Efficient Dicke-state distribution in a network of lossy channels

We analyze the generation of entanglement in a multipartite optical network. We generalize the twin-field strategy to the multipartite case and show that our protocol has advantageous rate-loss scalings of distributing $W$ states and Dicke states over the star networks. We provide precise theoretical formulas and quantitative performance analyses. Also, analysis of the same protocol using Gaussian states as resources, which is a typical setup in many experimental tests, is provided. We numerically study the feasibility of the protocol in realistic experimental conditions with imperfect detectors. We observe that the advantage depends on the chosen fidelity and is affected by the dark counts of the detector.

Figure 1

A network with $N=4$ parties connected to a central station $C$ by lossy channels, each of length $L$. The parties and the central station collaborate to generate Dicke states such that the generation rate scales favorably with the transmittance of the channel. $E_i$ denote the environments the photons can be lost to.

Figure 2

Generating one-photon entangled states in a network with two parties and a central station. The parties prepare entangled states similar to vacuum in their local sources and send one part of each to the central station that removes the information about the origin of potential photons. The detector's click denotes the creation of an entangled state between the parties. Success probability scales like the first power of the power transmittance of one channel that links one party with the central station. $E_i$ denote the environments the photons can be lost to.

Figure 3

Comparison of our repeaterlike protocol, the direct transmission, and the squashed entanglement bound for the distribution of the tripartite $W$ state defined in (6) in the star network. The generation rate is the probability of success of the experiment. The smaller rate of the repeaterlike scenario for short distances is implied by the low probability of generating one photon in the circuit, as $b$ is assumed to be small. However, the advantage of this method for larger distances comes from the fact that only one photon has to survive the loss.

Figure 4

Scheme of the eight-mode transform that uniformly mixes a single-photon state. The short, thick horizontal lines denote 50:50 beam splitters. The scheme realizes the Hadamard transform on the modes' photon creation operators.

Figure 5

Generation rate of an $N$-partite state $W$ state with fidelity $F$ in the setup in which the parties are connected to the central station by a lossy channel with transmittance $t=\sqrt{T}$. The dashed line shows the asymptote.

Figure 6

Comparison of the generation rate of $W_i$ states for $i=2,3,4$ with the direct method and using the repeaterlike method. Dashed and solid lines corresponding to $W_2$ have the highest generation rate, while the lines corresponding to $W_4$ have the lowest. For the repeaterlike scenario we assume fidelity $F=0.95$. We assume the loss rate is $0.2\mathrm{dB}/\mathrm{km}$. In the direct method we consider that a central station sends parts of the $i$-photon polarization $W$ state to each node.

Figure 7

Comparison of the generation rate of different four-mode states using the direct method and the repeaterlike method. For the repeaterlike method we assume fidelity $F=0.95$. We assume the loss rate is $0.2\mathrm{dB}/\mathrm{km}$. In the direct method we consider that a central station sends parts of a four-photon polarization $W$ state to each node. In this plot lines corresponding to higher fidelity have a lower generation rate.

Figure 8

Probability of a single detector click (generating the $W_2$ state from Gaussian states) as a function of distance for a loss rate of $0.2\mathrm{dB}/\mathrm{km}$ for different squeezings (0.87, 1.3, 1.74, 2.17, 2.61, 3.04, 3.47 dB). The lowest line corresponds to the smallest squeezing. For comparison the dashed line shows formula (15) with a fidelity of 0.99.

Figure 9

Fidelity of the $W_2$ state with the generated state when a single detector clicks as a function of distance for a loss rate of $0.2\mathrm{dB}/\mathrm{km}$ for different squeezing parameters (0.87, 1.3, 1.74, 2.17, 2.61, 3.04, 3.47 dB). The blue line (with the highest fidelity for small distances) corresponds to the smallest squeezing.

Figure 10

Generation rate of fixed-fidelity $W$ states from Gaussian states as a function of the distance. The dashed line shows the rate of generation of $W$ states using the direct method. The line corresponding to the lowest fidelity has the highest rate.

Figure 11

Rate of generation of $W_3$ with different fidelities. The dashed line denotes the direct method. The line corresponding to the lowest fidelity has the highest rate.

Figure 12

Rate of generation of $W_4$ with different fidelities. The dashed line denotes the direct method. The line corresponding to the lowest fidelity has the highest rate.

Figure 13

Generation rate for $W_N$ with fidelity $F=0.99$. The line corresponding to $W_2$ has the highest rate. The line for $W_3$ has the lowest.