Bell nonlocality and fully entangled fraction measured in an entanglement-swapping device without quantum state tomography

We propose and experimentally implement an efficient procedure based on entanglement swapping to determine the Bell nonlocality measure of Horodecki et al. [Phys. Lett. A 200, 340 (1995)] and the fully entangled fraction of Bennett et al. [Phys. Rev. A 54, 3824 (1996)] of an arbitrary two-qubit polarization-encoded state. The nonlocality measure corresponds to the amount of the violation of the Clauser-Horne-Shimony-Holt (CHSH) optimized over all measurement settings. By using simultaneously two copies of a given state, we measure directly only six parameters. This is an experimental determination of these quantities without quantum state tomography or continuous monitoring of all measurement bases in the usual CHSH inequality tests. We analyze how well the measured degrees of Bell nonlocality and other entanglement witnesses (including the fully entangled fraction and a nonlinear entropic witness) of an arbitrary two-qubit state can estimate its entanglement. In particular, we measure these witnesses and estimate the negativity of various two-qubit Werner states. Our approach could especially be useful for quantum communication protocols based on entanglement swapping.

Figure 1

Amount of entanglement measured with the negativity $N$ vs (a) the Bell nonlocality $M$, (b) entropic witness $E$, and (c) fully entangled fraction $F$ for ${10}^5$ two-qubit states randomly generated by a Monte Carlo simulation. Entangled states for which an entanglement witness is successful in detecting inseparability are marked with light cyan dots. The entangled states that are ignored by the respective witness are marked with dark cyan dots. The Werner ($\mathcal{W}$), Horodecki ($\mathcal{H}$), and pure ($\mathcal{P}$) states correspond to the upper solid, dashed, and lower solid curves, respectively. In particular, $F$ allows detecting the entanglement of the Werner states $\mathcal{W}$ for the whole range of their mixing parameter $p>1/3$. A given witness detects all the entangled states of the negativity above its respective threshold, i.e., $N_M=0.5607$, $N_E=0.4120$, and $N_F=0.2071.$

Figure 2

Experimental scheme for determining the Bell nonlocality measure and fully entangled fraction of polarization-encoded two-qubit states with linear optics via the elements of the $R$ matrix. This setup consists of narrow-band filters (F), half-wave plates (HWPs), quarter-wave plates (QWPs), beam dividers (BDs), detectors (D), a fiber beam splitter (FBS), polarization controllers (PCs), lenses (L), BBO crystals, mirrors, and motorized-translation stages (M). Note that this is an entanglement-swapping device, where the swapping is implemented by the FBS. The setup is powered by a laser system described in the text. The polarization-dispersion line (PDL) compensates the polarization dispersion introduced by the BBO crystals in the four-photon-source module (4PS). In this module, two copies of a two-qubit state ${\rho }_1$ and ${\rho }_2$ are prepared in the modes $(a_1,b_1)$ and $(a_2,b_2)$, respectively. We name the last two modules as belonging to Alice and Bob, respectively. In Alice's module, qubit $a_1$ is overlapped on an 50:50 beam splitter with qubit $a_2$ to implement the measurement of $S_{a_1{a}_2}=(I-4|{\mathrm{\Psi }}^{-}\rangle {\langle {\mathrm{\Psi }}^{-}\left|\right)}_{a_1{a}_2}$. In Bob's module, qubits $b_1$ and $b_2$ are projected onto the eigenstates of ${({\sigma }_i\otimes {\sigma }_j)}_{b_1{b}_2}$ for $i\ge j$ and $i,j=1,2,3$ by the respective polarizer (POL) and detected at the respective detector. The fourfold coincidence counts are then processed to estimate the values of $R_{i,j}$.

Figure 3

Experimental detection of entanglement witnesses for the Werner states $\mathcal{W}\left(p\right)$: (a) Bell nonlocality $M$, fully entangled fraction $F$, and entropic witness $E$ vs mixing parameter $p$. Our theoretical results for the perfect Werner states are marked with solid curves. The systematic deviation from the ideal case (solid curves) is caused by the fact that our experimentally created singlet state was not ideally pure and its purity reached circa $93\%$. Moreover, our experimentally created mixed state was not totally mixed. The ideal Werner states are separable for $p<1/3$. For these states the entanglement can be detected with $F$, $E$, and $M$ for $p_F>1/3$, $p_E>1/\sqrt{3}$, and $p_M>1/\sqrt{2}$, respectively. In (b) we show the geometric representation of the witnesses. For each witness the separable states are enclosed by a respective gray boundary. The coordinate system is given as $[r_1,r_2,r_3]=\mathrm{eig}\left(R\right)$. The Werner states for $0\le p\le 1$ are located on the diagonals connecting points $[0,0,0]$ and $[1,1,1]$. The entanglement is detected for non-negative values of a witness, i.e., for points outside of the gray areas. For $F$ (the most universal of the three witnesses) this area is the smallest.