Quantum State Engineering for Dissipative Quantum Computation Via a Two-Qubit System Plunged in a Global Squeezed Vacuum Field Reservoir

Abstract

Finding new strategies for preserving quantum resources is mandatory for reliable quantum technologies. Here, we address how to harness dissipation for quantum state engineering by considering a dissipative quantum channel consisting of a two-qubit plunged in a global reservoir. Despite the unwanted features of dissipation, we show that it can stabilize quantum resources and give rise to some key advantages for dissipative quantum state engineering and quantum computation. In particular, various classes of entangled states are generated by considering a general initial state for the two-qubit system. More importantly, our results demonstrate that preservation and enhancement of the quantum resources, i.e., coherence and entanglement may occur by adjusting the strength of the squeezed reservoir. Although the dissipation process may generally plague the coherence between two qubits, the squeezed reservoir can recover and enhance this quantity, especially, in the steady state regime. Besides, the stationary entangled states may be generated if a two-qubit system interacts with a global dissipative environment. In analogy with the universal quantum processor that produces unitary transformation, this dissipative two-qubit channel provides a well-established non-unitary transformation for universal quantum computation.

Appendices

Appendix A

$$\begin{aligned} M_{11}= \begin{bmatrix} 4a &{} 0 &{} 0 &{} 0 &{} 0 &{} -2b &{} -2b &{} 0\\ 0 &{} 3a+b &{} a+b &{} 0 &{} -2c &{} 0 &{} 0 &{} -2b\\ 0 &{} a+b &{} 3a+b &{} 0 &{} -2c &{} 0 &{} 0 &{} -2b\\ 2c &{} 0 &{} 0 &{} 2(a+b) &{} 0 &{} -2c &{} -2c &{} 0\\ 2c &{} -2c &{} -2c &{} 0 &{} 3a+b &{} 0 &{} 0 &{} 2c\\ -2a &{} 0 &{} 0 &{} -2c &{} 0 &{} 2(a+b) &{} a+b &{} 0\\ -2a &{} 0 &{} 0 &{} -2c &{} 0 &{} a+b &{} 2(a+b) &{} 0\\ 0 &{} 2c &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} a+3b\\ \end{bmatrix}, \end{aligned}$$

$$\begin{aligned} M_{12}= \begin{bmatrix} 0 &{} -2b &{} -2b &{} 0 &{} 4c &{} 0 &{} 0 &{} 0\\ -2c &{} 0 &{} 0 &{} -2b &{} 0 &{} 2c &{} 0 &{} 0\\ -2c &{} 0 &{} 0 &{} -2b &{} 0 &{} 0 &{} 2c &{} 0\\ 0 &{} -2c &{} -2c &{} 0 &{} 0 &{} 0 &{} 0 &{} 2c\\ a+b &{} 0 &{} 0 &{} 0 &{} 0 &{} -2b &{} -2b &{} 0\\ 0 &{} a+b &{} 0 &{} 0 &{} -2c &{} 0 &{} 0 &{} -2b\\ 0 &{} 0 &{} a+b &{} 0 &{} -2c &{} 0 &{} 0 &{} -2b\\ 0 &{} 0 &{} 0 &{} a+b &{} 0 &{} -2c &{} -2c &{} 0\\ \end{bmatrix}, \end{aligned}$$

$$\begin{aligned} M_{21}= \begin{bmatrix} 0 &{} -2c &{} -2c &{} 0 &{} a+b &{} 0 &{} 0 &{} 0\\ -2a &{} 0 &{} 0 &{} -2c &{} 0 &{} a+b &{} 0 &{} 0\\ -2a &{} 0 &{} 0 &{} -2c &{} 0 &{} 0 &{} a+b &{} -2c\\ 0 &{} -2a &{} -2a &{} 0 &{} 0 &{} 0 &{} 0 &{} a+b\\ 2c &{} 0 &{} 0 &{} 0 &{} 0 &{} -2c &{} -2c &{} 0\\ 0 &{} 2c &{} 0 &{} 0 &{} -2a &{} 0 &{} 0 &{} -2c\\ 0 &{} 0 &{} 2c &{} 0 &{} -2a &{} 0 &{} 0 &{} -2c\\ 0 &{} 0 &{} 0 &{} 4c &{} 0 &{} -2a &{} -2a &{} 0\\ \end{bmatrix}, \end{aligned}$$

$$\begin{aligned} M_{22}= \begin{bmatrix} 3a+b &{} 0 &{} 0 &{} 2c &{} 0 &{} -2b &{} -2b &{} 0\\ 0 &{} 2(a+b) &{} a+b &{} 0 &{} -2c &{} 0 &{} 0 &{} -2b\\ 2c &{} a+b &{} 2(a+b) &{} 0 &{} -2c &{} 0 &{} 0 &{} -2b\\ 2c &{} 0 &{} 0 &{} a+3b &{} 0 &{} -2c &{} -2c &{} 0\\ 0 &{} -2c &{} -2c &{} 0 &{} 2(a+b) &{} 0 &{} 0 &{} 2c\\ -2a &{} 0 &{} 0 &{} -2c &{} 0 &{} a+3b &{} a+b &{} 0\\ -2a &{} 0 &{} 0 &{} -2c &{} 0 &{} a+b &{} a+3b &{} 0\\ 0 &{} -2a &{} -2a &{} 0 &{} 0 &{} 0 &{} 0 &{} 4b\\ \end{bmatrix}, \end{aligned}$$

where we have defined the relations $a=\frac{\Gamma }{2}(\mathcal {N}+1)$, $b=\frac{\Gamma }{2}\mathcal {N}$ and $c=\frac{\Gamma }{2}\mathcal {M}$ and set $\theta =0$ for the sake of simplicity.

Appendix B

In order to solve the master equation (4), we assume the following inital state for the two-qubit system

$$\begin{aligned} \rho (0)=\begin{bmatrix} \rho _{11} &{} \rho _{12} &{} \rho _{13} &{} \rho _{14}\\ \rho _{21} &{} \rho _{22} &{} \rho _{23} &{} \rho _{24}\\ \rho _{31} &{} \rho _{32} &{} \rho _{33} &{} \rho _{34}\\ \rho _{41} &{} \rho _{42} &{} \rho _{43} &{} \rho _{44}\\ \end{bmatrix}. \end{aligned}$$

(21)

Considering small squeezing $r \ll 1$, one can set $b,c=0$ and find the following analytical expression for the density matrix of the two-qubi system

$$\begin{aligned} \rho _{11}(t)= & {} \rho _{11} e^{-4at}, \\ \nonumber \rho _{12}(t)= & {} \frac{1}{2}\rho _{12} (e^{-4at}+e^{-2at})+\frac{1}{2}\rho _{13} (e^{-4at}-e^{-2at}), \\ \nonumber \rho _{13}(t)= & {} \frac{1}{2}\rho _{12} (e^{-4at}-e^{-2at})+\frac{1}{2}\rho _{13} (e^{-4at}+e^{-2at}), \\ \nonumber \rho _{14}(t)= & {} \rho _{14} e^{-2at}, \\ \nonumber \rho _{22}(t)= & {} 2\rho _{11} e^{-4at}at+\frac{e^{-4at}}{4}(\rho _{22}+\rho _{23}+\rho _{32}+\rho _{33})+\frac{e^{-2at}}{2}(\rho _{22}-\rho _{33})\\ \nonumber{} & {} +\frac{1}{4}(\rho _{22}-\rho _{23}-\rho _{32}+\rho _{33}), \\ \nonumber \rho _{23}(t)= & {} 2\rho _{11} e^{-4at}at+\frac{e^{-4at}}{4}(\rho _{22}+\rho _{23}+\rho _{32}+\rho _{33})+\frac{e^{-2at}}{2}(\rho _{23}-\rho _{32}), \\ \nonumber \rho _{24}(t)= & {} -e^{-4at}(\rho _{12}+\rho _{13})+\frac{e^{-2at}}{2}(2\rho _{12}+2\rho _{13}+\rho _{24}+\rho _{34})+\frac{1}{2}(\rho _{24}-\rho _{34}), \\ \nonumber \rho _{33}(t)= & {} 2\rho _{11} e^{-4at}at+\frac{e^{-4at}}{4}(\rho _{22}+\rho _{23}+\rho _{32}+\rho _{33})-\frac{e^{-2at}}{2}(\rho _{22}-\rho _{33})\\ \nonumber{} & {} +\frac{1}{4}(\rho _{22}-\rho _{23}-\rho _{32}+\rho _{33}), \\ \nonumber \rho _{34}(t)= & {} -e^{-4at}(\rho _{12}+\rho _{13})+\frac{e^{-2at}}{2}(2\rho _{12}+2\rho _{13}+\rho _{24}+\rho _{34})-\frac{1}{2}(\rho _{24}-\rho _{34}), \\ \nonumber \rho _{44}(t)= & {} \rho _{11}(1-4e^{-4at}at)-e^{-4at}(\rho _{22}+\rho _{23}+\rho _{32}+\rho _{33})\\ \nonumber{} & {} +\frac{1}{2}(2\rho _{11}+\rho _{22}+\rho _{23}+\rho _{32}+\rho _{33}+2\rho _{44}). \end{aligned}$$

(22)

where $\rho _{ji}(t)= \rho _{ij}^*(t)$ and also $\rho _{11}(t)+\rho _{22}(t)+\rho _{33}(t)+\rho _{44}(t)=1$.