Hypothesis testing for an entangled state produced by spontaneous parametric down-conversion

Generation and characterization of entanglement are crucial tasks in quantum information processing. A hypothesis testing scheme for entanglement has been formulated. Three designs were proposed to test the entangled photon states created by the spontaneous parametric down conversion. The time allocations between the measurement vectors were designed to consider the anisotropic deviation of the generated photon states from the maximally entangled states. The designs were evaluated in terms of the $p$ value based on the observed data. It has been experimentally demonstrated that the optimal time allocation between the coincidence and anticoincidence measurement vectors improves the entanglement test. A further improvement is also experimentally demonstrated by optimizing the time allocation between the anticoincidence vectors. Analysis on the data obtained in the experiment verified the advantage of the entanglement test designed by the optimal time allocation.

Figure 1

Schematic of the entangled photon pair generation by spontaneous parametric down conversion. Cascade of the nonlinear crystals (NLC) generate the photon pairs. Group velocity dispersion and birefringence in the NLCs are pre-compensated with quartz plates and a Bereck compensator. Two-photon states are analyzed with half wave plates (HWP), quarter wave plates (QWP), and polarization beam splitters (PBS). Interference filters (IF) are placed before the single photon counting modules (SPCM).

Figure 2

Distribution of the counts obtained in one second on the vectors $\mid VH⟩$, $\mid HV⟩$, $\mid XD⟩$, $\mid DX⟩$, $\mid RR⟩$, and $\mid LL⟩$. Bars present the histograms of the measured numbers, and lines show the Poisson distribution with the mean values estimated from the experiment. Measurement time was $40\sec$ for each vector.

Figure 3

Calculated $p$ value as a function of the threshold. Dash-dotted line: (i) the modified visibility, dash line: (ii) design I, dotted line: (iii) design II, solid line: (iv) design III.