Interconversion of $W$ and Greenberger-Horne-Zeilinger states for Ising-coupled qubits with transverse global control

Interconversions of $W$ and Greenberger-Horne-Zeilinger states in various physical systems have recently been attracting considerable attention. We address this problem in the fairly general physical setting of qubit arrays with long-ranged (all-to-all) Ising-type qubit-qubit interaction, which are simultaneously acted upon by transverse Zeeman-type global control fields. Motivated in part by a recent Lie-algebraic result that implies state-to-state controllability of such a system for an arbitrary pair of states that are invariant with respect to qubit permutations, we present a detailed investigation of the state-interconversion problem in the three-qubit case. The envisioned interconversion protocol has the form of a pulse sequence that consists of two instantaneous ($\delta$-shaped) control pulses, each of them corresponding to a global qubit rotation, and an Ising-interaction pulse of finite duration between them. Its construction relies heavily on the use of the (four-dimensional) permutation-invariant subspace (symmetric sector) of the three-qubit Hilbert space. In order to demonstrate the viability of the proposed state-interconversion scheme, we provide a detailed analysis of the robustness of the underlying pulse sequence to systematic errors, i.e., deviations from the optimal values of its five characteristic parameters.

Figure 1

Pictorial illustration of the pulse sequence for realizing the $W$-to-GHZ state conversion, which consists of two instantaneous control pulses and an Ising-interaction pulse of finite duration $T$ between them. The first (second) control pulse is characterized by a vector in the $x-y$ plane, with the magnitude ${\alpha }_1$ (${\alpha }_2$) and polar angle ${\varphi }_1$ (${\varphi }_2$). Here $U_C({\alpha }_1,{\varphi }_1)$ and $U_C({\alpha }_2,{\varphi }_2)$ are the time-evolution operators corresponding to the control pulses at $t=0$ and $t=T$, respectively; $U_{ZZ}\left(\xi \right)$ corresponds to the interaction pulse, with $\xi \equiv JT$ being its dimensionless duration.

Figure 2

Pictorial illustration of the three branches of optimal values ${\varphi }_{1,0}$ and ${\varphi }_{2,0}$ of the angles that specify the directions of the global-rotation axes corresponding to the control pulses.

Figure 3

Deviation of the GHZ-state fidelity from unity (i.e., the infidelity) $1-{\mathcal{F}}_{\text{GHZ}}$ as a function of errors ${\varepsilon }_p$ in the values of the parameters that characterize the pulse sequence realizing the $W$-to-GHZ state conversion.

Figure 4

Comparison of the dependence of the infidelity $1-{\mathcal{F}}_{\text{GHZ}}$ on ${\varepsilon }_{{\varphi }_1}$ and ${\varepsilon }_{{\varphi }_2}$ for fixed and arbitrary values of $\phi$.

Figure 5

Deviation of the GHZ-state fidelity from unity (i.e., the infidelity) $1-{\mathcal{F}}_{\text{GHZ}}$ as a function of (a) errors in the parameters ${\varphi }_1$ and ${\varphi }_2$, i.e., deviations from their respective optimal values ${\varphi }_{1,0}=5\pi /6$ and ${\varphi }_{2,0}=\pi /3$, and (b) errors in the parameters ${\alpha }_1$ and ${\alpha }_2$, i.e., deviations from their respective optimal values ${\alpha }_{1,0}=\pi /4$ and ${\alpha }_{2,0}=arccos(1/3)/4$.

Figure 6

Deviation of the GHZ-state fidelity from unity (i.e., the infidelity) $1-{\mathcal{F}}_{\text{GHZ}}$ as a function of (a) errors in the Ising-pulse duration ${\varepsilon }_{\xi }$ and the angles ${\varepsilon }_{{\varphi }_1}={\varepsilon }_{{\varphi }_2}\equiv {\varepsilon }_{\varphi }$ specifying the two rotation axes and (b) errors in ${\varepsilon }_{\xi }$ and the two rotation angles ${\varepsilon }_{{\alpha }_1}={\varepsilon }_{{\alpha }_2}\equiv {\varepsilon }_{\alpha }$ corresponding to the instantaneous control pulses.