Experimental Detection of Entanglement with Optimal-Witness Families

We report an experiment in which one determines, with least tomographic effort, whether an unknown two-photon polarization state is entangled or separable. The method measures whole families of optimal entanglement witnesses. We introduce adaptive measurement schemes that greatly speed up the entanglement detection. The experiments are performed on states of different ranks, and we find good agreement with results from computer simulations.

Figure 1

Simulation results on the measurement of the set of six informationally complete entanglement witness families for $10^4$ randomly chosen two-qubit entangled states: pure states (bottom), and full-rank mixed states (top). The cumulative histograms compare between measurements performed with scheme $A$, scheme $B$, and scheme $C$.

Figure 2

Experimental setup. The polarization-entangled two-photon states are prepared by the method described in Ref. [20]. Upon emerging from the source, the two photons are guided with mirrors (M) to interfere at a $50:50$ beam splitter, with the temporal overlap controlled by a translation stage (TS). After passing through interference filters (IF), the photons are sorted by polarizing beam splitters and registered by one of the photodetectors, four on each side. The detector outputs are addressed to a time-to-digital converter, and coincidences between counts of any two detectors are recorded. Two sets of wave plates, each composed of a half-wave plate (HWP) and two quarter-wave plates (QWP), implement the polarization changes that correspond to the unitary operators of Table 1.

Figure 3

A comparison of schemes $B$ (left column) and $C$ (right column) for rank-one states (top row), rank-two states (middle row), and rank-four states (bottom row). The histograms report the percentage of entangled states detected against the number $n$ of witness families needed without performing state estimation; $⟨n⟩$ is the average value. Both the simulation data (left empty bars) and the experimental data (right full bars) show that, for the three kinds of quantum states considered, scheme $C$ provides further improvement over scheme $B$: It requires fewer families on average and the distributions are narrower. The similarity of the two histograms for the rank-one states is confirmed by their large fidelity $F$; similar values are obtained for the other histograms. Here, the simulation uses only states of the kind generated by the state preparation in the setup of Fig. 2, whereas no such restriction applies to the randomly chosen states for Fig. 1.