Witnessing Trustworthy Single-Photon Entanglement with Local Homodyne Measurements

Single-photon entangled states, i.e., states describing two optical paths sharing a single photon, constitute the simplest form of entanglement. Yet they provide a valuable resource in quantum information science. Specifically, they lie at the heart of quantum networks, as they can be used for quantum teleportation, swapped, and purified with linear optics. The main drawback of such entanglement is the difficulty in measuring it. Here, we present and experimentally test an entanglement witness allowing one to say whether a given state is path entangled and also that entanglement lies in the subspace, where the optical paths are each filled with one photon at most, i.e., refers to single-photon entanglement. It uses local homodyning only and relies on no assumption about the Hilbert space dimension of the measured system. Our work provides a simple and trustworthy method for verifying the proper functioning of future quantum networks.

Figure 1

Principle of the proposed entanglement witness.

Figure 2

Separable bound $S_{\mathrm{sep}}^{\max }$ as a function of the probability that at least one of the two protagonists gets more than one photon $p(n_A\ge 2\cup n_B\ge 2)$. The blue curve allows one to know the maximum value of the CHSH polynomial $S_{\mathrm{sep}}^{\max }$ that a state separable in the $\{|0⟩,|1⟩{\}}^{\otimes 2}$ subspace can reach. If $S_{\mathrm{obs}}>S_{\mathrm{sep}}^{\max }$, one can conclude that the projection of the state in the subspace with zero and one photon locally is entangled.

Figure 3

Experimental setup. A tunable single-photon entangled state is created by sending a heralded single photon on a tunable beam splitter based on a polarizing beam splitter (PBS) and a half-wave plate ($\lambda /2$). The proposed witness is then tested with two independent homodyne detections (Alice and Bob). The local oscillator is superposed to each mode via the first PBS. Its global phase is swept with a piezoelectric actuator. The relative phase $\Delta \phi$ is set with a combination of birefringent plates.

Figure 4

Observed CHSH values $S_{\mathrm{obs}}$ (the size of points accounts for statistical errors) and separable bound $S_{\mathrm{sep}}^{\max }$ as a function of beam-splitter angle $\theta$. When $S_{\mathrm{obs}}>S_{\mathrm{sep}}^{\max }$, one can conclude that the measured state is entangled.