Local Transformation of Two Einstein-Podolsky-Rosen Photon Pairs into a Three-Photon $W$ State

We propose and experimentally demonstrate a transformation of two Einstein-Podolsky-Rosen photon pairs distributed among three parties into a three-photon $W$ state using local operations and classical communication. We then characterize the final state using quantum state tomography on the three-photon state and on its marginal bipartite states. The fidelity of the final state to the ideal $W$ state is $0.778\pm 0.043$ and the expectation value for its witness operator is $-0.111\pm 0.043$ implying the success of the proposed local transformation.

Figure 1

Schematics of the experiment. (a) The concept of converting two EPR pairs into a $W$ state. EPR pairs are shared by Alice-Charlie and Bob-Charlie. Charlie’s operation is performed on his two qubits. (b) The experimental setup. The ultraviolet (UV) pulses (wavelength 395 nm, average power 380 mW, diagonal polarization) from a frequency-doubled mode-locked Ti:sapphire laser (wavelength 790 nm; pulse width 90 fs; repetition rate 82 MHz) make two passes through a pair of type-I phase matched $\beta$-barium borate (BBO) crystals (thickness 1.5 mm) stacked with their optical axes orthogonal to each other to produce two EPR photon pairs via spontaneous parametric down conversion (SPDC) [24, 25]. Extra BBOs with thickness 1.65 mm (Comp) are placed on the path of the each photon to compensate for walk-off effects. The spectral filtering of the photons is done by a narrow-band interference filter (IF, wavelength: 790 nm; bandwidth: 2.7 nm). All the detectors $D_1$, $D_4$, $D_5$, and $D_6$ are silicon avalanche photodiodes placed after single-mode optical fibers.

Figure 2

Observed two-photon interference by recording four-photon coincidences as a function of the optical delay. PDBS works as a $50:50$ BS for $V$-polarized photons. The best fit to the data is represented by the solid Gaussian curve which shows a coherence length of $l_c\simeq 110\mu \mathrm{m}$. The visibility is 0.885.

Figure 3

Real part of the reconstructed density matrices for the initial EPR pairs: (a) ${\rho }_{12}$ and (b) ${\rho }_{34}$.

Figure 4

(a) Real and imaginary parts of the reconstructed density matrix of the experimentally obtained $W$ state, and (b) that of the density matrix of the ideal $W$ state $|W_3⟩$.

Figure 5

Real part of the reconstructed reduced density matrices of the experimentally obtained $W$ state, (a) ${\rho }_{14}$, (b) ${\rho }_{16}$, and (c) ${\rho }_{46}$. (d) Real part of the reduced density matrix of the ideal state $|W_3⟩$ for which ${\rho }_{14}={\rho }_{16}={\rho }_{46}$.