Fig. 1. Hybrid entangled state transmission. (a) Hybrid VV-polarization entangled photon pair generated in the experiment: entanglement in polarization of the photon pair (blue ribbon) and entanglement between polarization and OAM of the single photon (green ribbon, VV state) are sketched. The inhomogeneous polarization patterns of the VV state |r⟩ (bottom) and |a⟩ (up) are explicitly shown. (b) Schematic of the experiment: hybrid VV-polarization entangled state is generated by an initial polarization entangled photon pair. One photon of the pair encodes the VV state by the action of a VP. The VV beam is transmitted through the air-core fiber. Finally, state detection shows that hybrid VV-polarization entanglement (blue and green ribbons) is preserved after fiber transmission.

Fig. 2. Experimental apparatus for the generation, distribution and analysis of the hybrid entangled states. Pairs of telecom polarization entangled photons are generated by exploiting a periodically poled titanyl phosphate crystal (ppKTP) in a Sagnac interferometer, which contains a dual-wavelength polarizing beam splitter (DPBS) and a dual half-wave plate (DHWP). Photons exiting along mode 1 are sent to a polarization analysis stage, composed of a QWP, an HWP and a PBS. Photons along mode 2 pass through a dichroic mirror (DC), which separates the pump from the photons. Photons in mode 2 impinge on a VP to generate a VV beam state and, in turn, the desired hybrid entangled state. The VV states are coupled to an air-core fiber and then measured with an OAM-polarization analysis stage composed of a second VP followed by a polarization analysis setup. To perform the measurements on the polarization and OAM degrees of freedom independently, an additional polarization measurement stage has to be inserted before the OAM-to-Gaussian conversion regulated by the second VP. Finally, both photons are coupled into single-mode fibers linked to avalanche photodiode single photon detectors (APDs).

Fig. 3. Two-qubit quantum tomographies. (a) Real (top) and imaginary (bottom) parts of the measured density matrix of the polarization entangled state generated by the source, before conversion in OAM. (b) Real (top) and imaginary (bottom) parts of the measured density matrix of the two-photon VV-polarization entangled state after the transmission of photon 2 through the OAM fiber. (c) Real (top) and imaginary (bottom) parts of the measured density matrix of the VV state on photon 2, transmitted through the OAM fiber. The OAM states |0⟩ and |1⟩ in the tomography are defined by the relations: |0⟩≡(|+7⟩+|−7⟩)/2 and |1⟩≡i(|−7⟩−|+7⟩)/2. Real and imaginary parts of the experimental density matrices are reconstructed via quantum state tomographies.

Fig. 4. CHSH measurement operators. Expectation values moduli of the measured operators that maximize the violation of the CHSH parameter S=⟨A1B1⟩−⟨A1B0⟩+⟨A0B1⟩+⟨A0B0⟩. The values are relative to the polarization entangled state generated by the source (green bars), the hybrid VV-polarization entangled state (blue bars), and the intrasystem entangled VV state embedded in the photon 2 and transmitted through the air-core fiber (yellow bars). All error bars are due to Poissonian statistics of the measured events.

Fig. 5. Three-qubit quantum tomography. Real and imaginary parts of the measured density matrix of the hybrid VV-polarization state in space {|pol⟩1|pol⟩2|oam⟩2} after the fiber transmission (right) and of the theoretical density matrix of state in Eq. (4) (left). The OAM states |0⟩ and |1⟩ in the tomography are defined by the relations: |0⟩≡(|+7⟩+|−7⟩)/2 and |1⟩≡i(|−7⟩−|+7⟩)/2. Real and imaginary parts of the experimental density matrices are reconstructed via quantum state tomography.

Table 1. CHSH violations. The CHSH violation parameters obtained from raw data (Sraw) and by subtracting for accidental coincidences (S) are reported for the polarization entangled state generated by the source, the hybrid VV-polarization entangled state (HyEnt), and the intrasystem entangled VV state embedded in the photon 2 and transmitted through the air-core fiber (intra).