Experimental measurement of a nonlinear entanglement witness by hyperentangling two-qubit states

We demonstrate that nonlinear entanglement witnesses can be made particularly useful for entanglement detection in hyperentangled or multilevel states. We test this idea experimentally on the platform of linear optics using a hyperentangled state of two photons. Instead of several simultaneous copies of two-photon entangled states, one can directly measure the witness on a single copy of a hyperentangled state. Our results indicate that hyperentanglement can be used for quick entanglement detection and it provides a practical test bed for experiments with nonlinear entanglement witnesses.

Figure 1

Conceptual scheme for measuring the collectibility of two-qubit states by hyperentangling. The initial state ${\widehat{\rho }}^{\left(p\right)}={\widehat{\rho }}^{\left(s\right)}$ is encoded both into polarization ($p$) and spatial ($s$) modes of photons A and B. Photon A is then subjected to projections ${\widehat{\mathrm{\Pi }}}_{Ap},{\widehat{\mathrm{\Pi }}}_{As}$ that are defined for polarization and spatial mode, respectively. Photon B is projected onto a singlet state $|{\psi }^{-}{\rangle }_{Bp,Bs}=\frac{1}{\sqrt{2}}\left(\right|01\rangle -{|10\rangle )}_{Bp,Bs}$, where the two degrees of freedom are addressed holistically at the same time.

Figure 2

Experimental setup for measuring collectibility of photonic two-qubit states by hyper-entangling the input photons. M: motorized translation; BD: beam displacer; HWP: half-wave plate; PBS: polarization beam splitter; QWP: quarter-wave plate; POL: polarizer; F: 10-nm interference filter; PC: polarization controller; FBS: fiber beam splitter; D: single-photon detector. Spatial modes are labeled by numbers 1–4.

Figure 3

Experimental results and theoretical values of entanglement witness for three characteristic quantum states. Note that measurement values are in good agreement with the theoretically predicted values. As expected, the witness gives a clearly negative values only in the case of the Bell state.

Figure 4

Experimental results and theoretical prediction of collectibility for the Werner states as a function of parameter $p$. The solid green line depicts the observed experimental results, the dashed black line is the theoretical prediction, and the light green area corresponds to the measurement uncertainty.

Figure 5

Experimental values of the ratio $R$ [see Eq. (10)] obtained for the quality analysis of HESs for Werner states. The solid green (dashed black) curve corresponds to pure (mixed) ${\widehat{\rho }}^{\left(p\right)}$. The collectibility has been measured for the encircled states.