Real-time deterministic generation of maximally entangled two-qubit and three-qubit states via bang-bang control

We exploit the notion of SWM- (simultaneous-weak-measurements)-induced quantum state reduction, enabling the use of measurement backaction to produce the desired quantum states. The probabilistic generation of maximally entangled Bell states and the $|\mathrm{GHZ}\rangle$ (Greenberger-Horne-Zeilinger) state are then performed as a consequence of this notion. By combining SWM-induced quantum state reduction with feedback control, we are able to deterministically generate any desired Bell state as well as the $|\mathrm{GHZ}\rangle$ state, without the knowledge of the initial state. The control-computation time is compensated for by the delay time in the time-delay bang-bang control, through which the real-time implementation of the control scheme is guaranteed. The effectiveness of the proposed scheme is illustrated numerically.

Figure 1

Illustration of the SWM-induced space ${\Phi }_{\mathrm{SWM}}$, which is the attractive set of filter states under the SWMs of commutative observables $A_1,A_2,...$, and $A_m$.

Figure 2

The setup for estimate feedback control of two atoms. Two cavities $C_1$ and $C_2,$ each of which contains a two-level atom, are connected in a closed loop through optical fibers. The off-resonant driving field $A_c$ generates an effective Hamiltonian $H_0$. The optical fields are measured continuously by the homodyne detectors $D=\{D_1,...,D_m\}$. The measurement records $y_t=[y_{1t},...,y_{mt}]$ are sent to a filter to extract the information of the system, and the filter state (estimate state) ${\rho }_t$ is then fed back via the controller $u_t$ and magnetic fields $L_1,L_2$ to modify the system Hamiltonian.

Figure 3

SWM-induced quantum state reduction of four arbitrary sample paths. (a) Concurrence. (b) Distance from ${\rho }_t$ to ${\rho }_d$. The time is in the units such that $\hslash =1$.

Figure 4

SWM-induced quantum state reduction on average over 100 sample paths. (a) Average concurrence. (b) Average distance. The time is in the units such that $\hslash =1$.

Figure 5

SWM-induced quantum state reduction of four arbitrary sample paths under bang-bang control. (a) Concurrence. (b) Distance from ${\rho }_t$ to ${\rho }_d$. (c) Time-delay control input $u\left({\rho }_{t-\tau }\right)$ with $\tau =0.25$ and $\gamma =0.2$. The time is in the units such that $\hslash =1$.

Figure 6

SWM-induced quantum state reduction on an average of over 100 sample paths under bang-bang control. (a) Average concurrence. (b) Average distance. The time is in the units such that $\hslash =1$.