Compact linear optical scheme for Bell state generation

The capability of linear optics to generate entangled states is exploited in photonic quantum information processing, however, it is challenging to obtain entangled logical qubit states. We report, to the best of our knowledge, the most compact scheme producing the dual-rail-encoded Bell states out of four single photons. Our scheme requires a five-mode interferometer and a single photon detector, while the previously known schemes use six-mode interferometers and two photon detectors. Using computer optimization, we have found a decomposition of the five-mode interferometer with a minimum number of beam splitters and phase-shift elements. Besides compactness, our scheme also offers a success probability of $1/9$, which is higher than $2/27$ provided by the six-mode counterparts. The analysis suggests that the elevated success probability is connected to a higher order of photon interference realized by our scheme, in particular, four-photon interference is implemented in our scheme, while three-photon interference was implemented in previous counterparts.

Figure 1

Three types of linear optical schemes for generation of dual-rail-encoded Bell states using four separable photons: (a) A scheme that uses a six-mode interferometer $U^{\left(6\right)}$ and two heralding photon detectors, ${\mathrm{D}}_1$ and ${\mathrm{D}}_2$. Both heralding detectors should measure single photons for the scheme to generate the Bell state. This type of scheme has been studied previously in Ref. [4]. (b) A scheme that uses a five-mode interferometer $U^{\left(5\right)}$ and one heralding photon detector D. The heralding detector should measure two photons for the scheme to generate the Bell state. (c) A scheme that uses a sequence of two five-mode interferometers and two detectors, ${\mathrm{D}}_1$ and ${\mathrm{D}}_2$. For successful heralding, both detectors should measure single photons.

Figure 2

The illustration of the optimization procedure: (a) the cost function $f$ dependence on the number of optimization steps, (b) the state infidelity $1-F$, and (c) probability $\tilde{p}$ in the course of the optimization. The dashed lines mark probability levels of $2/27$ and $1/9$. The following parameters were used in the optimization algorithm: $\mu ={10}^{-3}$, $\mu ={10}^{-4}$, $\mu ={10}^{-2}$ for the three schemes under study in the order they are present in Fig. 1; $\varepsilon ={10}^{-5}$ for all three schemes.

Figure 3

The explicit form of the six-mode scheme obtained by optimization. Here, $q_1$ and $q_2$ are the pairs of modes that encode logical qubits, $a_1$ and $a_2$ are auxiliary modes to be measured by the detectors [see Fig. 1]. The beam splitters are marked by their power transmissivities ${\tau }_j$. The dashed line separates the scheme into two parts, three-photon interference take place in each of them, provided that one photon is measured in the mode a1.

Figure 4

The explicit form of the five-mode scheme obtained by optimization [see Fig. 1]. The interferometer provides Bell state generation with probability of $1/9$ heralded by detection of two photons in the auxiliary mode a. The designations are the same as in Fig. 3.

Figure 5

The five-mode interferometers, $V_1^{\left(5\right)}$ (a) and $V_2^{\left(5\right)}$ (b), forming the scheme depicted in Fig. 1, which were obtained by optimization. The designations are the same as in Fig. 3.