Detection of nonlocal superpositions

We study in theory and experiment the verification of the Schmidt number of quantum correlated radiation fields. This measure of entanglement counts the number of global superpositions of product states which are needed to describe a compound quantum system. For unraveling a certain value of this measure, a family of entanglement and Schmidt number tests is constructed. The measurement of the whole class of witnesses is realized. By generating a path entangled two-photon state, we demonstrate the feasibility to directly count quantum superpositions.

Figure 1

Experimental setup for the entanglement test of a two-qutrit photonic state in a transverse path degree of freedom. (a) A cw diode laser (pump beam) at 405 nm is used to pump a bismuth borate (BiBO) crystal. As a result, two entangled photons are created for type-I SPDC. $L_p$ lens focuses the pump beam at the three-slit plane and generates a two-qutrit state when the entangled photons go through a three-slit aperture placed at the photon paths. Both a spatial light modulator (SLM) and a polarizing beam splitter (PBS) are used to implement amplitude modulation of the path states. Convergent lenses $L_s$ and $L_i$ are used to create a three-slit image or an interference pattern at the detection plane for the measurement operator realization. DM, dichroic mirror; M, mirror; D, single photon avalanche detector; C, coincidence counter. Subscripts $s$ and $i$ refer the signal and idler photon pair. (b) State preparation part of the setup. (c) Measurement implementation part of the setup exemplifying two-path superpositions: $|1\rangle ,|2\rangle$ for signal arm and $|0\rangle ,|1\rangle$ for idler arm.

Figure 2

Single path operator probabilities observed in the two-qutrit system.

Figure 3

At the image plane. Plot on the left in (a) corresponds to a double slit image for the photonic path states $|0\rangle$ and $|1\rangle$ and on the right is shown a gray level configuration used to implement a two-path superposition. (b) A double slit image is recorded by idler detector when the photon-path state $|1\rangle$ is attenuated by using a gray $\text{level}=0$ of the SLM. At the Fourier plane. On the left, (c) two-path superpositions are obtained by means of two-slit interference patterns. On the right, detector's positions at the path phase differences of $\{0,\pm \pi ,\pm \pi /2\}$ select the states related to ${\text{X}}_{lm}^{\pm }$ and ${\text{Y}}_{lm}^{\pm }$. Continuous curves represent the theoretical fits of the experimental data.