Autonomous multipartite entanglement engines

The generation of genuine multipartite entangled states is challenging in practice. Here we explore an alternative route to this task, via autonomous entanglement engines which use only incoherent coupling to thermal baths and time-independent interactions. We present a general machine architecture, which allows for the generation of a broad range of multipartite entangled states in a heralded manner. Specifically, given a target multiple-qubit state, we give a sufficient condition ensuring that it can be generated by our machine. We discuss the cases of Greenberger-Horne-Zeilinger, Dicke, and cluster states in detail. These results demonstrate the potential of purely thermal resources for creating multipartite entangled states useful for quantum information processing.

Figure 1

Autonomous thermal machine for the generation of $N$-qubit GHZ states. One qutrit is coupled to a hot thermal bath, while $N-1$ qutrits are coupled to cold thermal baths at equal temperatures. The energy-level structure is such that transitions in the hot qutrit are resonant with collective transitions of the cold qutrits, as indicated by arrows. All the cold systems have the same structure, i.e., ${\mathrm{\Delta }}_k^{\left(1\right)}={\mathrm{\Delta }}_c^{\left(1\right)}$ and ${\mathrm{\Delta }}_k^{\left(2\right)}={\mathrm{\Delta }}_c^{\left(2\right)}$ for $k=2,...,N$, and ${\mathrm{\Delta }}_c^{\left(1\right)}=({\mathrm{\Delta }}_h^{\left(2\right)}-{\mathrm{\Delta }}_h^{\left(1\right)})/(N-1)$ and ${\mathrm{\Delta }}_c^{\left(2\right)}={\mathrm{\Delta }}_h^{\left(2\right)}/(N-1)$. Local filters project the qutrits onto the qubit subspaces enclosed in dashed, gray boxes.

Figure 2

Fidelity of the generated state with the GHZ state vs the probability of successful filtering for different numbers of qutrits with one hot bath (solid lines) and two hot baths (dashed line). The curves are obtained by numerical optimization over the coupling parameters under the constraint $g,{\gamma }_k\le {10}^{-2}{\mathrm{\Delta }}_{\text{min}}$ where ${\mathrm{\Delta }}_{\text{min}}$ is the smallest energy gap in each case.

Figure 3

Fidelity of the filtered state with the GHZ state vs the bath temperatures, for $N=3$ and $g=1.6\times {10}^{-3}$, ${\gamma }_h={10}^{-4}$, ${\gamma }_c=5\times {10}^{-3}$, ${\mathrm{\Delta }}^{\left(1\right)}=1$, ${\mathrm{\Delta }}^{\left(2\right)}=2.5$. The units are dimensionless (Boltzmann's constant $k_B=1$).

Figure 4

Fidelity vs the filtering success probability for generation of $W$ states using one and two hot baths (solid) and cluster states using one hot bath (dashed). The results are obtained by constrained optimization over ${\gamma }_h,{\gamma }_c,g\le {10}^{-2}{\mathrm{\Delta }}_{\text{min}}$, where ${\mathrm{\Delta }}_{\text{min}}$ is the smallest energy gap in each case.

Figure 5

Flow diagram for population entering and leaving the state $|o\rangle$. Hot resets take the system from $|o\rangle$ to state $|o^{\prime }\rangle$ or $|\overline{\mathbf{n}}\rangle$, while cold resets take it to other states outside the support $S_{\overline{\psi }}$. The transition rates due to hot and cold resets are indicated.

Figure 6

Flow diagram for population entering and leaving the sets of states $S_k$ with $k$ cold qubits excited. The rates per state in the set of origin are indicated.

Figure 7

Nonlocality vs filtering success probability for $N=2,3,4$ in a GHZ machine with one hot system and $N-1$ cold systems. The results are obtained numerically by optimizing over ${\gamma }_h,g\le {10}^{-2}{\mathrm{\Delta }}_{\text{min}}$.

Figure 8

Nonlocality vs filtering success probability for the cluster state machine. The results are obtained by constrained optimization over ${\gamma }_h,{\gamma }_c,g\le {10}^{-2}{\mathrm{\Delta }}_{\text{min}}$.

Figure 9

Fidelity of the filtered state with the GHZ state vs the bath temperatures when using the Lindblad-type master equation (E1). The plot is for $N=3$ parties and the parameter settings are ${\mathrm{\Gamma }}_1={10}^{-4}$, ${\mathrm{\Gamma }}_2={\mathrm{\Gamma }}_3=5\times {10}^{-3}$, $g=1.6\times {10}^{-3}$, ${\mathrm{\Delta }}^{\left(1\right)}=1$ ${\mathrm{\Delta }}^{\left(2\right)}=2.5$.