Optimal local expansion of $W$ states using linear optics and Fock states

We derive the maximum success probability of circuits with passive linear optics for postselectively expanding an $N$-photon $W$ state to an $(N+n)$-photon $W$ state, by accessing only one photon of the initial $W$ state and adding $n$ photons in a Fock state. We show that the maximum success probability is achieved by a polarization-dependent beamsplitter and $n-1$ polarization-independent beamsplitters.

Figure 1

Local expansion circuits of $|W_N\rangle$ to $|W_{N+n}\rangle$. (a) The expansion circuit considered in this paper. This circuit is composed of only passive linear optics. (b) A lossless linear optical circuit equivalent to the expansion circuit in (a). Solid (dashed) arrows mean H- (V-) polarized modes.

Figure 2

A linear optical circuit which achieves the upper bound in Eq. (41).

Figure 3

The dependence of the maximum success probability $P_{\max }$ in Eq. (42) on $n$ and $N$.

Figure 4

Linear optical circuits which achieve (a) $P_{\mathrm{suc}}=n!(N+n)N^{-1}{H}_m$ and (b) $P_{\mathrm{suc}}=n!(N+n)N^{-1}{H}_{\mathrm{lossy}}$.