Operational Determination of Multiqubit Entanglement Classes via Tuning of Local Operations

We present a physical setup with which it is possible to produce arbitrary symmetric long-lived multiqubit entangled states in the internal ground levels of photon emitters, including the paradigmatic Greenberger-Horne-Zeilinger and $W$ states. In the case of three emitters, where each tripartite entangled state belongs to one of two well-defined entanglement classes, we prove a one-to-one correspondence between well-defined sets of experimental parameters, i.e., locally tunable polarizer orientations, and multiqubit entanglement classes inside the symmetric subspace.

Figure 1

Proposed experimental arrangement. $N$ excited emitters are aligned in a row, each of them defining a three-level $\Lambda$ system. A long-lived entangled state is obtained in the $N$ emitter qubits after detecting the $N$ spontaneously emitted photons with $N$ detectors equipped with polarizers. The final $N$-qubit state is tuned and determined by the polarizer orientations.

Figure 2

Pyramid of entanglement paths for the case of 3 emitters initialized in the excited state $|e,e,e⟩$. The figure illustrates the intermediate states, horizontally displayed, during three successive photodetection events realized by three detectors $d_1$, $d_2$, and $d_3$, equipped with general elliptical polarizers oriented along ${\mathit{ϵ}}_1$, ${\mathit{ϵ}}_2$, and ${\mathit{ϵ}}_3$ (${\mathit{ϵ}}_i\equiv {\alpha }_i{\mathit{\sigma }}_{+}+{\beta }_i{\mathit{\sigma }}_{-}$), respectively. After each detection, the unnormalized global state of the system is the sum of all states in the circles weighed by the tagged prefactors. Left-down arrows denote ${\mathit{\sigma }}_{+}$ transitions, and right-down arrows represent ${\mathit{\sigma }}_{-}$ ones. The final state components are shown inside colored frames and the colored arrows represent the quantum paths leading to these different components. Only the red (gray) circle states are obtained via a single quantum path, while the blue (dark gray) and green (light gray) ones are the result of three different interfering quantum paths.

Figure 3

Operational monitoring of tripartite entanglement classes by changing the polarization configurations. If the three polarizer orientations are all different, the final state belongs to the GHZ class; if two polarizer orientations are different, the final state belongs to the $W$ class; if all polarizers are identically oriented, the final state belongs to the $S$ class.