Generation of quantum entangled states of multiple groups of qubits distributed in multiple cavities

Provided that cavities are initially in a Greenberger-Horne-Zeilinger (GHZ) entangled state, we show that GHZ states of $N$-group qubits distributed in $N$ cavities can be created via a three-step operation. The GHZ states of the $N$-group qubits are generated by using $N$-group qutrits placed in the $N$ cavities. Here, “qutrit” refers to a three-level quantum system with the two lowest levels representing a qubit while the third level acting as an intermediate state necessary for the GHZ state creation. This proposal does not depend on the architecture of the cavity-based quantum network and the way for coupling the cavities. The operation time is independent of the number of qubits. The GHZ states are prepared deterministically because no measurement on the states of qutrits or cavities is needed. In addition, the third energy level of the qutrits during the entire operation is virtually excited and thus decoherence from higher energy levels is greatly suppressed. This proposal is quite general and can in principle be applied to create GHZ states of many qubits using different types of physical qutrits (e.g., atoms, quantum dots, NV centers, various superconducting qutrits, etc.) distributed in multiple cavities. As a specific example, we further discuss the experimental feasibility of preparing a GHZ state of four-group transmon qubits (each group consisting of three qubits) distributed in four one-dimensional transmission line resonators arranged in an array.

Figure 1

(a) 1D cavity-based quantum network. (b) 2D cavity-based quantum network. (c) 3D cavity-based quantum network. In (a)–(c), each short line represents an optical fiber or other auxiliary system, which is used to couple two adjacent cavities. In addition, each cavity is a 1D or 3D cavity, hosting one group of qutrits (red dots).

Figure 2

(a) Illustration of the dispersive interaction between cavity $l$ and the $\left|e\right.〉\leftrightarrow \left|f\right.〉$ transition of qutrits $\left\{1_l,2_l,...,{\left(m-1\right)}_l\right\}$, with coupling constant $g_l$ and detuning ${\mathrm{\Delta }}_l={\omega }_{fe}-{\omega }_{c_l}>0$. Here, ${\omega }_{fe}$ is the $\left|e\right.〉\leftrightarrow \left|f\right.〉$ transition frequency of the qutrits and ${\omega }_{c_l}$ is the frequency of cavity $l$. (b) Illustration of the resonant interaction between cavity $l$ and the $\left|g\right.〉\leftrightarrow \left|e\right.〉$ transition of qutrit $m_l$ with coupling constant $g_{r,l}$. (c) Illustration of the resonant interaction between a classical pulse and the $\left|g\right.〉\leftrightarrow \left|e\right.〉$ transition of qutrits $\left\{1_l,2_l,...,{\left(m-1\right)}_l\right\}$ in cavity $l$. Note that the level structures in (a), (b), and (c) are different. The level spacings of qutrits in (a) are adjusted such that $\left|e\right.〉\leftrightarrow \left|f\right.〉$ transition is dispersively coupled to cavity $l$. The level spacings in (b) are adjusted such that the $\left|g\right.〉\leftrightarrow \left|e\right.〉$ transition is resonant with cavity $l$. The level spacings in (c) are adjusted such that qutrits are decoupled from cavity $l$ during the pulse. A blue double-arrow vertical line in (a) and (b) represents the frequency of cavity $l$, while a blue double-arrow vertical line in (c) represents the pulse frequency.

Figure 3

(a) Sequence of operations for step 1. (b) Sequence of operations for step 2. (c) Sequence of operations for step 3. Here, ${\tau }_1$ and ${\tau }_2$ are the qutrit-cavity interaction times, while ${\tau }_3$ is the qutrit-pulse interaction time, as described in the text. In addition, ${\tau }_a$ is the typical time required to adjust the qutrit level spacings. Note that the operation sequence in (a)–(c) follows from left to right.

Figure 4

1D quantum network consisting of four one-dimensional transmission line resonators (TLRs) arranged in an array. Each TLR hosts three SC transmon qutrits (red dots), and adjacent TLRs are coupled through SC transmon qutrits ($q_1,q_2,q_3$).

Figure 5

(a) Dispersive interaction between cavity $l$ and the $\left|e\right.〉\leftrightarrow \left|f\right.〉$ transition of qutrits $\left\{1_l,2_l\right\}$ with coupling strength $g_l$ and detuning ${\mathrm{\Delta }}_l={\omega }_{fe}-{\omega }_{c_l}>0$, as well as the unwanted off-resonant interaction between cavity $l$ and the $\left|g\right.〉\leftrightarrow \left|e\right.〉$ transition of qutrits $\left\{1_l,2_l\right\}$ with coupling strength ${\tilde{g}}_l$ and detuning ${\tilde{\mathrm{\Delta }}}_l={\omega }_{eg}-{\omega }_{c_l}>0$. (b) Resonant interaction between cavity $l$ and the $\left|g\right.〉\leftrightarrow \left|e\right.〉$ transition of qutrit $3_l$ with coupling constant $g_{r,l}$, as well as the unwanted off-resonant interaction between cavity $l$ and the $\left|e\right.〉\leftrightarrow \left|f\right.〉$ transition of qutrit $3_l$ with coupling constant ${\tilde{g}}_{r,l}$ and detuning ${\mathrm{\Delta }}_{r,l}$. (c) Resonant interaction between a classical pulse and the $\left|g\right.〉\leftrightarrow \left|e\right.〉$ transition of qutrits $\left\{1_l,2_l\right\}$ with Rabi frequency ${\mathrm{\Omega }}_l$, as well as the unwanted off-resonant interaction between the pulse and the $\left|e\right.〉\leftrightarrow \left|f\right.〉$ transition of qutrits $\left\{1_l,2_l\right\}$ with Rabi frequency ${\tilde{\mathrm{\Omega }}}_l$ and detuning ${\mathrm{\Delta }}_p={\omega }_{fe}-{\omega }_p$. Here, ${\omega }_p$ is the pulse frequency.

Figure 6

Fidelity versus $g_1$. The parameters used in the numerical simulation are referred to the text.

Figure 7

Fidelity versus ${\kappa }^{-1}$ for $g_1/2\pi =14.15\mathrm{MHz}$ and ${\mathrm{\Omega }}_l/2\pi =45\mathrm{MHz}$. Other parameters used in the numerical simulation are the same as those used in Fig. 6.