Introduction

Entanglement is an essential aspect of quantum information science, the investigation of which has resulted in new fundamental understandings about the nature1. Practically, entanglement is recognized as a valuable resource in the field of quantum information processing, with potential applications in areas such as cryptography2,3 and computing4,5. To study and utilize entanglement, it is a prerequisite to find reliable procedures to construct entangled quantum systems. One of the promising approaches to this task is to exploit the indistinguishability of quantum particles6. Various works suggested theoretical and experimental entanglement generation schemes based on the identicality of particles and postselection. Along these lines, refs. 7,8,9 showed that two spatially overlapped indistinguishable identical particles can carry bipartite entanglement. The quantitative relation of particle indistinguishability and spatial overlap to the bipartite entanglement was rigorously analyzed in refs. 10,11,12. For the case of multipartite entanglement, schemes for GHZ and W states with identical particles have been theoretically suggested13,14,15,16,17,18,19,20,21,22 and experimented23,24. Ref. 22 presented a comprehensive graph-theoretic approach to embrace the schemes to generate the entanglement of identical particles in linear quantum networks (LQNs) with postselection.

On the other hand, considering that the postselected schemes are highly sensitive to particle loss in circuits25,26 and the multipartite correlations can be created by the postselection bias27,28, there have been several attempts to generate the entanglement of identical particles without postselection. Specifically, the heralded generation of entangled states of photons was studied for bipartite29,30,31,32 and multipartite systems33,34,35,36,37,38. While postselected schemes involve identifying desired outcomes after conducting operations, heralded schemes use real-time signals or measurements in ancillary spatial modes as ‘heralds’ to sort out successful runs during the experiment (see, e.g., ref. 37 for a more detailed explanation). This property of heralded operations renders more tolerable schemes from photon loss25, however, it is usually more challenging to find proper circuits to obtain the heralded entanglement of an arbitrary N-partite systems than the postselected ones37.

In this work, we introduce a systematic method to overcome the difficulty of obtaining heralded entanglement generation schemes for an arbitrary N-partite system. Our method employs the sculpting protocol introduced in ref. 39, which generates an N-partite entangled state by applying N single-boson subtraction operators (which we name the ‘sculpting operator’) to a 2N boson initial state. By setting the initial state to have the even distribution of the bosons in different 2N states, the spatially overlapped sculpting operation (i.e, a single-boson subtraction operator \({\hat{A}}^{(l)}\) is a summation of subtractions on different spatial modes) generates the N-partite entanglement. And various sculpting operators result in different entangled states. Since linear bosonic systems with heralding detectors can realize boson subtraction operators40,41,42,43, we can design N-partite genuine entanglement generation schemes with this theoretical process.

Therefore, our methodology to find heralded entanglement generation schemes consists of two steps:

  1. 1.

    Find sculpting operators that generate multipartite entangled states

  2. 2.

    Find linear optical circuits that correspond to the sculpting operators

As we will show in this work and the follow-up paper44, this two-step division provides an organized strategy to the scheme-searching problem. Our primary focus here lies in demonstrating our graph approach to the first step, with a brief explanation of the second step at the end of RESULTS. The comprehensive treatment of the second step is covered in ref. 44. This follow-up work elaborates on translation rules from sculpting operators to optical operators, based on which we actually propose several heralded optical schemes generating various forms of multipartite entanglement.

In the sculpting protocol, the difficulty of designing a circuit for an N-partite entangled state is translated to the difficulty of finding a suitable sculpting operator to ‘chisel’ the state. However, former research on the sculpting protocol39,45 provides only proof-of-concept schemes demonstrating the method and lacks a systematic way of linking the features of sculpting operators to the expected final states. We show that a graph picture of the sculpting protocol provides a straightforward strategy for finding appropriate sculpting operators for the multipartite entanglement. We map multi-boson systems with sculpting operators into bipartite graphs (bigraphs), for which we develop techniques to understand key properties of the entanglement generation process. Our list of correspondence relations between sculpting protocols and graphs is a variation of that given in ref. 10, which provided a systematic method to analyze and design LQNs for obtaining entanglement with postselection. In this graph picture, we have found a special type of bigraphs, which we name ‘effective perfect matching (EPM) bigraphs’. These bigraphs are highly useful because they can directly correspond to sculpting operators that generate entanglement. Fig 1 describes our graph strategy to search for genuinely entangled states.

Fig. 1: Our strategy of finding heralded schemes for generating multipartite entanglement.
figure 1

Previous research on generating entangled states has primarily relied on trial and error methods via the direct route depicted by the blue to red boxes, or by taking a detour through the yellow box. However, we present a more systematic approach to designing circuits for various entangled states by mapping the elements of many-boson systems onto graphs, as illustrated by the routes of the green arrows.

Full size image

With our graph-theoretic approach, we present sculpting operators that generate qubit N-partite GHZ and W states, and an N = 3 Type 5 entangled state (the superposition of GHZ and W states46). The GHZ and W schemes are significantly more efficient than those given in ref. 39. And contrary to the schemes in refs. 25,37, they work for an arbitrary number of parties. In addition, our N = 3 Type 5 entangled state generation scheme illustrates that our approach can be extended to find more generalized forms of entangled states. To top it off, by generalizing the bigraph used to obtain qudit GHZ states, we also present a quditN-partite GHZ state generation scheme with dN bosons. To our knowledge, our scheme requires much less bosons than any known heralded schemes as in ref. 47. Our theoretical schemes can be realized in any many-boson system, e.g., linear optical systems with polarization qubit encoding and heralded detections. These outcomes showcase the effectiveness of our method in finding simple solutions for the generation of intricate entangled states.

Results

Sculpting protocol for qubit entanglement

We first formalize the sculpting protocol39 that converts the boson identity into entanglement. While N-partite entangled state was constructed in ref. 39 based on 2N modes with the dual-rail qubit encoding, we re-explain it based on N spatial modes and consider the qubit state as a two-dimensional internal degree of freedom of bosons. This way of expression not only embraces the dual-rail encoding, but also provides a more intuitive description of qubit states in the system.

Since in our setup, each boson in jth spatial mode (j ∈ {1, 2,...,N}) has a two-dimensional internal degree of freedom s(∈{0, 1}), boson creation and annihilation operators are denoted as \({\hat{a}}_{j,s}^{{\dagger} }\) and \({\hat{a}}_{j,s}\) respectively. Then, as an input state, we distribute 2N bosons into N spatial modes so that each spatial mode has two bosons with orthogonal internal states (0 and 1, see Fig. 2). Therefore, the initial state is given by

$$\begin{array}{rc}\left\vert Sy{m}_{N}\right\rangle \equiv {\hat{a}}_{1,0}^{{\dagger} }{\hat{a}}_{1,1}^{{\dagger} }{\hat{a}}_{2,0}^{{\dagger} }{\hat{a}}_{2,1}^{{\dagger} }\cdots {\hat{a}}_{N,0}^{{\dagger} }{\hat{a}}_{N,1}^{{\dagger} }\left\vert vac\right\rangle =\prod\limits_{j=1}^{N}({\hat{a}}_{j,0}^{{\dagger} }{\hat{a}}_{j,1}^{{\dagger} })\left\vert vac\right\rangle .\end{array}$$
(1)
Fig. 2: The initial state \(\left\vert Sy{m}_{N}\right\rangle\) of 2N bosons in N spatial modes.
figure 2

Each spatial mode has two bosons, one in the internal state \(\left\vert 0\right\rangle\) and the other in \(\left\vert 1\right\rangle\).

Full size image

Following the former works39,45, we call it the maximally symmetric state of 2N bosons. Rewriting \(\left\vert Sy{m}_{N}\right\rangle\) in the mode occupation representation as

$$\left\vert Sy{m}_{N}\right\rangle =\left\vert (1,1),(1,1),\cdots (1,1)\right\rangle$$
(2)

and the particle number distribution in the 2N states as a vector, we see that \(\left\vert Sy{m}_{N}\right\rangle\) is majorized by all the other Fock states of 2N bosons (i.e., since each mode has one particle, the particle number distribution vector of \(\left\vert Sy{m}_{N}\right\rangle\) is written as \({\overrightarrow{n}}_{Sym}=(1,1,\cdots \,,1)\). This vector is majorized by any particle number distribution vector of 2N dimension. See ref. 48, Section II for the rigorous definitions and analyses). Several research papers showed that this kind of state is very resourceful in many quantum computation protocols48,49,50.

What we need to obtain in the sculpting protocol is a state in which each spatial mode has one boson whose internal state corresponds to the qubit state. For such a final state, we need to annihilate N single bosons from the initial state \(\left\vert Sy{m}_{N}\right\rangle\). The N single-boson subtraction operator, which we name the sculpting operator, is expressed in the most general form as

$$\begin{array}{ll}\,\prod\limits_{l=1}^{N}\sum\limits_{j=1}^{N}({k}_{j,0}^{(l)}{\hat{a}}_{j,0}+{k}_{j,1}^{(l)}{\hat{a}}_{j,1}) \,\equiv \prod\limits_{l=1}^{N}{\hat{A}}^{(l)}\equiv {\hat{A}}_{N}\\ \,({k}_{j,s}^{(l)}\in {\mathbb{C}}\,{{{\rm{and}}}}\,\mathop{\sum}\limits_{j,s}| {k}_{j,s}^{(l)}{| }^{2}=1).\end{array}$$
(3)

We see that the one-boson subtraction operator \({\hat{A}}^{(l)}\) can superpose among different spatial modes. Such an operation has been implemented probabilistically in several bosonic experimental setups40,41,42,43.

For a later convenience, we rewrite \({\hat{A}}^{(l)}\) as

$$\begin{array}{rc}{\hat{A}}^{(l)}=\sum\limits_{j=1}^{N}({k}_{j,0}^{(l)}{\hat{a}}_{j,0}+{k}_{j,1}^{(l)}{\hat{a}}_{j,1})\equiv \sum\limits_{j=1}^{N}{\alpha }_{j}^{(l)}{\hat{a}}_{j,{\psi }_{j}^{(l)}}\end{array}$$
(4)

where \({\alpha }_{j}^{(l)}\in {\mathbb{C}}\) with \(\sum\limits_{j}| {\alpha }_{j}^{(l)}{| }^{2}=1\) and \(\left\vert {\psi }_{j}^{(l)}\right\rangle\) a normalized qubit state.

Applying the sculpting operator \({\hat{A}}_{N}\) to the initial state \(\left\vert Sy{m}_{N}\right\rangle\), we obtain the final state,

$${\left\vert {{\Psi }}\right\rangle }_{fin}={\hat{A}}_{N}\left\vert Sy{m}_{N}\right\rangle .$$
(5)

The entangled structure of \({\left\vert {{\Psi }}\right\rangle }_{fin}\) comes from two features of bosons: First, bosons are not additionally distinguishable other than their spatial modes and qubit states. Second, \({\hat{A}}_{N}\) generates the spatial overlap among bosons (See Supplementary Note 1 for the implication of these statements with a detailed analysis). Therefore, we can state that the indistinguishability of identical particles and spatial overlap are two essential elements for the entanglement generation with sculpting protocol, as in postselected schemes9,12,24,51.

There is an essential restriction that we need to impose on a sculpting operator \({\hat{A}}_{N}\): it must be a sum of operators that annihilates one particle per spatial mode so that the final total state \({\left\vert {{\Psi }}\right\rangle }_{fin}\) must consist of states with one particle per spatial mode. Then the internal state of one particle encodes the qubit information. This also means that any term of \({\hat{A}}_{N}\) that subtracts both particles from a given spatial mode must vanish. Otherwise, as the total number of subtracted particles is fixed, there remains a term with two particles in the same spatial mode in \(\left\vert {{\Psi }}\right\rangle\). We call this restriction the ‘no-bunching restriction’. We can find such an \({\hat{A}}_{N}\) that satisfies the restriction by controlling the probability amplitudes of it.

For the simplest N = 2 example, \({\hat{A}}_{2}\) is written as

$$\begin{array}{lll}{\hat{A}}_{2} &=&{\hat{A}}_{2}^{(1)}{\hat{A}}_{2}^{(2)}\\ &=&({\alpha}_{1}{\hat{a}}_{1{\psi}_{1}}+{\alpha}_{2}{\hat{a}}_{2{\psi}_{2}})({\beta}_{2}{\hat{a}}_{1{\phi}_{1}}+{\beta}_{2}{\hat{a}}_{2{\phi}_{2}})\end{array}$$
(6)

and the final state is given by

$$\begin{array}{lll}{\left\vert {{\Psi}}\right\rangle }_{fin}&=&{\hat{A}}_{2}\left\vert Sy{m}_{2}\right\rangle \\ &=&({\alpha}_{1}{\beta }_{2}{\hat{a}}_{1{\psi}_{1}}{\hat{a}}_{2{\phi}_{2}}+{\alpha}_{2}{\beta}_{1}{\hat{a}}_{2{\psi}_{2}}{\hat{a}}_{1{\phi}_{1}})\left\vert Sy{m}_{2}\right\rangle \\ &&+\, {\alpha}_{1}{\beta}_{1}{\hat{a}}_{1{\psi}_{1}}{\hat{a}}_{1{\phi}_{1}}\left\vert Sy{m}_{2}\right\rangle+{\alpha}_{2}{\beta}_{2}{\hat{a}}_{2{\psi}_{2}}{\hat{a}}_{2{\phi}_{2}}\left\vert Sy{m}_{2}\right\rangle .\end{array}$$
(7)

In the above equation, \({\alpha }_{1}{\beta }_{1}{\hat{a}}_{1{\psi }_{1}}{\hat{a}}_{1{\phi }_{1}}\left\vert Sy{m}_{2}\right\rangle\) must vanish, otherwise, the term has two particles in the second spatial mode which violates the no-bunching condition. For the same reason, \({\alpha }_{2}{\beta }_{2}{\hat{a}}_{2{\psi }_{2}}{\hat{a}}_{2{\phi }_{2}}\left\vert Sy{m}_{2}\right\rangle\) also must vanish.

In the dual rail encoding setup39,45, the no-bunching condition appears as a seemingly different form. In that setup, repetitive annihilations on the same spatial mode naturally vanish. However, since two spatial modes combine to constitute one subsystem for the case, valid final states are only restricted to those with one boson per two spatial modes. This exactly corresponds to the no-bunching restriction in our setup.

All things considered, we summarize the sculpting protocol as follows:

Sculpting protocol

  1. 1.

    Initial state: We prepare the maximally symmetric state \(\left\vert Sy{m}_{N}\right\rangle\) of 2N bosons, i.e., each boson has different states (either spatial or internal) with each other as Eq. (36) (see Fig. 2).

  2. 2.

    Operation: We apply the sculpting operator \({\hat{A}}_{N}\) of the form (4) to the initial state \(\left\vert Sy{m}_{N}\right\rangle\). The sculpting process must satisfy the no-bunching condition.

  3. 3.

    Final state: The final state can be fully separable, partially separable, or genuinely entangled.

It is worth mentioning that the degree of entanglement for an N-partite pure state can be categorized into three classes: fully separable, partially separable, and genuinely entangled52 (see ref. 22, 3.1 for a quick summary of these concepts). A state \(\left\vert \psi \right\rangle\) (\(\in {{{\mathcal{H}}}}={\otimes }_{j = 1}^{N}{{{{\mathcal{H}}}}}_{i}\)) is fully separable if it can be written as \(\left\vert \psi \right\rangle =\left\vert {\psi }_{1}\right\rangle \otimes \left\vert {\psi }_{2}\right\rangle \otimes \cdots \otimes \left\vert {\psi }_{N}\right\rangle\) where \(\left\vert {\psi }_{j}\right\rangle \in {{{{\mathcal{H}}}}}_{j}\) for all j = 1, 2,...,N. It is genuinely entangled if it cannot be separable under any bipartition of \({{{\mathcal{H}}}}\). It is partially separable when it is neither fully separable nor genuinely entangled. The states we target in the work are genuinely entangled states such as GHZ and W states.

Another crucial remark is that a sculpting operator can generate an N-partite entangled state with K ancillary spatial modes. Then the above sculpting protocol can be slightly varied to an (N + K)-partite intial state with the sculpting operator \({\hat{A}}_{N+K}\). We can see such cases in generating W states.

Most of the technical difficulty to find \({\hat{A}}_{N}\) for a specific entanglement state comes from Step 2, for it is critical to control the probability amplitudes so that the sculpting operator satisfies the no-bunching restriction. There has been no systematic technique to find a suitable \({\hat{A}}_{N}\) that simultaneously satisfies the no-bunching condition and generates a non-trivial entanglement state39. As we will explain in the following subsections, our graph techniques facilitate a powerful tool to overcome this limitation.

Graph picture of boson systems with sculpting operators

We now present a list of correspondence relations between the fundamental elements of the sculpting protocol and those of graphs. With the mapping, we can replace key physical properties and restrictions on the sculpting operators with those on graphs, which renders a handy guideline to the operator-finding process for genuinely entangled states.

Ref. 22 proposed a list of correspondence relations between linear quantum networks (LQNs) and graphs for providing a systematic method to analyze and design networks for obtaining entanglement without postselection. Since our sculpting protocol also consists of linear transformations of boson subtraction operators, a similar graph mapping dictionary can be imposed to find a suitable \({\hat{A}}_{N}\) that generates genuine entanglement. Indeed, we can map spatially overlapped subtraction operators into graph elements with a variation of the correspondence relations in ref. 22, which leads to a practical graph-theoretic method to analyze our system.

The correspondence relations of elements between bosonic systems with sculpting operators and bigraphs can be enumerated in Table 1. \({\alpha }_{j}^{(l)}\) and \({\psi }_{j}^{(l)}\) are defined as in Eq. (4). A brief glossary in graph theory can be found in ref. 22, Appendix A.

Table 1 Correspondence relations of a sculpting operator to a sculpting bigraph
Full size table

In our graph picture, a subtraction operator\({\hat{A}}^{(l)} = \sum_{j=1}^N{\alpha}_{j}^{(l)}{\hat{a}}_{j,{\psi}_{j}^{(l)}}\) is denoted as an unlabelled vertex in V. Dynamical variables specifying the operator, such as spatial distributions and internal states, are encoded as weighted edges connecting V to labelled vertices in U. Below, unlabelled and labelled vertices are drawn as dots (•) and circles () respectively. The array of subtraction operators (dots) is on the right-hand side of the array of spatial modes (circles). We can consider a more comprehensive mapping including creation operators, which is given in Supplementary Note 2. However, Table 1 suffices to analyze the crucial properties of sculpting operators for generating entanglement.

As a proof of concept, we analyze the simplest N = 2 examples with \({\hat{A}}_{2}={\hat{A}}^{(1)}{\hat{A}}^{(2)}\). Let us write \({\hat{A}}^{(1)}\) and \({\hat{A}}^{(2)}\) as

$$\begin{array}{rc}{\hat{A}}^{(1)}=&{\alpha }_{1}{\hat{a}}_{1{\psi }_{1}}+{\alpha }_{2}{\hat{a}}_{2{\psi }_{2}},\\ {\hat{A}}^{(2)}=&{\beta }_{1}{\hat{a}}_{1{\phi }_{1}}+{\beta }_{2}{\hat{a}}_{2{\phi }_{2}}.\end{array}$$
(8)

Then \({\hat{A}}^{(1)}\) applied to the system is mapped to a bipartite graph (bigraph)

(9)

Now, by applying \({\hat{A}}^{(2)}\), the total sculpting operator \({\hat{A}}_{2}\) corresponds to

(10)

Note that the physical system is invariant under the exchange of two unlabelled vertices (dots), i.e., \({\hat{A}}_{2}\) can be also expressed as

(11)

This represents nothing but the commutation relation \([{\hat{A}}^{(1)},{\hat{A}}^{(2)}]=0\).

When \({\hat{A}}_{2}\) is expanded as

$$\begin{array}{ll}{\hat{A}}_{2}\,=&{\hat A^{(1)}}{\hat A^{(2)}}\\ \,\quad=&{\alpha }_{1}{\beta }_{1}{\hat{a}}_{1{\psi }_{1}}{\hat{a}}_{1{\phi }_{1}}+{\alpha }_{1}{\beta }_{2}{\hat{a}}_{1{\psi }_{1}}{\hat{a}}_{2{\phi }_{2}}\\ \,\qquad&+{\alpha }_{2}{\beta }_{1}{\hat{a}}_{2{\psi }_{2}}{\hat{a}}_{1{\phi }_{1}}+{\alpha }_{2}{\beta }_{2}{\hat{a}}_{2{\psi }_{2}}{\hat{a}}_{2{\phi }_{2}},\end{array}$$
(12)

each term corresponds to a possible collective path (a possible connection of dots to circles in which each dot is uniquely connected to one circle) of the annihilation operators, e.g., the bigraph (10) has four possibilities for two annihilation operators to be applied to the spatial modes. Therefore, the expansion of \({\hat{A}}_{2}\)(12) is expressed with collective paths as

(13)

i.e., \({\hat{A}}_{2}\) is a superposition of the above four collective paths.

For the sculpting operator \({\hat{A}}_{2}\) to obey the no-bunching condition, we must set the amplitudes so that the first two collective paths in (13) vanish when they are applied to. We can achieve such a sculpting operator by setting

$$\begin{array}{ll}\,{\alpha }_{j}={\beta }_{j}=\frac{1}{\sqrt{2}}\,\,(j\in \{1,2\}),\\ \,\left\vert {\psi }_{1}\right\rangle =\left\vert {\phi }_{2}\right\rangle =\frac{1}{\sqrt{2}}(\left\vert 0\right\rangle +\left\vert 1\right\rangle )\equiv \left\vert +\right\rangle ,\\ \,\left\vert {\psi }_{2}\right\rangle =\left\vert {\phi }_{1}\right\rangle =\frac{1}{\sqrt{2}}(\left\vert 0\right\rangle -\left\vert 1\right\rangle )\equiv \left\vert -\right\rangle .\end{array}$$
(14)

Then it is direct to check that

(15)

i.e., by fixing amplitudes as Eq. (14), we obtain a Bell state as the final state.

From the above N = 2 example, we can understand the role of the no-bunching condition in the graph picture. Since the bigraph expression of \({\hat{A}}_{N}\) such as (10) is expanded with a summation of all the possible collective paths of annihilation operators as Eq. (13), we have to control the complex weights of the edges so that any collective path with more than two edges in the same circle does not contribute to the final state. This property can be understood with the concept of perfect matchings (PMs), which are independent sets of edges in which every vertex of U is connected to exactly one vertex of V (see ref. 22, Appendix A), as follows:

Property 1

For a specific sculpting operator \({\hat{A}}_{N}\), the final state \({\left\vert {{\Psi }}\right\rangle }_{fin}={\hat{A}}_{N}\left\vert Sy{m}_{N}\right\rangle\) must be fully determined by the addition of the perfect matchings (PMs) of the bigraph corresponding to \({\hat{A}}_{N}\).

Indeed, we see that the two collective paths in Eq. (15) are the two perfect matchings of the bigraph (10). The above property is useful for understanding given sculpting operators in several aspects, which we explain in Method with a general sculpting-operator-finding strategy based on this property. From now on, a bigraph that corresponds to a sculpting operator is called a sculpting bigraph.

Qubit entanglement: GHZ and W states

Here we will provide sculpting operators that generate qubit N-partite GHZ state, N-partite W state, and a superposition of N = 3 GHZ and W states, using Property 1. Our operator solutions are more efficient and more feasible to construct in many-boson systems than those given in ref. 39, especially for the W state case.

To find the sculpting operators for those entangled states, we define a especially convenient type of bigraphs, which we dub ‘effective perfect matching bigraphs (EPM)’. We restrict our attention to sculpting bigraphs whose edge weights of internal states are only among \(\{\left\vert 0\right\rangle ,\left\vert 1\right\rangle ,\left\vert +\right\rangle ,\left\vert -\right\rangle \}\) with \(\left\vert \pm \right\rangle \equiv \frac{1}{\sqrt{2}}(\left\vert 0\right\rangle \pm \left\vert 1\right\rangle )\).

Among the creation and annihilation operators in the above basis, we can easily see the following identity

$$\begin{array}{ll}\forall j\,\in \{1,2,\cdots \,,N\},\\ \,{\hat{a}}_{j,\pm }{\hat{a}}_{j,0}^{{\dagger} }{\hat{a}}_{j,1}^{{\dagger} }\left\vert vac\right\rangle =\pm {\hat{a}}_{j,\pm }^{{\dagger} }\left\vert vac\right\rangle ,\end{array}$$
(16)

holds, which directly results in the following identities:

$$\begin{array}{ll}\,{\hat{a}}_{j,+}{\hat{a}}_{j,-}{\hat{a}}_{j,0}^{{\dagger} }{\hat{a}}_{j,1}^{{\dagger} }\left\vert vac\right\rangle =0,\\ \,{\hat{a}}_{j,0}^{n}{\hat{a}}_{j,0}^{{\dagger} }{\hat{a}}_{j,1}^{{\dagger} }\left\vert vac\right\rangle ={\hat{a}}_{j,1}^{n}{\hat{a}}_{j,0}^{{\dagger} }{\hat{a}}_{j,1}^{{\dagger} }\left\vert vac\right\rangle =0.\quad (n\ge 2)\end{array}$$
(17)

The above identities are translated into our bigraph language as

(18)

Here, the internal state edge weights \(\{\left\vert 0\right\rangle ,\left\vert 1\right\rangle ,\left\vert +\right\rangle ,\left\vert -\right\rangle \}\) are denoted as edge colors {Black, Dotted, Red, Blue} respectively for convenience. The amplitude edge weights are omitted. The translation from Eq. (17) to (18) can be explained more clearly with directed bigraphs (see Supplementary Note 2).

Then, we define effective PM bigraphs (EPM bigraphs) as bigraphs whose edges always attach to the circles as one of the above forms. An example of EPM bigraphs is the N = 2 bigraph (10) with restrictions (14), i.e.,

(19)

because all the edges are attached to the circles as the first form of (18).

From the identities (18), we can see a crucial property of EPM bigraphs as follows:

Property 2

If a sculpting bigraph is an EPM bigraph, then the final state is always fully determined by the PMs of the bigraph.

The combination of Properties 1 and 2 provides a convenient strategy to find sculpting operators that generate a specific entangled state. Since we can express an entangled state with an addtion of PMs, if we can draw an effective PM bigraph that has the same PMs, the bigraph corresponds to a sculpting bigraph that generates the entangled state. We will show that various qubit N-partite genuinely entangled states can be generated with such bigraphs.

First, the sculpting bigraph that generates the N-partite GHZ state is given by

(20)

where the edge weights represent the probability amplitudes and edge colors Red and Blue represent the internal states \(\left\vert +\right\rangle\) and \(\left\vert -\right\rangle\). This bigraph was also used in ref. 22 to obtain the GHZ state in LQNs (see bigraph (30) of ref. 22).

The sculpting operator \({\hat{A}}_{N}\) corresponding to (20) is

$$\begin{array}{ll}{\hat{A}}_{N}=\frac{1}{\sqrt{{2}^{N}}}\,({\hat{a}}_{1,+}-{\hat{a}}_{2,-})({\hat{a}}_{2,+}-{\hat{a}}_{3,-})\cdots \\ \,\qquad\times ({\hat{a}}_{N-1,+}-{\hat{a}}_{N,-})({\hat{a}}_{N,+}-{\hat{a}}_{1,-})\\\quad\,\,\,\,=\frac{1}{\sqrt{{2}^{N}}}\,\mathop{\prod}\limits_{j=1}^{N}({\hat{a}}_{j,+}-{\hat{a}}_{j{\oplus }_{N}1,-}),\end{array}$$
(21)

where ⊕ N in the last line is defined as the addition mod N.

It is simple to verify that the bigraph (20) corresponds to a sculpting operator that generates the GHZ state. First, the edges of (20) attach to circles as the first of the three graphs in (18). Therefore we see that only the PMs contribute to the final state. Second, the bigraph has two PMs

(22)

and

(23)

(the above equality holds since the dots are identical), which constructs the GHZ state.

In the operator form, with the identity (16), we see that the final state is explicitly given by

$$\begin{array}{rc}{\hat{A}}_{N}\left\vert Sy{m}_{N}\right\rangle &=\frac{1}{\sqrt{{2}^{N}}}\left(\prod\limits_{j=1}^{N}{\hat{a}}_{j,+}+\prod\limits_{j=1}^{N}{\hat{a}}_{j,-}\right)\left\vert Sy{m}_{N}\right\rangle \\ &=\frac{1}{\sqrt{{2}^{N}}}\left(\prod\limits_{j=1}^{N}{\hat{a}}_{j,+}^{{\dagger} }+\prod\limits_{j=1}^{N}{\hat{a}}_{j,-}^{{\dagger} }\right)\left\vert vac\right\rangle\\ &=\frac{1}{\sqrt{{2}^{N-1}}}\left\vert GH{Z}_{N,2}\right\rangle .\end{array}$$
(24)

From the normalization factor, we directly see that the success probability becomes 1/2N−1.

Note that we can find other sculpting bigraphs for the GHZ state based on (20). While the GHZ state is invariant under the permutation of spatial modes, the bigraph (20) is not. Therefore, any bigraph with the permuted vertex labels of (20) also generates the GHZ state, i.e.,

(25)

under a permutation σ (∈SN). Since the GHZ state is also invariant under the qubit state flip, the exchange of blue and red edges also gives the GHZ state. However, such graph transformations are already included in the above permutation.

It is worth comparing our GHZ solution (21) with the solution given in ref. 39,

$${\hat{A}}_{N}=\frac{1}{\sqrt{{2}^{N}}}\prod\limits_{l=1}^{N}\left(\sum\limits_{j=1}^{N}{\hat{a}}_{j,0}+\sum\limits_{j=1}^{N}{e}^{\frac{2\pi i}{N}(j-l)}{\hat{a}}_{j,1}\right).$$
(26)

Most importantly, the consecutive annihilations in (26) do not remove particles from orthogonal modes. Hence they are very challenging to realize with experimental setups. In contrast, the procedure (21) is based on orthogonal modes, so that a single unitary change of basis is sufficient to prepare all the modes from which a single particle is to be removed. On top of that, each mode in (26) is a weighted superposition of all the initial ones. Understanding the operator from the graph picture, (26) corresponds to a bigraph with NN edges. On the other hand, (20) corresponds to a bigraph with only 2N edges. Therefore, the scheme described by (20) is more effective in the sense that each annihilation operator used there is constructed by superposing just two modes with internal state basis changes.

Second, a sculpting bigraph for N-partite W state can be conceived with one ancillary spatial mode as

(27)

Edge color Red, Blue, and Black respectively represent the internal state \(\left\vert +\right\rangle ,\left\vert -\right\rangle\) and \(\left\vert 0\right\rangle\), and ∣α∣2 + ∣β∣2 = 1. Circle A denotes the ancillary spatial mode. Note that this bigraph shares the same permutation symmetry as the W state, i.e., invariance under the permutation of spatial modes.

The sculpting operator corresponding to (27) is given by

$$\begin{array}{ll}\,{\hat{A}}_{N+A}&=\left(\alpha {\hat{a}}_{1+}+\beta {\hat{a}}_{A0}\right)\left(\alpha {\hat{a}}_{2+}+\beta {\hat{a}}_{A0}\right)\cdots \left(\alpha {\hat{a}}_{N+}+\beta {\hat{a}}_{A0}\right)\\ \,&\times \frac{1}{\sqrt{N}}\left({\hat{a}}_{1-}+{\hat{a}}_{2-}+\cdots +{\hat{a}}_{N-}\right)\\ &=\frac{1}{\sqrt{N}}\left(\prod\limits_{j=1}^{N}(\alpha {\hat{a}}_{j+}+\beta {\hat{a}}_{A0})\right)\sum\limits_{k=1}^{N}{\hat{a}}_{k-}.\end{array}$$
(28)

The initial state is prepared in a slightly varied way as

$$\left\vert Sy{m}_{N+A}\right\rangle \equiv \left(\prod\limits_{m=1}^{N}{\hat{a}}_{m0}^{{\dagger} }{\hat{a}}_{m1}^{{\dagger} }\right){\hat{a}}_{A0}^{{\dagger} }\left\vert vac\right\rangle ,$$
(29)

one ancillary boson at the ancillary spatial mode.

It is as manifest as for the GHZ state case to see the above sculpting bigraph (27) generates a W state. First, the bigraph is an effective bigraph since edges attach to circles as the first and second graphs in (18). Second, the above bigraph has N PMs

(30)

which corresponds to the W state with an ancillary system.

In the operator form, with the first identity of Eq. (16) again, we have

$$\begin{array}{lll}{\hat{A}}_{N+A}\left\vert Sy{m}_{N+A}\right\rangle &=&\frac{{\alpha }^{N-1}\beta }{\sqrt{N}}\left({\hat{a}}_{1-}{\hat{a}}_{2+}\cdots {\hat{a}}_{N+}+{\hat{a}}_{1+}{\hat{a}}_{2-}\cdots {\hat{a}}_{N+}\right.\\&& \left.+\, \cdots +{\hat{a}}_{1+}{\hat{a}}_{2+}\cdots {\hat{a}}_{N-}\right) {\hat{a}}_{A0}{\hat{a}}_{A0}^{{\dagger} }\prod\limits_{m=1}^{N}{\hat{a}}_{m0}^{{\dagger} }{\hat{a}}_{m1}^{{\dagger} }\left\vert vac\right\rangle \\ &=&\frac{{\alpha }^{N-1}\beta }{\sqrt{N}}\left({\hat{a}}_{1-}^{{\dagger} }{\hat{a}}_{2+}^{{\dagger} }\cdots {\hat{a}}_{N+}^{{\dagger} }+{\hat{a}}_{1+}^{{\dagger} }{\hat{a}}_{2-}^{{\dagger} }\cdots {\hat{a}}_{N+}^{{\dagger} }\right.\\ &&\left.+\, \cdots +{\hat{a}}_{1+}^{{\dagger} }{\hat{a}}_{2+}^{{\dagger} }\cdots {\hat{a}}_{N-}^{{\dagger} }\right)\left\vert vac\right\rangle \\&=&{\alpha }^{N-1}\beta \left\vert {W}_{N}\right\rangle.\end{array}$$
(31)

The success probability is ∣αN−1β∣2, whose maximal value becomes \(\frac{{(N-1)}^{N-1}}{{N}^{N}}\) when \(| \alpha | =\sqrt{\frac{N-1}{N}}\) and \(| \beta | =\frac{1}{\sqrt{N}}\).

We can also check that this bigraph can be used to generate a W state in LQNs with postselection. Indeed, by drawing a bigraph that corresponds to the schemes suggested in ref. 15,16, we obtain the same form of bigraph with (30).

Comparing with the W state generation scheme suggested in ref. 39, we can easily see that our current scheme has accomplished an outstanding improvement. The scheme in ref. 39 starts from 4N bosons in 2N modes and goes through two steps of sculpting to generate the final N-partite W state. On the other hand, using the graph mapping technique, we have obtained a much more efficient N-partite W-state generation scheme just with 2N + 1 bosons in N + 1 spatial modes and one simple step of sculpting.

The genuinely entangled states that we have discussed so far have some convenient symmetries, which admit relatively simple sculpting bigraphs for generating them. However, we can also conceive less symmetric entangled states with the support of ancillary modes. It is always achieved by any EPM bigraph that connects all the dots to with black or dotted edges.

As an example, we present an EPM bigraph for a tripartite system that generates a superposition of the N = 3 GHZ and W states, which is called the N = 3 Type 5 state in ref. 46. The EPM bigraph used here includes three ancillae to construct such a sculpting operator:

(32)

In the above bigraph, the amplitude weights are omitted under the assumption that they are nonzero and satisfy the normalization conditions.

The corresponding sculpting operator of the above bigraph is given by

$$\begin{array}{rc}&{\hat{A}}_{3+A,B,C} =\frac{({\hat{a}}_{1+}+{\hat{a}}_{A0})}{\sqrt{2}}\frac{({\hat{a}}_{2+}+{\hat{a}}_{B0})}{\sqrt{2}}\frac{({\hat{a}}_{3+}+{\hat{a}}_{C0})}{\sqrt{2}}\\ &\qquad\qquad\qquad\qquad\quad \times \frac{({\hat{a}}_{C0}-{\hat{a}}_{1-})}{\sqrt{2}}\frac{({\hat{a}}_{A0}+{\hat{a}}_{B0}-{\hat{a}}_{2-})}{\sqrt{3}}\frac{({\hat{a}}_{B0}+{\hat{a}}_{C0}-{\hat{a}}_{3-})}{\sqrt{3}}.\end{array}$$
(33)

When the above sculpting operator is applied to the initial state

$$| Sy{m}_{3+A,B,C}\equiv {\hat{a}}_{A0}^{{\dagger} }{\hat{a}}_{B0}^{{\dagger} }{\hat{a}}_{C0}^{{\dagger} }\prod\limits_{m=1}^{3}{\hat{a}}_{m0}^{{\dagger} }{\hat{a}}_{m1}^{{\dagger} }\left\vert vac\right\rangle ,$$
(34)

we have

$$\begin{array}{ll}&{\hat{A}}_{3+A,B,C}\left\vert Sy{m}_{3+A,B,C}\right\rangle\\ &=\frac{1}{12}\left({\hat{a}}_{1+}{\hat{a}}_{2+}{\hat{a}}_{3+}-{\hat{a}}_{1-}{\hat{a}}_{2+}{\hat{a}}_{3+}+{\hat{a}}_{1-}{\hat{a}}_{2+}{\hat{a}}_{3-}\right.\\ &\qquad+\left.{\hat{a}}_{1-}{\hat{a}}_{2-}{\hat{a}}_{3+}-{\hat{a}}_{1-}{\hat{a}}_{2-}{\hat{a}}_{3-}\right)\prod\limits_{m=1}^{3}{\hat{a}}_{m0}^{{\dagger} }{\hat{a}}_{m1}^{{\dagger} }\left\vert vac\right\rangle \\ &=\frac{1}{12}\left({\hat{a}}_{1+}^{{\dagger} }{\hat{a}}_{2+}^{{\dagger} }{\hat{a}}_{3+}^{{\dagger} }+{\hat{a}}_{1-}^{{\dagger} }{\hat{a}}_{2+}^{{\dagger} }{\hat{a}}_{3+}^{{\dagger} }+{\hat{a}}_{1-}^{{\dagger} }{\hat{a}}_{2+}^{{\dagger} }{\hat{a}}_{3-}^{{\dagger} }\right.\\&\qquad+\left.{\hat{a}}_{1-}^{{\dagger} }{\hat{a}}_{2-}^{{\dagger} }{\hat{a}}_{3+}^{{\dagger} }+{\hat{a}}_{1-}^{{\dagger} }{\hat{a}}_{2-}^{{\dagger} }{\hat{a}}_{3-}^{{\dagger} }\right)\left\vert vac\right\rangle \\ &=\frac{1}{12}(\left\vert +++\right\rangle +\left\vert -++\right\rangle +\left\vert -+-\right\rangle +\left\vert --+\right\rangle +\left\vert ---\right\rangle ).\end{array}$$
(35)

See that the five states with nonzero amplitudes in the final line of the above equation constitute the set of bases that can transform into any tripartite state under local operations46. The final state we just obtained is categorized as N = 3 Type 5 state46,53.

This N = 3 example shows that the graph method has the potential to design other general forms of multipartite entangled states with heralding detectors.

Qudit entanglement: GHZ state

Our graph picture also provides a useful insight to find sculpting operators for the general qudit systems. We will present in this subsection a sculpting bigraph for the qudit N-partite GHZ state, which has a generalized form of the qubit GHZ bigraph (39).

The qudit state is represented by a d-dimensional internal degree of freedom s(∈{0,1,...,d}) of bosons. To construct N partite qudit genuinely entangled states, we initially distribute dN bosons into N spatial modes so that exactly d bosons with mutually orthogonal internal states belong to a spatial mode (see Fig. 3). Hence, the initial state is given by

$$\left\vert Sy{m}_{N,d}\right\rangle \equiv \prod\limits_{j=1}^{N}({\hat{a}}_{j,0}{\hat{a}}_{j,1}\cdots {\hat{a}}_{j,d})\left\vert vac\right\rangle .$$
(36)

Here \(\left\vert Sy{m}_{N,d}\right\rangle\) denotes the N spatial mode maximally symmetric state with a d-dimensional internal degree of freedom.

Fig. 3: The initial state \(\left\vert Sy{m}_{N.d}\right\rangle\) of dN bosons in N spatial modes.
figure 3

Each spatial mode has d bosons, which have mutually orthogonal internal states \(\left\vert 0\right\rangle ,\left\vert 1\right\rangle ,\cdots \,,\left\vert d-1\right\rangle\).

Full size image

The sculpting operator

$${\hat{A}}_{N}=\prod\limits_{l=1}^{(d-1)N}{\hat{A}}^{(l)}$$
(37)

must be set to extract (d−1) bosons per spatial mode so that one boson per spatial mode in the final state determines the qudit state of each subsystem. All in all, the sculpting protocol is modified for qudits as follows:

Sculpting protocol of qudits

  1. 1.

    Initial state: We prepare the maximally symmetric state \(\left\vert Sy{m}_{N,d}\right\rangle\) of dN bosons, i.e., each boson has different states (either spatial or internal) with each other as Eq. (36). See Fig. 3.

  2. 2.

    Operation: We apply the sculpting operator \({\hat{A}}_{N}\) to the initial state \(\left\vert Sy{m}_{N,d}\right\rangle\). The sculpting operator must be set to extract (d−1) bosons per spatial mode.

  3. 3.

    Final state: The final state can be fully separable, partially separable, or genuinely entangled.

Now we provide a sculpting operator that generates the N-partite GHZ state of d-level systems, denoted as \(\left\vert GH{Z}_{N,d}\right\rangle\), by generalizing the qubit GHZ sculpting operator (20).

First, by generalizing the d = 2 basis set \(\{\left\vert +\right\rangle ,\left\vert -\right\rangle \}\) for the internal states of the sculpting operators, we choose the arbitrary d-dimensional basis set \({\{\left\vert \tilde{k}\right\rangle \}}_{k = 0}^{d-1}\) where

$$\begin{array}{r}\vert \tilde{k}\rangle =\frac{1}{\sqrt{d}}\left(\left\vert 0\right\rangle +{\omega }^{k}\left\vert 1\right\rangle +{\omega }^{2k}\left\vert 2\right\rangle +\cdots +{\omega }^{(d-1)k}\left\vert d-1\right\rangle \right)\end{array}$$
(38)

\((\omega ={e}^{i\frac{2\pi k}{d}})\) for the internal states of the sculpting operators.

Second, we use an overlap of (d−1) copies of the graph (20) for the sculpting bigraph, i.e., the following bigraph corresponds to the sculpting operator for the GHZ state:

(39)

where a gray circle represents a group of (d−1) identical vertices that have the same edges. For example, when N = 3 and d = 4, the above graph is explicitly drawn as

(40)

The sculpting operator that corresponds to the bigraph (39) is given by

$$\begin{array}{rc}&{\hat{A}}_{N,d}={\left(\frac{1}{\sqrt{2}}\right)}^{(d-1)N}{({\hat{a}}_{1,\tilde{0}}-{\hat{a}}_{2,\widetilde{d-1}})}^{d-1}{({\hat{a}}_{2,\tilde{0}}-{\hat{a}}_{3,\widetilde{d-1}})}^{d-1}\\ &\qquad \times {({\hat{a}}_{3,\tilde{0}}-{\hat{a}}_{4,\widetilde{d-1}})}^{d-1}\cdots \times {({\hat{a}}_{N,\tilde{0}}-{\hat{a}}_{1,\widetilde{d-1}})}^{d-1}.\end{array}$$
(41)

We verify that the above operator constructs the qudit N-partite GHZ state in Supplementary Note 3. To catch the sense of how the graph (39) works, we explicitly explain the qutrit case (d = 3) here.

For a qutrit system, the basis set (38) is given by

$$\begin{array}{rc}&\left\{\left\vert \tilde{0}\right\rangle \right.=\frac{1}{\sqrt{3}}(\left\vert 0\right\rangle +\left\vert 1\right\rangle +\left\vert 2\right\rangle ),\\ &\left\vert \tilde{1}\right\rangle =\frac{1}{\sqrt{3}}(\left\vert 0\right\rangle +{e}^{i\frac{2\pi }{3}}\left\vert 1\right\rangle +{e}^{i\frac{4\pi }{3}}\left\vert 2\right\rangle ),\\ &\left\vert \tilde{2}\right\rangle =\left.\frac{1}{\sqrt{3}}(\left\vert 0\right\rangle +{e}^{i\frac{4\pi }{3}}\left\vert 1\right\rangle +{e}^{i\frac{2\pi }{3}}\left\vert 2\right\rangle )\right\}.\end{array}$$
(42)

Then, we directly check that the following identities hold:

$$\begin{array}{ll}{({\hat{a}}_{j,\tilde{0}})}^{2}{\hat{a}}_{j,0}^{{\dagger} }{\hat{a}}_{j,1}^{{\dagger} }{\hat{a}}_{j,2}^{{\dagger} }\left\vert vac\right\rangle \,=\frac{2}{\sqrt{3}}{\hat{a}}_{j,\tilde{0}}^{{\dagger} }\left\vert vac\right\rangle ,\\ {\hat{a}}_{j,\tilde{0}}{\hat{a}}_{j,\tilde{2}}{\hat{a}}_{j,0}^{{\dagger} }{\hat{a}}_{j,1}^{{\dagger} }{\hat{a}}_{j,2}^{{\dagger} }\left\vert vac\right\rangle \,=-\frac{1}{\sqrt{3}}{\hat{a}}_{j,\tilde{1}}^{{\dagger} }\left\vert vac\right\rangle ,\\ {({\hat{a}}_{j,\tilde{2}})}^{2}{\hat{a}}_{j,0}^{{\dagger} }{\hat{a}}_{j,1}^{{\dagger} }{\hat{a}}_{j,2}^{{\dagger} }\left\vert vac\right\rangle \,=\frac{2}{\sqrt{3}}{\hat{a}}_{j,\tilde{2}}^{{\dagger} }\left\vert vac\right\rangle .\end{array}$$
(43)

(note that the above identities are also obtained from Supplementary Note Eq. (12) with d = 3). From the second identity of the above, we can see that

$$\begin{array}{rc}&{({\hat{a}}_{j,\tilde{0}})}^{2}{\hat{a}}_{j,\tilde{2}}{\hat{a}}_{j,0}^{{\dagger} }{\hat{a}}_{j,1}^{{\dagger} }{\hat{a}}_{j,2}^{{\dagger} }\left\vert vac\right\rangle \\ &={\hat{a}}_{j,\tilde{0}}{({\hat{a}}_{j,\tilde{2}})}^{2}{\hat{a}}_{j,0}^{{\dagger} }{\hat{a}}_{j,1}^{{\dagger} }{\hat{a}}_{j,2}^{{\dagger} }\left\vert vac\right\rangle =0\end{array}$$
(44)

also holds. The graph (39) for d = 3 is now drawn as

(45)

and the corresponding sculpting operator becomes

$$\begin{array}{rc}&{\hat{A}}_{N,3}={\left(\frac{1}{\sqrt{2}}\right)}^{2N}{({\hat{a}}_{1,\tilde{0}}-{\hat{a}}_{2,\widetilde{2}})}^{2}{({\hat{a}}_{2,\tilde{0}}-{\hat{a}}_{3,\widetilde{2}})}^{2}\\ &\qquad \qquad \times {({\hat{a}}_{3,\tilde{0}}-{\hat{a}}_{4,\widetilde{2}})}^{2}\times \cdots \times {({\hat{a}}_{N,\tilde{0}}-{\hat{a}}_{1,\widetilde{2}})}^{2}.\end{array}$$
(46)

Then, the final state is given by

$$\begin{array}{ll}&{\hat{A}}_{N,3}\left\vert Sy{m}_{N,3}\right\rangle\\ &={\left(\frac{1}{\sqrt{2}}\right)}^{2N}\left(\prod\limits_{r=1}^{N}{({\hat{a}}_{r,\tilde{0}})}^{2}+\prod\limits_{s=1}^{N}(-2{\hat{a}}_{s\tilde{0}}{\hat{a}}_{s\tilde{2}})+\prod\limits_{t=1}^{N}{({\hat{a}}_{t,\tilde{2}})}^{2}\right)\\ &\quad\times \prod\limits_{p=1}^{N}{\hat{a}}_{p,0}^{{\dagger} }{\hat{a}}_{p,1}^{{\dagger} }{\hat{a}}_{p,2}^{{\dagger} }\left\vert vac\right\rangle\\ &={\left(\frac{1}{\sqrt{3}}\right)}^{N}\left(\prod\limits_{r=1}^{N}{\hat{a}}_{r,\tilde{0}}^{{\dagger} }+\prod\limits_{s=1}^{N}{\hat{a}}_{s\tilde{1}}^{{\dagger} }+\prod\limits_{t=1}^{N}{\hat{a}}_{t,\tilde{2}}^{{\dagger} }\right)\left\vert vac\right\rangle \\ &={\left(\frac{1}{\sqrt{3}}\right)}^{N-1}\left\vert GH{Z}_{N,3}\right\rangle . \end{array}$$
(47)

The second line is obtained by Eq. (44) and the third by Eq. (43). The success probability is 1/3N−1.

To the best of our knowledge, our scheme needs much fewer bosons than any other heralded schemes for the qudit GHZ state. For example, the scheme in ref. 47 that need (2d + 1) particles for the d-level bipartite GHZ state (i.e., the Bell state) and 25 particles for the 3-level tripartite GHZ state in their optimized method. Our sculpting schemes just need 2d and 9 particles respectively.

Heralded scheme of Sculpting protocols in linear optics: Bell state example

In the former subsections, we have presented sculpting bigraphs that generate genuinely multipartite entangled states. Therefore, if we know how to build spatially overlapped subtraction operators by heralding, we can directly design heralded entanglement generation circuits by combining these operators as the structure of sculpting bigraphs.

There are general schemes in optics40,41,42,43 to establish subtraction operators of bosons. Based on such methods, ref. 39 proposed an optical scheme for constructing sculpting operators. More recently, ref. 45 suggested an experimental scheme with arithmetic subtractions of trapped ions (which are near-deterministic operations established in ref. 54 that work unitarily except when the system is in the vacuum) to generate the GHZ state with the sculpting operator (26) in ref. 39. Refs. 39 and 45 used dual-rail encoding and binomial encoding respectively.

Here, we briefly explain an alternative linear optical circuit designed to implement spatially overlapped subtraction operators by heralding, which become building blocks of heralded schemes that generate entanglement. Then, as a proof of concept, we design a heralded Bell state generation scheme by utilizing the heralded subtraction operator and the Bell state bigraph (19). The optical elements used in the circuit can be applied to more general multipartite entangled states. For a more comprehensive explanation and solutions on the construction of heralded schemes from our sculpting schemes, see ref. 44.

Since all the qubit sculpting schemes in our work correspond to EPM bigraphs whose final states are determined by the identities (16) and (17), we need to find heralded optical circuits that correspond to those identities. The basic elements of our optical schemes are polarizing beam splitters (PBSs) and half-wave plates (HWPs), hence highly feasible. It also implies that our translation rule can be applied to any bosonic system with operators that play the roles of PBSs and HWPs.

In our setup, we encode the internal boson states \(\{\left\vert 0\right\rangle ,\left\vert 1\right\rangle ,\left\vert +\right\rangle ,\left\vert -\right\rangle \}\) as the polarization of photons \(\{\left\vert D\right\rangle ,\left\vert A\right\rangle ,\left\vert H\right\rangle ,\left\vert V\right\rangle \}\) where (D = diagonal, A = antidiagonal, H = horizotal, V = vertical). The polarized states have the following relations:

$$\begin{array}{rc}&\langle D| A\rangle =\langle H| V\rangle =0,\\ &\left\vert H\right\rangle =\frac{1}{\sqrt{2}}(\left\vert D\right\rangle +\left\vert A\right\rangle ),\quad \left\vert V\right\rangle =\frac{1}{\sqrt{2}}(\left\vert D\right\rangle -\left\vert A\right\rangle ).\end{array}$$
(48)

Then the initial state is given by

$$\left\vert Sy{m}_{N}\right\rangle =\prod\limits_{j=1}^{N}{\hat{a}}_{j,D}^{{\dagger} }{\hat{a}}_{j,A}^{{\dagger} }\left\vert vac\right\rangle .$$
(49)

We first consider the heralded optical circuit for the identity (16), which can be rewritten as

$${\hat{a}}_{\pm }\frac{({\hat{a}}_{+}^{{\dagger} 2}-{\hat{a}}_{-}^{{\dagger} 2})}{2}\left\vert vac\right\rangle =\pm {\hat{a}}_{\pm }^{{\dagger} }\left\vert vac\right\rangle .$$
(50)

We can perform the above operation with the following optical circuit:

(51)

where the one-photon detection at 21 and 22 correspond to the action of \({\hat{a}}_{+}\) and \({\hat{a}}_{-}\), i.e, the final states \(\left\vert +\right\rangle\) and \(\left\vert -\right\rangle\) in the spatial mode 1, respectively. In the above circuit, PBSs transform photons as

and HWPs as {H, V}↔{D, A}.

Step-by-step explanation

  1. 1.

    The first HWP rotates the photon state basis:

    $${\hat{a}}_{H}^{{\dagger} }{\hat{a}}_{V}^{{\dagger} }\to {\hat{a}}_{D}^{{\dagger} }{\hat{a}}_{A}^{{\dagger} }=\frac{1}{2}({\hat{a}}_{H}^{{\dagger} 2}-{\hat{a}}_{V}^{{\dagger} 2})$$
    (52)
  2. 2.

    The first PBS divides the photon paths according to the internal states:

    $$\frac{1}{2}({\hat{a}}_{H}^{{\dagger} 2}-{\hat{a}}_{V}^{{\dagger} 2})\to \frac{1}{2}({\hat{a}}_{1,H}^{{\dagger} 2}-{\hat{a}}_{2,V}^{{\dagger} 2})$$
    (53)

    where 1 and 2 denote the upper and lower paths of the PBS.

  3. 3.

    The dashed purple box subtracts \({\hat{a}}_{1,H}^{{\dagger} }\) or \({\hat{a}}_{2,V}^{{\dagger} }\) by heralding in the last PBS and send the remaining one to the initial mode:

    $$\begin{array}{ll}\,\displaystyle\frac{1}{2}({\hat{a}}_{1,H}^{{\dagger} 2}-{\hat{a}}_{2,V}^{{\dagger} 2})\\ \,{\underrightarrow {HWP}\atop}\displaystyle\frac{1}{2}({\hat{a}}_{1,D}^{{\dagger} 2}-{\hat{a}}_{2,A}^{{\dagger} 2})\\ \,{\underrightarrow {2nd\,PBS}\atop}\displaystyle\frac{1}{4}\left({({\hat{a}}_{1,H}^{{\dagger} }+{\hat{a}}_{2,V}^{{\dagger} })}^{2}-{({\hat{a}}_{2,H}^{{\dagger} }-{\hat{a}}_{1,V}^{{\dagger} })}^{2}\right)\\ \,{\underrightarrow {3rd\,PBS}\atop}\displaystyle\frac{1}{4}\left({({\hat{a}}_{1,H}^{{\dagger} }+{\hat{a}}_{22,V}^{{\dagger} })}^{2}-{({\hat{a}}_{21,H}^{{\dagger} }-{\hat{a}}_{1,V}^{{\dagger} })}^{2}\right)\end{array}$$
    (54)

    After the operation of the last (3rd) PBS, we postselect only cases when one particle arrives at the spatial mode 21 or 22. Then the final state will be \({\hat{a}}_{1,H}^{{\dagger} }{\hat{a}}_{22,V}^{{\dagger} }\) or \({\hat{a}}_{1,V}^{{\dagger} }{\hat{a}}_{21,H}^{{\dagger} }\). Therefore, the detection of a photon with V in the lower mode 22 (H in the upper mode 21) heralds the final state \({\hat{a}}_{H}^{{\dagger} }\) (\({\hat{a}}_{V}^{{\dagger} }\)).

In the above process, the dashed purple box plays the role of the heralded subtraction operator.

We can deform the subtractor to design a heralded optical circuit for spatially overlapped subtraction operators such as \(({\hat{a}}_{1,+}-{\hat{a}}_{2,-})\). Instead of attaching both wires of the subtractor to the same spatial mode as in (51), we now attach two wires to different spatial modes 1 and 2 as in

(55)

so that it plays the role of a spatially overlapped subtraction operator. A crucial difference of the subtraction operator in the above from that in (51) is the need for an HWP between two PBSs, which makes possible the subtraction of different internal states from different input modes.

As a proof of concept that the above-heralded operator plays the role of spatially overlapped subtraction operator, we provide a Bell-state generation scheme by employing the heralded subtraction operator based on the sculpting bigraph (19),

(56)

which corresponds to

$$\frac{1}{2}({\hat{a}}_{1,H}-{\hat{a}}_{2,V})({\hat{a}}_{2,H}-{\hat{a}}_{1,V})$$
(57)

in our optical setup.

We implement two heralded subtraction operators of the form (55) following the structure of the sculpting bigraph (19) that corresponds to (57), which results in the following circuit:

By postselecting only the cases when each detector observes one photon, we generate Bell states in spatial modes 11 and 21 as expected. Note that the wires of heralded subtraction operators are attached as the structure of the Bell sculpting bigraph (19) (see the wires in the dashed purple box). Supplementary Note 4 explains the state evolution in detail.

In the case of W state and N = 3 Type 5 state, we require subtraction operators for the superposition of larger than 2 spatial modes in the ancillae. This can be achieved by generalizing (55). Ref. 44 provides a more thorough analysis and optical circuits specific to these states. It is worth noting that any bosonic system with linear operators that transforms the spatial and internal states as PBSs and HWPs can execute the same entanglement generation schemes as those given here.

For the case of qudit entanglement generation with sculpting operators, we need a bosonic system that has a higher level of internal degree of freedom. For example, photons can have the orbital angular momentum (OAM) that can encode qudit information. Therefore, we can generate the GHZ qudit entanglement given in Sec. Qudit entanglement: GHZ state with OAM beam splitters55 and OAM-only Fourier transformation operators56. OAM beam splitters change the outgoing paths of photons with respect to the internal states, hence a d-level generalization of the PBS. OAM-only Fourier transformation operators transform the computational basis of the internal states, hence a d-level generalization of the HWP.

Discussions

Our strategy for finding entanglement generation schemes based on linear bosonic systems with heralding involves a two-step process: first, we find a theoretical sculpting operator that generates an entangled state. Second, we construct a concrete experimental circuit for such a sculpting operator. For the first step of finding sculpting operators, we have exploited graph techniques by imposing the correspondence relations of bosonic systems to bigraphs. We have shown that the graph picture of bosonic systems facilitates a powerful tool to find proper sculpting operators. For the second step, we have explained that a spatially overlapped subtraction operator can be installed in linear optical networks with heralding. This operator allows us to design circuits for generating heralded entanglement by combining them based on sculpting bigraphs. As the simplest example, we have presented a Bell state circuit with the heralded subtraction operators, which can be extended to other sculpting schemes that we have proposed (see ref. 44). Our current results suggest several interesting future research directions.

First, our formalism and strategy can be extended to encompass more complex qudit systems. We have introduced EPM bigraphs for finding qubit solutions, which can be generalized to qudit cases. We can suggest a more complete demonstration on the construction of designing qudit heralded schemes with more solutions for qudit entanglement. We can encode such qudit entangled states as the orbital angular momentum (OAM) of photons with OAM beam splitters55 and OAM-only Fourier transformation operators56.

Second, our sculpting protocol can identify other interesting multipartite entangled states. Since our graph approach provides a handy guideline for coming up with useful sculpting operators, we expect that it will be used to find heralded schemes for other crucial entanglements. For example, ref. 57 has recently found sculpting schemes for generating a special type of graph state, i.e., caterpillar graph states, with fewer photons than fusion gates58. Therefore, one of our next goals will be to find, e.g., more general types of graph states, cyclically symmetric states59, and N-particle N-level singlet states60.

Methods

Sculpting-operator-finding strategy

Here we briefly explain the advantage of Property 1 that links the final entangled state and perfect matchings (PMs) of sculpting bigraphs. Even if we have found sculpting operators that generate entanglement by combining it with Property 2 in the main content, there are several reasons that Property 1 itself provides useful insight to analyze sculpting operators that do not correspond to effective PM diagrams.

First, for a given bigraph that corresponds to a sculpting operator, we can immediately read the possible final state from the PMs of the bigraph. For the N = 2 example, we can expect from Eq. (13) that \({\hat{A}}_{2}\) has the potential to generate the Bell state before fixing the amplitudes.

Second, we can apply all the PM diagram techniques developed in ref. 22 to our system. Since the final states in both approaches correspond to the summation of PMs in a given bigraph, necessary conditions for a bigraph to carry genuine entanglement in LQNs (see Theorem 1 of ref. 22) are also valid to our protocol.

Third, in the same context as the second reason, we can consider bigraphs that generate entanglement in LQNs22 as strong candidates for sculpting operators that generate entanglement in our protocol.

Based on these advantages, we can build a strategy to find a sculpting operator for a genuinely entangled state:

Sculpting-operator-finding strategy

  1. 1.

    Write down all the states that consist of the entangled state that we want to generate. Draw the PMs that correspond to the states.

  2. 2.

    Draw a bigraph that has the above PMs. We choose a bigraph with minimal edges so that it has minimal collective paths.

  3. 3.

    Examine whether we can set the edge weights so that only PMs among the collective paths contribute to the final state.

  4. 4.

    If we can find such an edge weight solution, it corresponds to the sculpting operators that generate the entangled state we expect. If we cannot, we try other bigraphs with the same PMs but more edges.

In Step 2, we can use, e.g., a method suggested in ref. 22, 3.2 to find bigraphs for a specific set of PMs. Note that Step 2 provides a significant benefit since reducing edges in bigraphs means reducing the number of possible no-bunching restrictions that we have to consider. Furthermore, a sculpting operator found in that way usually can be constructed more efficiently since a smaller number of edges implies a small amount of resources to create operator superpositions. One can understand the edge number as the coherence number61 of a quantum state, which is a coherence monotone that quantifies the amount of coherence in a quantum system. Therefore, we can consider in a general sense that a system corresponding to a bigraph with more edges needs more quantum resources.