Composite control for protecting two nonorthogonal qubit states against decoherence

Quantum feedback control (QFBC) and quantum feedforward control (QFFC) are two of the major techniques for protecting two nonorthogonal qubit states against decoherence. In this paper, we propose a quantum composite control scheme for protecting such states, where QFBC and QFFC are combined. Note that the combination is deliberately devised, other than simply one plus one (i.e., the trivial combination). We show the performance of our scheme is better than that of both the previous control schemes and the trivially combined one for most states and noises in the sense that, for given success probability, our scheme can achieve the biggest fidelity, and for given fidelity, our scheme has the biggest success probability. Furthermore, our scheme can be experimentally implemented with current technologies.

Figure 1

Schematic of our composite control scheme for protecting two nonorthogonal qubit states.

Figure 2

Schematics of the QFBC, QFFC, QCC, and TCC schemes. $M, M_B$, and $N_m$ are measurements. $m, m_B$, and $n$ are corresponding sets of measurement results. $U_m$ and $R_n$ are operations, and $U_m^{-1}$ is the reversal of $U_m$. $N_m$ is a partial measurement (only part results are remained), which is almost the reversal of the measurement $M_m$. A control box (blue) is a process of performing a measurement and then making an operation based on the measurement result. A reversion (green) is to perform the reversion of the control box before noise, i.e., the reversed operation and measurement.

Figure 3

In exact protecting schemes, the fidelity $f^{{}^{\prime }}$ and the fidelity improvement ${\zeta }_f$ as a function of the success probability $g^{{}^{\prime }}$ and the angle $\theta$, whose unit is rad, for different noises (a) $r=0.3$, (b) $r=0.6$, and (c) $r=0.9$, with $\varphi =\frac{\pi }{2}$. The $Z$ axis represents the fidelity of our $\mathrm{scheme}{f}^{{}^{\prime }}$, and the color denotes the fidelity improvement ${\zeta }_f$ of our scheme over the QFFC scheme.

Figure 4

In optimal protecting schemes, the fidelity $F^{\mathrm{dif}}$ as a function of the noise $r$ and the angle $\theta$, the unit of which is rad, for different initial phases (a) $\varphi =0$, (b) $\varphi =\frac{\pi }{4}$, and (c) $\varphi =\frac{\pi }{2}$; and the corresponding fidelity of our QCC scheme ${\left(f\right|g=1)}^{\mathrm{opt}}$ for (d) $\varphi =0$, (e) $\varphi =\frac{\pi }{4}$, and (f) $\varphi =\frac{\pi }{2}$.

Figure 5

The optimal success probability $g$ as a function of the fidelity $f$ in optimal protecting schemes with given $\varphi =\frac{\pi }{2}, \theta =1.1310$, and $r=0.82$ by the QCC scheme (solid red curve with “$\times$” markers), the TCC one (solid blue curve with “$•$” markers), the QFFC one (solid purple curve with “$\circ$” markers), the QFBC one (dash-dotted green line), the Helstrom one (dashed yellow line) and the DN one (solid black line).

Figure 6

Bloch vectors of the original states $|{\mathrm{\Psi }}_{+}\rangle$ and $|{\mathrm{\Psi }}_{-}\rangle$ (solid black arrows), the parts of states after the feedforward control process (dashed red arrows for $|{\mathrm{\Psi }}_{+}\rangle$, dashed green arrows for state $|{\mathrm{\Psi }}_{-}\rangle$), the states after the feedforward control process ${\tilde{\rho }}_N^{+}$ (dash-dotted red arrow) and ${\tilde{\rho }}_N^{-}$ (dash-dotted green arrow), and the states after our scheme ${\tilde{\rho }}_{\mathrm{fin}}^{+}$ (dash-double-dotted red arrow) and ${\tilde{\rho }}_{\mathrm{fin}}^{-}$ (dash-double-dotted green arrow).