Efficient quantum filtering for quantum feedback control

We discuss an efficient numerical scheme for the recursive filtering of diffusive quantum stochastic master equations. We show that the resultant quantum trajectory is robust and may be used for feedback based on inefficient measurements. The proposed numerical scheme is amenable to approximation, which can be used to further reduce the computational burden associated with calculating quantum trajectories and may allow real-time quantum filtering. We provide a two-qubit example where feedback control of entanglement may be within the scope of current experimental systems.

Figure 1

(a) Average purity for the initial pure state (solid black line) and initial completely mixed state (dashed black line) calculated using the Euler-Milstein method with 5000 steps per cycle. (b) The $1-\mathrm{average}$ fidelity, where the fidelity is calculated between an initial pure state with 5000 steps per cycle and an initial completely mixed state using the Euler-Milstein method with 1000 steps [(blue) crosses], 500 steps [(red) circles], and 250 steps [(green) asterisks] per cycle. (c) The $1-\mathrm{average}$ fidelity, where the fidelity is calculated between an initial pure state with 5000 steps per cycle and an initial completely mixed state using the proposed method with 250 steps [(blue) crosses], 100 steps [(red) circles], and 50 steps [(green) asterisks] per cycle;- all values averaged over 1000 realizations.

Figure 2

(a) Average concurrence (dashed lines) and average negativity (solid lines) for the three control strategies for a completely mixed initial state, averaged over 500 realizations and 250 steps per cycle using the proposed integration scheme: two-qubit $YI$ and $IY$ controls [(blue) crosses], one-qubit $XI$ controls [(red) circles], and no controls, i.e., free rotation about the qubit $X$ axes [(green) asterisks]. (b) Concurrence (dashed lines) and negativity (solid lines) for one realization of the three control strategies in (a).

Figure 3

Average concurrence (dashed lines) and average negativity (solid lines) for the two-qubit controls rotating towards the $YI/IY$ [(blue) crosses], $XI/IY$ [(red) circles], and $XI/IX$ [(green) asterisks] axes as functions of the Hamiltonian rotation (angular) frequency $\omega /{\omega }_0$, after 50 cycles (${\omega }_0$) averaged over 500 realizations and 250 steps per cycle using the proposed integration scheme.

Figure 4

(a) Average concurrence for the two-qubit controls rotating towards $YI/IY$ for $\omega ={\omega }_0$ using the approximate update operator, (10), with 500 steps per cycle (solid black line), 50 steps per cycle [(blue) crosses], 20 steps per cycle [(red) circles], 10 steps per cycle [(green) asterisks], and 5 steps per cycle [(purple) diamonds]. (b) The $1-\mathrm{average}$ fidelity values corresponding to the concurrence values in (a); all values averaged over 500 realizations, with fidelities compared against solutions using the proposed integration scheme with 500 steps per cycle.

Figure 5

(a) Average concurrence for the two-qubit controls rotating towards $YI/IY$ for $\omega ={\omega }_0$ with 500 steps per cycle (solid black line) using a full measurement record and 50 steps per cycle using a measurement record truncated to 6 bits per measurement [(blue) crosses], 4 bits per measurement [(red) circles], 3 bits per measurement [(green) asterisks], and 2 bits per measurement [(purple) diamonds]. (b) The $1-\mathrm{average}$ fidelity values corresponding to the concurrence values in (a); all values averaged over 500 realizations, with fidelities compared against solutions using the proposed integration scheme with 500 steps per cycle.