Full Characterization of a Three-Photon Greenberger-Horne-Zeilinger State Using Quantum State Tomography

We have performed the first experimental tomographic reconstruction of a three-photon polarization state. Quantum state tomography is a powerful tool for fully describing the density matrix of a quantum system. We measured 64 three-photon polarization correlations and used a “maximum-likelihood” reconstruction method to reconstruct the Greenberger-Horne-Zeilinger state. The entanglement class has been characterized using an entanglement witness operator and the maximum predicted values for the Mermin inequality were extracted.

Figure 1

Experimental Setup. An ultraviolet laser pulse makes two passes through a type-II phase-matched BBO crystal. This probabilistically produces two pairs of photons into the four modes (1-4). A HWP and compensation crystal (COMP) are placed in each path to counter walk-off effects. The polarization of one photon from each pair are rotated by 90 ° using HWPs to create a pair of $|{\varphi }^{+}⟩$ states. These two independent photon pairs are further entangled using the PBS provided that the photons in mode 2 and 3 overlap temporally. The three-photon GHZ state was produced between modes $A$, $B$, and $1$ when photon 4 is successfully projected onto the state $|D{⟩}_4$ signalled by a click at the trigger detector (TRIG.). Tomographic projections were performed using QWPs and polarizers (POL.) for photons at detectors $A$ and $B$ and a half-wave or quarter-wave plate in front of a fixed horizontal polarizer at detector 1. An additional quarter-wave plate was used in mode $A$ to cancel an unwanted phase shift from the PBS.

Figure 2

Fourfold coincidences versus pump mirror position. The source produces two pairs of photons in the state $|{\varphi }^{+}⟩$—one pair into modes 1 and 2 and the other into modes 3 and 4. For this measurement, the photons in modes 1 and 4 were successfully projected onto the state $|D⟩$. As the pump mirror is displaced, the overlap between the photons in modes 2 and 3 is changed. If the photons do not overlap, then the outputs of PBS have no correlations in the $D/A$-basis. However, if the photons overlap perfectly, then the outputs of the PBS are left in the state $|{\varphi }^{+}⟩$ which has strong correlations. The changes in the correlations manifest as an interference dip when the output modes $A$ and $B$ of the PBS are measured in the states $|A⟩$ and $|D⟩$. The visibility of the dip is $(69\pm 5)\%$.

Figure 3

Density matrix of a three-photon GHZ state. The real (left hand) and imaginary (right hand) of the density matrix were reconstructed from 64 linearly independent triggered threefold coincidence measurements. Large diagonal elements in the $|HHH⟩$ and $|VVV⟩$ positions along with large positive coherences indicate that this state has the qualities of the desired GHZ state. Those elements that have negative values are shown as white-topped bars while those that are positive are blue topped.