Figure 1

(Color online) Schematic setup: three superconducting qubits (indicated by boxes between center conductor and ground planes) are coupled to a mode of a coplanar microwave resonator (consisting of a center conductor and two neighboring ground planes). The measurement of the phase shift of a transmitted microwave beam can be utilized to rapidly synthesize, e.g., maximally entangled multiqubit states such as GHZ and $W$ states.Reuse & Permissions

Figure 2

(Color online) Generation of the Bell state $|{\Psi }^{+}⟩=\left(|10⟩+|01⟩\right)/\sqrt{2}$ (the two-qubit $W$ state): (a) quantum trajectories illustrating the different phase-shift signal traces $X\left(t\right)$. Three traces have been selected, corresponding to the possible outcomes of the measurement given the same input state $|{\Psi }_0⟩$. At time $\overline{\Gamma }{t}_0=3$, Hadamard gates are applied to both qubits, starting from the ground state. As in every real measurement of field quadratures, the signal $X\left(t\right)$ is smoothed by doing a windowed average over a suitable time span, $\overline{\Gamma }{\tau }_{avg}=1.0$. Part (b) displays the excitation numbers, state synthesis fidelity and the entanglement (log negativity) for the one trajectory of plot (a) that ended up in the desired state.Reuse & Permissions

Figure 3

(Color online) Generation of two-qubit Bell state $|{\Phi }^{+}⟩=\left(|00⟩+|11⟩\right)/\sqrt{2}$: (a) Probability density of the integrated (cumulative) phase-shift signal $\overline{X}\left(t\right)=t^{-1}{\int }_0^{t}X\left(t^{\prime }\right)d{t}^{\prime }$ from 6000 runs of the simulation. At time $t_0$ Hadamard gates are applied to both qubits. Part (b) displays the probability density of the entanglement measure $E_N$, the log negativity. Note that neither of these plots can be obtained from the standard, nonstochastic master equation (see main text), i.e., quantum jump trajectory simulations are essential.Reuse & Permissions

Figure 4

(Color online) Generation of three-qubit $W$ states: (a) quantum trajectories for the different states that can arise from the given input state $|{\Psi }_0⟩$. At time $\overline{\Gamma }{t}_0=3$, Hadamard gates are applied to all qubits. Windowed averaging is performed as in Fig. 2. Part (b) displays the excitation numbers, state synthesis fidelity and the log negativity for the one trajectory of plot (a) that ended up in the desired $W$ state. Here ${\text{tr}}_i\rho$ denotes the partial trace over qubit number $i$, and the resulting pairwise entanglement happens to be the same for all choices of qubit pairs in this example. Note that in the target state all pairs of qubits are mutually entangled which is characteristic for the $W$ state and the reason for the robustness of its entanglement compared to the GHZ state.Reuse & Permissions

Figure 5

(Color online) generation of three-qubit GHZ states: (a) quantum trajectories corresponding to the different states that can arise from the given input state. At time $t_0$, Hadamard gates are applied to both qubits. Note that among the unwanted outcomes there are two-qubit $|{\Psi }^{+}⟩$ Bell states. These are actually generated with a success rate of $\eta =3/4$ which is higher than in the original two-qubit scheme. Part (b) displays the excitation numbers, state synthesis fidelity and the log negativity for all pairs of qubits for the trace of part (a) that ended up in the desired GHZ state. Note that once the GHZ state is reached, all pairwise entanglement is lost. This is a typical feature of GHZ states, which contain only genuine three-particle entanglement. Part (c) shows the evolution for the particular trajectory that reaches the Bell state between qubits 2 and 3, which can be generated very efficiently as a by-product using this three-qubit GHZ scheme.Reuse & Permissions

Figure 6

(Color online) Effects of adding decoherence to the dynamics. The situation is identical to the simulation of Fig. 2, with the target state $|{\Psi }^{+}⟩$, except for the added relaxation rate ${\gamma }_1=0.01\overline{\Gamma }$ and pure dephasing rate ${\gamma }_{\varphi }=0.02\Gamma$. We can observe that the subspace of choice is stabilized before the eventual decay due to relaxation. However, even before the sudden jump due to relaxation, one observes a slow decay of the fidelity and entanglement between the qubits, due to the pure dephasing rate ${\gamma }_{\varphi }$ (dashed lines).Reuse & Permissions

Figure 7

(Color online) Effects of adding decoherence and relaxation to the creation of the Bell state $|{\Phi }^{+}⟩\equiv \frac{1}{\sqrt{2}}\left(|00⟩+|11⟩\right)$. The situation is identical to the simulation of Fig. 3, except for the added relaxation rate ${\gamma }_1=0.1\overline{\Gamma }$ and pure dephasing rate ${\gamma }_{\varphi }=0.05\Gamma$. Part (a) shows the probability density of the time-averaged (cumulative) phase-shift signal $\overline{X}\left(t\right)$ with an example trajectory superimposed. Note the buildup of finite probability at finite signal values, before relaxation back to zero phase shift, which represents the vacuum state $|00⟩$ at long times. Part (b) shows the probability density of the entanglement (log negativity). We can observe that the entanglement is lost on a time scale given by $T_2={\left({\gamma }_1/2+{\gamma }_2\right)}^{-1}$. Note in particular the sharply defined, exponentially decaying envelope that defines a strict upper bound for the entanglement at any given time. This is due to the pure dephasing.Reuse & Permissions

Figure 8

(Color online) Probability density of the time-averaged cumulative (integrated) phase-shift signal $\overline{X}$ as an illustration of the effect of parameter spread in the couplings. Apart from the deviation in the couplings from the ideal values, the setup is identical to the example in which we aimed for a three-qubit $W$ state, as seen in Fig. 4. Hadamard gates are applied to all qubits at time $t_0=3{\overline{\Gamma }}^{-1}$. During the following time interval of length ${\overline{\Gamma }}^{-1}$ all trajectories are projected onto the $W$ state, $|\overline{W}⟩$, $|000⟩$, or $|111⟩$. Meanwhile the competing projection on the individual number states of the qubits becomes more pronounced and dominates the dynamics on a time scale $\delta {\Gamma }_2^{-1}=50{\overline{\Gamma }}^{-1}$. This is exactly the time scale on which we can be sure to identify all the product base states by their phase-shift values individually.Reuse & Permissions