Limits to multipartite entanglement generation with bosons and fermions

Many-photon interference in linear-optics setups can be exploited to generate and to detect multipartite entanglement. Without recurring to any interparticle interaction, many entangled states have been created experimentally, and a panoply of theoretical schemes for the generation of various classes of entangled states is available. Here, we present a unifying framework that accommodates the present experiments and theoretical protocols for the creation of multiparticle entanglement via interference. A general representation of the states that can be created is provided for bosons and fermions for any particle number and for any dimensionality of the entangled degree of freedom. Using this framework, we derive an upper bound on the generalized Schmidt number of the states that can be generated, and we establish bounds on the dimensionality of the manifold of these states. We show that—at the expense of a smaller success probability—more states can be created with bosons than with fermions and give an intuitive interpretation of the state representation and of the established bounds in terms of superimposed many-particle paths.

Figure 1

Scattering setup for $N=4$ qutritlike ($d=3$) particles. The input state corresponds to $\vec{r}=(1,0,0,0,1,0,0,0,1,1,0,0),\vec{d}\left(\vec{r}\right)=(1,5,9,10)$, and $|{\Phi }^{1q}\left(c\right)\rangle =|1,2,3,1\rangle$.

Figure 2

Entanglement manipulation via many-particle scattering. Given the $N$-qudit state $|{\Phi }^{1q}\left(c\right)\rangle$ [Eq. (1)], the physical second-quantized state is given by Eq. (9), $|{\Psi }^{2q}\left(c\right)\rangle ={\widehat{\Omega }}_a|{\Phi }^{1q}\left(c\right)\rangle$ with the map operator ${\widehat{\Omega }}_a$ given by Eq. (8). The scattering of the particles through the setup is described by $\widehat{M}\left(W\right)$ [Eq. (14)], and the projection on the subspace of postselected states is mediated by ${\widehat{P}}_{1,b}$ [Eq. (10)]. The mapping to a first-quantized state occurs via ${\widehat{\Omega }}_b^{-1}$ [Eq. (8)]. The sketch on the right-hand side shows the physical situation for $d=4$ and $N=2$. The initial state contains only one component; it is injected into the scattering setup. The final state contains many coherently superposed components. Some of them (marked by red crosses) are not interpretable as $N$-qudit states. The projector ${\widehat{P}}_{1,b}$ describes the postselection mechanism, i.e., it eliminates the unwanted components and leaves only the postselected components (green crotchets).

Figure 3

Superposition of many-particle paths. (a) A state of $N=2$ particles that carry a ($d=4$)-dimensional degree of freedom is characterized by the coefficient tensor $\tilde{g}$, which indicates the amplitude of each basis state. (b) Each basis state (here, $|1,1\rangle$) is fed by the many-particle paths that contribute to the respective event. (c) In general, there are up to $N!$ of these paths; here, we show two exemplary paths (solid green and dashed blue lines) for $N=4$ particles.

Figure 4

GHZ entanglement projection. The photon pairs (1,2), (3,4), and (5,6) are initially polarization entangled as described by Eq. (A4). By the projection of photons 1, 3, and 5 onto the GHZ state [at detectors $D1$, $D3$, and $D5$ by the implementation of the scattering matrix (A8)], photons 2, 4, and 6 are also left in the GHZ state. Courtesy of C.-Y. Lu et al. [94].

Figure 5

Creation of a family of entangled four-photon states. Photons are created via spontaneous parametric down-conversion (SPDC), and injected into single mode (SM) fibres. Two input modes $ⓐ$ and $ⓑ$ are combined at the PBS and filtered by interference filters (IF). The resulting modes $ⓒ$ and $ⓓ$ are, subsequently, split into four modes $ⓔ$, $ⓕ$, $ⓖ$, and $ⓗ$ by two additional beam splitters (BSs). The initial state (A9) is, thus, scattered on a setup, which implements (A10). The final quantum-information state (A11) depends on the polarization-rotation parameter $\gamma$. Courtesy of W. Wieczorek et al. [38].