Effects of spectral entanglement in polarization-entanglement swapping and type-I fusion gates

We examine how spectral entanglement in polarization-entangled photon states generated from bulk-crystal, spontaneous parametric down-conversion affects the success of entanglement swapping and type-I fusion gates. We quantify the success of the entanglement swapping and fusion gates by calculating the bipartite concurrence and residual tangle, respectively, in terms of the joint spectral probability amplitudes of the initial broad-bandwidth polarization-entangled states. We find that both polarization-entanglement measures depend strongly on the initial spectral entanglement, as well as on the configuration of the independent sources. Specifically, when spectral differences correlate with polarization, the optimal source configuration is different for the two protocols. We conclude that this distinction is founded in how the underlying Bell-state measurement and quantum-erasure techniques respond differently to distinguishing spectral information.

Figure 1

(Color online) A comparison of how spectral properties correlate with polarization (left panel) and spatial degrees of freedom (right panel). Boxes (red) and circles (blue) indicate the different marginal spectra for the horizontal $\left(H\right)$ and vertical $\left(V\right)$ polarization states. The inset relationships define the different types of correlations. The black boxes signify entanglement sources and any additional linear optical elements.

Figure 2

The concurrence of photon pair (1,2) as a function of the spectral entanglement $K$ when spectral differences correlate with polarization. The curves are labeled according to the value of the aspect ratio $a=\sigma ∕{\sigma }^{\prime }$.

Figure 3

An entanglement swapping experiment, where the interference of photons 2 and 3 at a 50:50 beam splitter (BS) is analyzed using two polarization beam splitters (PBS) and four detectors ($h_2$, $v_2$, $h_3$, and $v_3$). Certain coincidences between detectors (semicircles) signal the projection of photons 1 and 4 into a Bell state. Black boxes signify entanglement sources and any additional linear optical elements.

Figure 4

(Color online) Four configurations of entanglement sources distinguished by the relationships between the joint spectral amplitudes. Boxes (red) and circles (blue) denote the different marginal spectra of the horizontal $\left(H\right)$ and vertical $\left(V\right)$ polarization states. In (a) and (b) spectral differences correlate with polarization, while in (c) and (d) spectral differences correlate with path. Black boxes signify entangled photon sources and any additional linear optical elements.

Figure 5

(Color online) The concurrence $C_{14}$ obtained after entanglement swapping as a function of the spectral entanglement $K$. The upper panel is the concurrence using the source configuration in Fig. 4a, and the lower panel is the concurrence using the source configuration in Fig. 4b. In each panel, the curves do not cross and are labeled in descending order by the value of the aspect ratio $a=\sigma ∕{\sigma }^{\prime }$. For entanglement swapping the upper panel also quantifies the concurrence obtainable with the source configuration in Fig. 4c. The solid line in both panels corresponds with the concurrence obtained from the source configuration in Fig. 4d, which is independent of $a$.

Figure 6

A type-I fusion experiment where the interference of photons 2 and 3 at a polarizing beam splitter (PBS) is analyzed by rotating the output in mode 2 by $\pi ∕4$ and using a polarization beam splitter (PBS) and two detectors ($h_2$ and ${\mathrm{v}}_2$). Single-photon detection events project photons 1, 3, and 4 into a linear cluster state. The black boxes signify the entangled photon sources and any additional linear optical elements.

Figure 7

(Color online) The coherence magnitude $C$ versus the angle $\phi$ and the number of sequential gate operations $N$. The curves are labeled by the number of gate operations $N=1$ to 8 and represent the results both of entanglement swapping and type-I fusion.