Graph-theoretic insights on the constructability of complex entangled states

The most efficient automated way to construct a large class of quantum photonic experiments is via abstract representation of graphs with certain properties. While new directions were explored using Artificial intelligence and SAT solvers to find such graphs, it becomes computationally infeasible to do so as the size of the graph increases. So, we take an analytical approach and introduce the technique of local sparsification on experiment graphs, using which we answer a crucial open question in experimental quantum optics, namely whether certain complex entangled quantum states can be constructed. This provides us with more insights into quantum resource theory, the limitation of specific quantum photonic systems and initiates the use of graph-theoretic techniques for designing quantum physics experiments.


Introduction
Recent years have seen dramatic advances in quantum optical technology [16,17] as photons are at the core of many quantum technologies and in the experimental investigation of fundamental questions about our universe's local and realistic nature.Due to the peculiar behaviour of multi-particle interference, designing experimental setups to generate multipartite entanglement in photonics is challenging.The most efficient automated way to construct a large class of such quantum photonic experiments is via abstract representation of graphs with certain properties [6,10,7] allowing new possibilities for quantum technology [13,5,1].The construction of such graphs has been a challenging mathematical open problem as one needs to carefully tune the edge weights to satisfy exponentially many equations [9].
Recently, new directions were explored using Artificial intelligence and SAT solvers to find such graphs, which could be used to design quantum photonic experiments [11,15,3].However, using these methods, it is computationally infeasible to find solutions in large graphs as the search space grows exponentially.Therefore, more advanced analytical methods are necessary.In this work, we introduce the technique of local sparsification on experiment graphs, using which we answer a crucial open question in experimental quantum optics, namely whether certain complex entangled quantum states can be constructed.The two main ideas behind our technique are: 1) To develop an edge pruning algorithm which helps to construct quantum optical experiments with as few resources as possible.
2) Detect a special sparse subgraph in the pruned experiment graph whose edge count bounds the dimension of some multi-particle entangled quantum states Our ideas are general and might be useful to understand the experimental designs to construct several other quantum states like NOON states, cluster states, W states and Dickes states.With more structural insights into the graphs used for creating high-dimensional The edges {1, 2}, {3, 4}, {5, 6}, {3, 6} are of colour green (mode number 1) and {1, 6}, {2, 3}, {4, 5} are of colour red (mode number 0).The edges {4, 6} and {3, 5} are bi-chromatic, where the halves starting at the vertices 4, 5 are of the colour red, and the remaining halves are of the colour green.multi-particle entanglement, we believe our techniques can be used to resolve a conjecture on the constructability of certain complex entangled quantum states by Cervera-Lierta, Krenn, and Aspuru-Guzik [3].This would give us more insights into quantum resource theory and the limitation of specific quantum photonic systems.

Graph representation of quantum optics experiments
In 2017, Krenn, Gu and Zeilinger [10] have discovered (and later extended [7,6]) a bridge between experimental quantum optics and graph theory.Here, large classes of quantum optics experiments (including those containing probabilistic photon pair sources, deterministic photon sources and linear optics elements) can be represented as an edge-coloured edgeweighted graph.Additionally, every edge-coloured edge-weighted graph can be translated into a concrete experimental setup.This technique has led to the discovery of new quantum interference effects and connections to quantum computing [6].Furthermore, it has been used as the representation of efficient AI-based design methods for new quantum experiments [11,15].

Mathematical formulation of the problem
We first define some commonly used graph-theoretic terms.For a graph G, let V (G), E(G) denote the set of vertices and edges, respectively.For S ⊆ V (G), G[S] denotes the induced subgraph of G on S. N, C denote the set of natural and complex numbers, respectively.The cardinality of a set S is denoted by |S|.For a positive integer r, [r] denotes the set {1, 2 . . ., r}.
We assume all graphs to be simple throughout unless otherwise mentioned.Usually, in an edge colouring, each edge is associated with a natural number.But in such edge colourings, the edges are assumed to be monochromatic.But in the experiment graph, we are allowed to have bi-coloured edges, i.e. one half coloured by a certain colour and the other half coloured by a different colour (as shown in Figure 1).So, we develop some new notation to describe bi-coloured edges.
An edge colouring c associates a coloured edge {(v 1 , i), (v 2 , j)} to an uncoloured edge {v 1 , v 2 } for some i, j ∈ N.For an edge e = {(v 1 , i), (v 2 , j)}, we say that {i, j} is the colour of e.We will use the notation c(e, v 1 ) to denote i and c(e, v 2 ) to denote j in this case.When c(e, v 2 ) = c(e, v 1 ), we call the edge to be monochromatic and represent the colour of the edge by c(e).If c(e, v 2 ) = c(e, v 1 ), we call the edge to be a bi-chromatic edge.The colour degree of u with respect to the colour i, d(u, i) is the number of edges e incident on u such that c(e, u) = i.A weight assignment w assigns every edge e a weight w(e) ∈ C \ {0}.
We call a subset P of edges in a graph a perfect matching if each vertex in the graph has exactly one edge in P incident on it.Definition 1.The weight of this perfect matching P , w(P ) is the product of the weights of all its edges e∈P w(e) A vertex colouring vc associates a colour i to each vertex in the graph for some i ∈ N. We use vc(v) to denote the colour of vertex v in the vertex colouring vc.We say that each perfect matching P induces a vertex colouring vc if for each vertex v, vc(v) is equal to c(e, v), where e is the unique edge in P incident on v.We say that this vertex colouring vc is induced by P .Note that different perfect matchings can induce the same vertex colouring.A vertex colouring is defined to be feasible if it is induced by at least one perfect matching.

Definition 2. The weight of a vertex colouring vc, w(vc) is the sum of the weights of all perfect matchings P inducing vc.
The weight of a vertex colouring which is not feasible, is zero by default.

Definition 3. An experiment graph is said to be valid, if:
1.All feasible monochromatic vertex colourings have a weight of 1.

All non-monochromatic vertex colourings have a weight of 0.
An example of a valid experiment graph is shown in Figure 1 Definition

The dimension of a valid experiment graph G, µ(G) is the number of feasible monochromatic vertex colourings having a weight of 1.
In graph theoretic terms, Theorem 6 is equivalent to stating that µ The reader may note that the number of particles and the dimension of a GHZ state corresponds to the number of vertices and dimensions in the valid experiment graph, respectively.For a detailed description of the quantum physical meaning of this setup, refer to [9].

Progress on the problem
Krenn and Gu conjectured that physically it is not possible to generate a GHZ state of dimension d > 2 with more than n = 4 photons with perfect quality and finite count rates without additional resources [9], that is, when G is not isomorphic to K 4 , then µ(G) ≤ 2.
While proving their conjecture would immediately lead to new insights into resource theory in quantum optics, finding a counter-example would uncover new peculiar quantum interference effects of a multi-photonic quantum system.However, when multi-edges and bi-chromatic edges are allowed, even for a valid experiment graph with just 4 vertices, the question of whether the dimension is bounded or not looks surprisingly challenging.So a general bound on the dimension as the function of the number of vertices of the experiment graph remains elusive.This motivated researchers to look at the problem by restricting the presence of bi-chromatic edges and multi-edges.
Absence of destructive interference.When destructive interference is absent, the problem reduces to a simpler problem on unweighted coloured graphs.Bogdanov [2] proved that the dimension of a 4 vertex experiment graph is at most 3, and an n ≥ 6 vertex graph is at most 2. Chandran and Gajjala [4] gave a structural classification of experiment graphs of dimension 2. They also proved that if the maximum dimension achievable on a graph without destructive interference is not one, the maximum dimension achievable remains the same even if destructive interference is allowed.Vardi and Zhang [18,19] proposed new colouring problems which are inspired by these experiments and investigated their computational complexity.
Absence of multi-edges.Chandran and Gajjala [4] proved that the dimension of an n vertex experiment graph is at most n − 3 even when bi-chromatic edges are present for simple graphs.Their techniques can be extended to get the same bound when multi-edges are present and bi-chromatic edges are absent.
Absence of bi-chromatic edges.Ravsky [14] proposed a claim connecting this problem to rainbow matchings.Using a result of Kostochka and Yancey [8], he showed that the dimension of an n vertex experiment graph is at most n − 2. Neugebauer [12] used these ideas and did computational experiments.For valid experiment graphs with a small number of vertices, Cervera-Lierta, Krenn, and Aspuru-Guzik [3] translated this question into a boolean equation system and found that the system does not have a solution using SAT solvers.In particular, they show that for graphs with monochromatic edges, GHZ states with n = 6, d ≥ 3 and n = 8, d ≥ 4 cannot exist.The authors further conjectured the following more general claim Conjecture 5.It is not possible to generate an n > 4 vertex experiment graph with dimension d ≥ n 2 .
In a simple valid experiment graph with n vertices, as every vertex has at most n − 1 neighbours and each colour has at least one monochromatic edge incident on each vertex (from Observation 8), a trivial bound of n − 1 can be obtained on the dimension.A more careful argument would also give the same bound when multi-edges are allowed and bi-chromatic edges are absent [12].So far, all results either improve an additive factor over this trivial bound or work only on graphs with at most 8 vertices.In this work, using the technique of local sparsification for experiment graphs, we overcome this barrier.

Theorem 6. It is not possible to generate an n > 4 vertex experiment graph with dimension
This translates to saying that it is not possible to produce a GHZ state of n particles, with ≥ n √ 2 dimensions using this graph approach without additional quantum resources (such as auxiliary photons).Note that our bound holds even when bi-chromatic edges are allowed.

Edge pruning
We introduce the concept of an edge minimum valid experiment graph.A valid experiment graph G is said to be edge minimum if there is no other graph G such that Such graphs correspond to the experiments to create GHZ states with minimum resources and are of interest to experimental physicists.The reader may notice that proving Theorem 6 for Edge minimum valid experiment graphs implies Theorem 6 is true for all valid experiment graphs.
If the monochromatic vertex colouring of colour i is not feasible, then all edges with at least half of it coloured i can be discarded as such a mode number i would not help in increasing the dimension of the corresponding GHZ state.So, in the edge minimum valid experiment graphs, all monochromatic vertex colourings are feasible A graph is matching covered if every edge of it is part of at least one perfect matching.If an edge e is not part of any perfect matching M , then we call the edge e to be redundant.By removing all redundant edges from the given graph G, we get its unique maximum matching covered sub-graph.Observation 7. Edge minimum valid experiment graphs are matching covered.
Proof.Suppose that there is an edge minimum valid experiment graph G which is not matching covered.Consider its matching covered subgraph H.By definition, every perfect matching of G is contained in H. Therefore, the weight of a vertex colouring in H and G are equal; hence, H is also a valid experiment graph and µ(G) = µ(H).Since G is not matching covered, we know that |E(H)| < |E(G)|.This contradicts the edge minimality of G. Observation 8.In a valid experiment graph G, for any vertex v ∈ V (G) and colour i ∈ [µ(G)], there exists a monochromatic edge e incident on v such that c(e) = i.
Proof.By definition, the weight of the monochromatic vertex colouring with colour i is non-zero.Therefore, there must be a monochromatic perfect matching P in which all edges are of colour i.By the definition of perfect matching, an edge e ∈ P exists, which is incident on v. Therefore, e must be of colour i. Proof.Since G is a subgraph of G, every perfect matching in G would also be contained in G. Consider a perfect matching P of G, which is not contained in G .Let vc be the vertex colouring induced by P .We now claim that vc is not feasible in G .
We first prove that vc(v) = c(e) and vc(u) = c(e).As P is not contained in G , there must be an edge e ∈ P such that e ∈ G and e / ∈ G .By construction, such an e = e, must be incident on v and has c(e , v) = c(e).Therefore, P induces the colour c(e) on v and hence vc(v) = c(e).Since e is already incident on v, it must be the case that e / ∈ P .As d(u, c(e)) = 1 and e / ∈ P , it follows that vc(u) = c(e).We now claim that a vertex colouring vc with vc(u) = c(e) and vc(v) = c(e) is infeasible in G .Suppose not.We know that there must be a perfect matching P inducing the vertex colouring vc.By construction, e is the only edge of colour c(e) incident on v in G .Therefore, e ∈ P .As e is also incident on u, P induces the colour c(e) on u.It follows that vc(u) = c(e).Contradiction.
Therefore, if a vertex colouring vc is feasible in G , then vc is induced by the same set of perfect matchings in both G and G .It follows that the weight of any feasible vertex colouring vc in G is equal to the weight of the vc in G; hence, G is a valid experiment graph.
Observe that none of the removed edges could have been part of a monochromatic perfect matching in G. Therefore, the monochromatic vertex colouring of colour i in G has a weight equal to the weight of monochromatic vertex colouring of colour i in G, which is 1 for all i ∈ [µ].It follows that µ(G) = µ(G ).This contradicts the edge minimality of G as G is a valid experiment graph with n vertices, µ(G) = µ(G ) and has fewer edges than G.

4
Proof of the Theorem 6

Proof sketch
Towards a contradiction, suppose there is exists an edge minimum simple valid experiment graph G with n vertices and dimension If the dimension of the graph µ = n − 1, then it must be the case that on each vertex v, d(v, i) = 1 for every i ∈ [n − 1] (from Observation 8).Similarly, as shown in Observation 12, if the dimension is µ, it must be the case that d(v, i) = 1 for at least 2µ − n + 1 values of i ∈ [µ].We then use Lemma 10 to guarantee that at least 2µ − n + 1 colour isolated edges are incident on v.As there are at least 2µ − n + 1 colour-isolated edges incident on each vertex v, we get that there must be at least 0.5n(2µ − n + 1) colour-isolated edges in G.A simple averaging argument over µ colours would now guarantee that for some i ∈ [µ], there must be a large matching M with (2µ − n)n 2µ i-coloured isolated edges.
In Section 4.3, we find some structural properties over a special subgraph on the vertices spanned by M , called the representative sparse graph.Using this, we prove that µ ≤ n − |M |.

Existence of a large special matching
Proof.Suppose not.Then there are 2µ − n − i colours whose colour degree is exactly one for some vertex v ∈ V (G) for some i ∈ [0, 2µ − n].Therefore, there are n − µ + i colours with a colour degree of at least 2 on v.It follows that there are 2µ Proof.From Lemma 10, we know that for every colour i with d(u, i) = 1, there is a colour isolated edge of colour i incident on u.Along with Observation 12, this implies that there are at least 2µ − n + 1 colour isolated edges are incident on each vertex.Therefore, there are at least (2µ − n + 1) n 2 colour isolated edges in G. Theorem 14.For some i ∈ [µ], there exist at least Proof.From Observation 13, we know that there are at least (2µ − n + 1) n 2 colour isolated edges.Since there are µ colours, by a simple averaging argument, we get that for some colour i ∈ [µ], there exist at least (2µ − n)n 2µ i-coloured isolated edges.

Detecting a sparse subgraph
We will assume that the large matching exhibited in Theorem 14 has colour 1 for the remainder of this section.Let R represent the set of vertices whose colour degree with respect to the colour 1 is one.Let U be the set of vertices whose colour degree with respect to the colour 1 is at least two.
|R| must be integral, |R| ≥ 4. Therefore, there exists at least two 1-coloured isolated edges.We now pick an arbitrary 1-coloured isolated edge {u, v}.For the remainder of this section, we base our analysis on the edges incident on the vertices u, v.We now construct a special subgraph on G[R] called the representative sparse graph χ using the edges incident on u, v.
We first construct the graph χ u in the following way over R: For every colour i ∈ [µ], if there is no vertex w ∈ U such that c({u, w}, u) = i, then add an arbitrary monochromatic edge of colour i (its existence is guaranteed from Observation 8) to E(χ u ).
Similarly, we construct the graph χ v in the following way over R: For every colour i ∈ [µ], if there is no vertex w ∈ U such that c({v, w}, v) = i, then add an arbitrary monochromatic edge of colour i (its existence is guaranteed from Observation 8) to E(χ v ).
We define the graph χ over R with the edge set E(χ) = E(χ u ) ∪ E(χ v ).An example graph and its representative sparse graph are presented in Figure 3.
Proof.For any colour i ∈ [µ], there is either a vertex w ∈ U such that c(u, {u, w}) = i or there is a monochromatic edge of colour i in χ u .Therefore, Since every edge in χ is incident either on u or on v, it easy to see that |E(χ)| ≤ 2|R| − 3.But this can be strengthened by using the following structural observation.
Proof.Suppose |{{u, u }, {u, v }, {v, u }, {v, v }} χ| ≥ 3. Without loss of generality, let the edges {u, u }, {u, v }, {v, v } be present in E(χ) with colours i, j, k.By definition of the representative sparse graph, we know that i = j and i, j, k are not equal to 1.Note that k might be equal to i or j.
Consider the vertex colouring vc in which u, u get the colour i, v, v get the colour k, and the vertices in Let vc be the vertex colouring in which every vertex in V (G) gets the colour 1.

Observation 18.
w(vc) w(vc Proof.Let M be the set of all 1-coloured isolated edges.As every vertex in R has exactly one 1-coloured edge incident on it, every perfect matching P inducing vc must contain all edges in M .Let W denote the weight of the monochromatic vertex colouring of colour 1 on G[U ].As the vertices in U match with themselves in every perfect matching P , it is easy to see that w(vc As every vertex in R \ {u, v, u , v } has exactly one 1-coloured edge incident on it, every perfect matching P inducing vc must contain all edges in M \ {{u, v}, {u , v }}.By definition of the representative sparse graph, the vertex u can obtain the colour i only through an edge {u, w} such that w ∈ R.But the vertices R \ {u, v, u , v } are already matched and c({u, v}, u) = 1 = i and c({u, v }, u) = j = i.Therefore, the edge {u, u } must be present in every perfect matching P that induces vc.Again, by definition of the representative sparse graph, the vertex v can obtain the colour k only through an edge {v, w} such that w ∈ R.But all the vertices in R \ {v, v } are already matched.Therefore, the edge {v, v } must be present in every perfect matching P .The remaining vertices in U should match among themselves in every perfect matching P that induces vc.It is now easy to see that the weight of the vertex colouring vc is This is a contradiction to our assumption that µ > n √ 2 and hence, µ ≤ n √ 2 .

1 Figure 2
Figure 2The vertex colourings induced by the perfect matchings of Figure1.

Definition 9 .Lemma 10 .
If an edge e = {u, v} is monochromatic and d(u, c(e)) = d(v, c(e)) = 1, then e is said to be colour isolated.For an edge e ∈ E(G) incident on u ∈ V (G), if d(u, c(e)) = 1, then e is a colour-isolated edge.Proof.Let e = {u, v}.From Observation 8, we know that e must be a monochromatic edge and hence d(v, c(e)) ≥ 1. Towards a contradiction, suppose e is not a colour-isolated edge.It follows that d(v, c(e)) ≥ 2. We now construct G , a subgraph of G, by removing each edge e = e, which is incident on v and has c(e , v) = c(e).We show that G is a valid experiment graph by showing that the weight of any feasible vertex colouring vc in G is equal to the weight of vc in G. Observation 11.G is a valid experiment graph.