Eq. (\ref{['eq:ellipse']}) and (\ref{['eq:hyperbola']}) for the $XX$, SSH and $iXY$ models, with system-size $N=16$ for the $XX$ and the SSH model, while $N=8$ for the $iXY$ model. Solid lines and dashed-dot lines correspond to the nH and corresponding Hermitian models respectively. All the axes are dimensionless.
• Figure 4: Illustration of fidelity $\mathcal{F}_N(t)$ with respect to time is presented for (a) $N=16, h_1 = 0.5, h_2 = 0.19$, and (b) $N=25, h_1 = 0.05, h_2 = 0.05$. Both these parameter sets considered belong to the broken regime of the non-Hermitian $XX$ Hamiltonian. (a) The first maxima of fidelity obtained via non-Hermitian evolving Hamiltonian with $\mathcal{\hat{P}\hat{T}}$-symmetry rises above the classical limit of $2/3$ but remains below the fidelity through the corresponding Hermitian model. In contrast, in (b), it beats the performance via the Hermitian model, thereby establishing the advantage of non-Hermiticity. Note that the nH $XX$ model in this case lacks $\mathcal{\hat{P}\hat{T}}$-symmetry and the number of sites is odd. All the axes are dimensionless.
• Figure 5: Profile of entanglement $\mathcal{E}(t)$ (orange colored dashed-dot lines), and the corresponding fidelity $\mathcal{F}_N(t)$ (solid blue lines), with respect to time $t$. (a) Unbroken, and (b) broken regimes of the $XX$ Hamiltonian. The maxima(s) of both $\mathcal{F}_N(t)$ and $\mathcal{E}(t)$ are aligned when $\mathcal{F}_N (t) > \mathcal{F}_c$, thereby underscoring that the presence of entanglement is necessary for successful QST. Here, $N=16$ and all the axes are dimensionless.