Deterministic conversions between Greenberger-Horne-Zeilinger states and $W$ states of spin qubits via Lie-transform-based inverse Hamiltonian engineering

In this paper, we propose a protocol to realize the conversions between Greenberger-Horne-Zeilinger (GHZ) states and $W$ states of spin qubits. By analyzing and simplifying the dynamics of the system, the control fields are designed via the inverse Hamiltonian engineering based on the Lie transforms. Moreover, the states' conversions between multitype GHZ states and $W$ states can be executed deterministically and reversibly, which makes the protocol resource saving and flexible. We show in numerical simulations that the state conversions are robust against the systematic errors, random noise, and frequency mismatching of the control fields. Therefore, the protocol may provide some useful prospectives to the research of quantum mechanics and applications of quantum information processing based on GHZ states and $W$ states.

Figure 1

(a) ${\mathrm{\Gamma }}_3/\pi$ versus ${\mathrm{\Gamma }}_1/\pi$. (b) ${\mathrm{\Omega }}_1\left(t\right), {\mathrm{\Omega }}_2\left(t\right)$, and ${\mathrm{\Omega }}_3\left(t\right)$ versus $t/T$ for $t\in [0,T]$. (c) The population $P_W^{\mathrm{eff}}\left(t\right) \left[P_W^{\mathrm{full}}\left(t\right)\right]$ of $W$ state $|W\rangle$ and the population $P_{+}^{\mathrm{eff}}\left(t\right) \left[P_{+}^{\mathrm{full}}\left(t\right)\right]$ of GHZ state $|{\text{GHZ}}_{+}\rangle$ with effective (full) Hamiltonian $H_e\left(t\right) \left[H\right(t\left)\right]$ versus $t/T$, where $J=50/T$ is set for the full Hamiltonian $H\left(t\right)$. The red solid line: $P_W^{\mathrm{eff}}\left(t\right)$; the blue dashed-dotted line: $P_W^{\mathrm{full}}\left(t\right)$; the green dashed line: $P_{+}^{\mathrm{eff}}\left(t\right)$; and the purple dotted line: $P_{+}^{\mathrm{full}}\left(t\right)$. (d) The red solid line: The infidelity of the conversion from $|{\text{GHZ}}_{+}\rangle$ to $|W\rangle$ versus $J$; the blue dashed line: the infidelity of the conversion from $|W\rangle$ to $|{\text{GHZ}}_{+}\rangle$ versus $J$.

Figure 2

(a) ${\mathrm{\Omega }}_1\left(t\right),{\mathrm{\Omega }}_2\left(t\right)$, and ${\mathrm{\Omega }}_3\left(t\right)$ versus $t/T$ for $t\in [0,T]$. (b) The population $P_{W^{\prime }}^{\mathrm{eff}}\left(t\right) \left[P_{W^{\prime }}^{\mathrm{full}}\left(t\right)\right]$ of W state $|W^{\prime }\rangle$ and the population $P_{+}^{\mathrm{eff}}\left(t\right)$ ($P_{+}^{\mathrm{full}}\left(t\right)$) of GHZ state $|{\text{GHZ}}_{+}\rangle$ with effective (full) Hamiltonian $H_e\left(t\right) \left[H\right(t\left)\right]$ versus $t/T$, where $J=50/T$ is set for the full Hamiltonian $H\left(t\right)$. The red solid line: $P_{W^{\prime }}^{\mathrm{eff}}\left(t\right)$; the blue dashed-dotted line: $P_{W^{\prime }}^{\mathrm{full}}\left(t\right)$; the green dashed line: $P_{+}^{\mathrm{eff}}\left(t\right)$; and the purple dotted line: $P_{+}^{\mathrm{full}}\left(t\right)$.

Figure 3

(a) ${\mathrm{\Omega }}_1\left(t\right),{\mathrm{\Omega }}_2\left(t\right)$, and ${\mathrm{\Omega }}_3\left(t\right)$ versus $t/T$ for $t\in [0,T]$. (b) The population $P_W^{\mathrm{eff}}\left(t\right) \left[P_W^{\mathrm{full}}\left(t\right)\right]$ of $W$ state $|W\rangle$ and the population $P_{-}^{\mathrm{eff}}\left(t\right) \left[P_{-}^{\mathrm{full}}\left(t\right)\right]$ of GHZ state $|{\text{GHZ}}_{-}\rangle$ with effective (full) Hamiltonian $H_e\left(t\right) \left[H\right(t\left)\right]$ versus $t/T$, where $J=50/T$ is set for the full Hamiltonian $H\left(t\right)$. The red solid line: $P_W^{\mathrm{eff}}\left(t\right)$; the blue dashed-dotted line: $P_W^{\mathrm{full}}\left(t\right)$; the green dashed line: $P_{-}^{\mathrm{eff}}\left(t\right)$; and the purple dotted line: $P_{-}^{\mathrm{full}}\left(t\right)$.

Figure 4

(a) The fidelity of the conversion from $|{\text{GHZ}}_{+}\rangle$ to $|W\rangle$ versus ${\eta }_j$ ($j=1,2,3$). (b) The fidelity of the conversion from $|{\text{GHZ}}_{+}\rangle$ to $|W^{\prime }\rangle$ versus ${\eta }_j$.

Figure 5

(a) The infidelity of the conversion from $|{\text{GHZ}}_{+}\rangle$ to $|W\rangle$ under AWGN versus simulation counts. (b) The infidelity of the conversion from $|{\text{GHZ}}_{+}\rangle$ to $|W^{\prime }\rangle$ under AWGN versus simulation counts.

Figure 6

(a) The fidelity of the conversion from $|{\text{GHZ}}_{+}\rangle$ to $|W\rangle$ versus ${\mathrm{\Delta }}_1/{\mathrm{\Omega }}_{max}$ and ${\mathrm{\Delta }}_3/{\mathrm{\Omega }}_{max}$. (b) The fidelity of the conversion from $|{\text{GHZ}}_{+}\rangle$ to $|W^{\prime }\rangle$ versus ${\mathrm{\Delta }}_1/{\mathrm{\Omega }}_{max}$ and ${\mathrm{\Delta }}_3/{\mathrm{\Omega }}_{max}$.

Figure 7

(a) The fidelity of the conversion from $|{\text{GHZ}}_{+}\rangle$ to $|W\rangle$ versus $t/T$ with AWGN and ${\kappa }_j=1$. (${\mathrm{b}}_1$) ${\mathrm{\Omega }}_1^{\text{no}}\left(t\right)$ and $\mathrm{awgn}({\mathrm{\Omega }}_1\left(t\right),1)$ versus $t/T$. (${\mathrm{b}}_2$) ${\mathrm{\Omega }}_2^{\text{no}}\left(t\right)$ and $\mathrm{awgn}({\mathrm{\Omega }}_2\left(t\right),1)$ versus $t/T$. (${\mathrm{b}}_3$) ${\mathrm{\Omega }}_3^{\text{no}}\left(t\right)$ and $\mathrm{awgn}({\mathrm{\Omega }}_3\left(t\right),1)$ versus $t/T$.