Heralded Linear Optical Generation of Dicke States

TL;DR

This work presents a linear-optical, heralded method to generate arbitrary Dicke states $|D_n^k angle$ by embedding their permutation symmetry into a Dicke digraph within the linear quantum graph (LQG) framework. By converting the digraph into an effective bigraph and then mapping to a dual-rail optical network with multiport interferometers, the scheme realizes heralded generation with a verifiable success signal, avoiding destructive postselection. The authors derive the exact heralded success probability $P_{ ext{suc}} = inom{n}{k}rac{(k!)^{4}(n-k)^{n-k}}{2^{2n}n^{n+2k-1}(k+1)^{n-1}}$ and show feasibility for realistic photonic parameters, with prospects for rate enhancement via multiplexing and feed-forward. Overall, the approach combines graph-based design with linear optics to enable practical, scalable Dicke-state resources for quantum technologies.

Abstract

Entanglement is a fundamental feature of quantum mechanics and a key resource for quantum information processing. Among multipartite entangled states, Dicke states $|D_n^k\rangle$ are distinguished by their permutation symmetry, which provides robustness against particle loss and enables applications for quantum communication and computation. Although Dicke states have been realized in various platforms, most optical implementations rely on postselection, which destroys the state upon detection and prevents its further use. A heralded optical scheme is therefore highly desirable. Here, we present a linear-optical heralded scheme for generating arbitrary Dicke states $|D_n^k\rangle$ with $3n+k$ photons through the framework of the linear quantum graph (LQG) picture. By mapping the scheme design into the graph-finding problem, and exploiting the permutation symmetry of Dicke states, we overcome the structural complexity that has hindered previous approaches. Our results provide a resource-efficient pathway toward practical heralded preparation of Dicke states for quantum technologies.

Figures (6)

• Figure 1: Dicke digraph $D^k_n$ and its Directed Cycle Covers (DCCs). (a) The Dicke digraph $D^k_n$. Edge weights are omitted, implying that all outgoing edges from a vertex have equal amplitude weights. Two-headed arrows for $S_s \leftrightarrow A_a$ represent the combination of the two directed edges $S_s \rightarrow A_a$ and $A_a \rightarrow S_s$. (b) DCCs of $D_n^k$ shown in (a). These include three types of cycles: (i) a system self-loop $(j \to j)$; (ii) a 2-cycle on a system–ancilla pair $(S_j \to j \to S_j)$; or (iii) an alternating $S\leftrightarrow T$ cycle of even length. The alternating cycles arise from the remaining vertices $U$ and $V$ forming a complete directed bipartite subgraph, where each choice of perfect matchings $S \to T$ and $T \to S$ specifies an alternating cycle cover.
• Figure 2: Example of the mechanism for the Dicke digraph with $(n,k)=(4,2)$. (a) (b) The corresponding system vertices take self-loops, while the remaining system vertices form 2-cycles. $U=\{S_1,S_2\}$ and $W=\{T_1,T_2\}$ form a complete balanced directed bipartite subgraph. (c) Four disjoint cycle covers are possible in the subgraph, which correspond to the same operator monomial and generate $|1100\rangle$. (d) By the permutation symmetry among $(j,S_j)$ ($j\in\{1,2,3,4\}$) of the Dicke digraph, we can see that there are six more subgraphs that give permuted states of $|1100\rangle$, hence Dicke state $|D_4^2\rangle$.
• Figure 3: Conversion of a Dicke digraph into a sculpting bigraph. (a--c) Local subgraph transformation from $G_{du}$ to $G_{ub}$. Node indices, colors and edge weights (including signs) are preserved. (d) Bigraphs $D^2_4$ and (e) $D^3_6$ constructed from the conversion.
• Figure 4: Transformation rules from bigraphs to linear optical networks. We decomposed $D_n^k$ into circle-dot pairs on the same line in the bigraph. Each dashed box on the LHS is an open subgraph, whose open edges are attached to dots or circles following the designated labels. Then we obtain the circuit elements in the dotted boxes on the RHS from the translation rules in Fig. 2 of Ref. chin2024heralded. By connecting the open wires in the circuit elements, we can uniquely construct the linear optical circuit that generates $|D_n^k\rangle$ by heralding.
• Figure 5: Linear optical network from Dicke digraph $D_4^2$, split by several steps. Note that this figure is drawn with the positions of the $4$-partite port and the $3$-partite port swapped for clarity.