Fig. 2

Logic circuit and experimental setup. a The logic circuit for implementing U. b The integrated experimental setup. One photon is sent to Bob, and the other is sent to Alice. On Bob’s side, each one of the gates g _m before the Sagnac-like interferometer (SLI) is realized using a combination of a quarter-wave plate (QWP), a half-wave plate (HWP) and a QWP; the photons are measured along the z direction using an HWP and a polarized beam splitter (PBS). On Alice’s side, in the single-qubit protocol, the photons are detected directly to provide a coincidence signal. While, in the two-qubit protocol, Alice measures her photons along a direction \(\vec n_i\) that is chosen based on the result b|g _m received from Bob. The photons on both sides are detected by single-photon detectors (SPD). Finally, Alice’s measurement result is sent to coincidence units, unit0 and unit1, to coincide with the corresponding results from port₀ and port₁, respectively. c The unit used to prepare the investigated entangled states. The polarization Sagnac interferometer is used to prepare the maximally entangled state \(\left| {\mathrm{\Phi }} \right\rangle\) to be fed into the dual-wavelength PBS and HWP, i.e., HWP2. An additional unit M, in which the dashed gray part inserted with a long enough birefringent crystal (BC) assists in preparing the maximally mixed component \({\mathbb {I}}{\mathrm{/}}4\), is placed at the port to Alice to produce the mixed state ρ _AB . Two moveable shutters are used to adjust the parameter η. BS beam splitter, DM dichroic mirror. d Experimental realization of U with the SLI constructed from a homemade beam splitter, with half of it coated as a PBS and the other half coated as a non-polarized beam splitter (NBS)