Direct evaluation of pure graph state entanglement

The REE of an entangled state ρ is defined as follows [5]:

$$\begin{eqnarray*} &&E_{\rm R}(\rho):=\min_{\omega\in \mathrm{SEP}}S(\rho||\omega), \end{eqnarray*}$$

where S: = Tr[ρ log ρ − ρ log ω] is the quantum relative entropy and the minimization is taken over all separable states ω. Since we investigate only pure states in this paper the relative entropy takes a simplified form of S = −Tr[ρ log ω]. The separable state that achieves the minimum of the relative entropy is referred to as the closest separable state (CSS). Even though the relative entropy is not a true metric we can interpret E_R(ρ) as the shortest distance between the entangled state ρ and the set of separable states SEP.

The geometric measure [6–8] for a pure state |ψ〉 can be defined as

$$\begin{eqnarray*} &&E_{\rm G}(|\psi\rangle):=\min_{|\phi\rangle\in {\rm PROD}}-\log|\langle\phi|\psi\rangle|^2, \end{eqnarray*}$$

where the minimization is taken over all product states |ϕ〉. Note that in the case of geometric measure the minimization is taken over pure states and not all mixed states like in the case of REE. Therefore it is usually easier to compute E_G rather than E_R. We call the product state that achieves the maximum overlap with |ψ〉 the closest product state. For general entangled states, E_G gives the lower bound to E_R, but for pure stabilizer states the measures are equal [20].

Consider a pure state $|\psi \rangle \in \mathcal {H}_1\otimes \cdots \otimes \mathcal {H}_N$ of an N-partite system that can be written as

$$\begin{eqnarray*} &&|\psi\rangle=\sum_{i=1}^{R}\xi_i|\psi_i^1\rangle\otimes\cdots\otimes|\psi_i^N\rangle, \end{eqnarray*}$$

where $\xi _i\in \mathbb {C}$. The Schmidt measure [9] is then defined as

$$\begin{eqnarray*} &&E_{\rm S}(|\psi\rangle)=\log R_{\min}, \end{eqnarray*}$$

where $R_{\min }$ is the minimal number of terms in the expansion of |ψ〉 over all linear decompositions into product states. Note that E_S is not continuous, but this does not pose a problem as the set of graph states is also discrete.

A graph is a pair G = (V,E) [21]. The elements of V ={1,...,N} are the vertices and the elements of E⊆[V ]² are the edges connecting the vertices. Pictorially one represents graphs as a set of dots, representing the vertices, connected by lines according to E, representing the edges. We consider only simple graphs, that is, graphs where the vertices are not connected to themselves so (a_i,a_i) ≠ E, and the graph contains no multiple edges between the same set of vertices. Furthermore, we concentrate only on connected graphs; that is, any two vertices a_i,a_j∈V are connected by a path in G.

The neighbourhood of a vertex a∈V , denoted by N_a, is defined as the set of all vertices that are adjacent to vertex a, N_a: = {b∈V |(a,b)∈E}. The degree d(a) of a vertex a is in our case the size of a's neighbourhood |N_a|. A vertex colouring is a map c:V →S such that c(v) ≠ c(w) when vertices v and w are adjacent. The elements of S are called the colours. A graph is called bipartite if it is two-colourable. If a graph cannot be coloured by only two colours, then it is called non-bipartite.

One crucial quantity that we make use of extensively is the independent set. An independent set is a set of vertices where no pair is adjacent. A maximum independent set is the largest independent set for a given graph and is denoted by α(G). A closely related concept is that of a vertex cover. A vertex cover is a set of vertices such that each edge of the graph is incident to at least one vertex in the vertex cover. The minimum vertex cover, β(G), is then naturally the smallest such set of G. The relationship between α(G) and β(G) is that they are complements of each other, which means that α(G) + β(G) = V . This means that finding one set automatically gives the other. However, identifying either the maximum independent set or the minimum vertex cover is a known NP-hard problem [22].

Finally, we mention a very useful transformation of a graph known as local complementation [23]. The local complement (LC) of G at vertex a, denoted by τ_a(G), is obtained by complementing the subgraph of G induced by the neighbourhood N_a, and leaving the rest of the graph unchanged. It is equivalent to adding a fully connected graph of vertices N_a, denoted by G(N_a), to the original graph modulo 2

$$\begin{eqnarray*} &&\tau_a:G\mapsto\tau_a(G):=G+G(N_a). \end{eqnarray*} \noindent$$

Applying local complementation to a vertex a adds edges between its neighbours where the addition is performed modulo 2 as demonstrated in figure 1.

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Consider a graph G and its associated graph state |G〉 which can be prepared by placing a qubit at each vertex in the state $|+\rangle =\frac {1}{\sqrt {2}}(|0\rangle +|1\rangle )$ and applying the entangling control-Z gate CZ_ij between all vertices i and j that are adjacent [24, 25]:

$$\begin{equation} |G\rangle=\prod_{(i,j)\in E}CZ_{ij}|+\rangle^{\otimes V}. \end{equation} \noindent \tag{ 1 }$$

An alternative and more efficient way of describing the graph states is by using the stabilizer formalism. The graph state is the unique, common eigenvector in $(\mathbb {C}^2)^V$ to the set of independent commuting observables {g_i}^N_i=1 given by [26]

$$\begin{equation} g_i:=X_i\bigotimes_{j\in N_i}Z_j. \end{equation} \noindent \tag{ 2 }$$

The eigenvalues of the correlation operators in equation (2) are +1 for all i∈V . The Abelian subgroup $\mathcal {S}$ of the Pauli group $\mathcal {P}^V$ generated by the set of all the correlation operators {g_i|i∈V } is called the stabilizer of |G〉. We can generate a basis for $(\mathbb {C}^2)^V$ using the graph state |G〉 by defining the set of states

$$\begin{eqnarray*} &&g_i|G_{k_{1}\ldots k_{i}\ldots k_{N}}\rangle=(-1)^{k_{i}}|G_{k_{1}\ldots k_{i}\ldots k_{N}}\rangle, \end{eqnarray*} \noindent$$

where k₁...k_N is a binary string. The original graph state given by equation (1) is simply |G_0...0〉. These states are all locally equivalent as $|G_{k_{1}\ldots k_{N}}\rangle =\prod _{i=1}^NZ_i^{k_{i}}|G_{0\ldots 0}\rangle$ and therefore they all have the same amount of entanglement. The projector onto the graph state can be written in the following nice form:

$$\begin{equation} |G\rangle\langle G|=\frac{1}{2^N}\sum_{\sigma\in\mathcal{S}}\sigma, \end{equation} \noindent \tag{ 3 }$$

where the sum is taken over all elements of the stabilizer $\mathcal {S}$.

A graph state |G〉 corresponds uniquely to some graph G. However, a situation may arise when two distinct graphs G and G' represent two graph states that are related by the some unitary operation |G'〉 = U|G〉. Their stabilizers transform accordingly $\mathcal {S}'=U\mathcal {S}U^{\dagger }=\{U\sigma U^{\dagger}|\sigma \in \mathcal {S}\}$. So it can easily happen that two seemingly different graphs represent graph states with the same amount of entanglement. In this paper, we consider such unitary operations that permute the elements of the Pauli group and therefore map stabilizers to other stabilizers. These transformations are called local Clifford operations, $C_1^V=\{U\in \mathbf {U}(2)^V|U\mathcal {P}^VU^{\dagger}=\mathcal {P}^V\}$. So two graph states |G〉 and |G'〉 are LC-equivalent iff they are related by some local Clifford unitary U∈C^V ₁. In fact, the action of local Clifford unitaries on graph states can be described graphically using local complementation transformations on the corresponding graph [27]. Applying the local complementation transformation to vertex a of graph G yields a new graph τ_a(G). The corresponding graph states |G〉 and |τ_a(G)〉 are then LC-equivalent and are related by the local Clifford operation |τ_a(G)〉 = U^τ_a|G〉 given by

$$\begin{eqnarray*} &&U_a^{\tau}=\sqrt{-\mathrm{i}X_a}\otimes\sqrt{\mathrm{i}Z_{N_{a}}}. \end{eqnarray*} \noindent$$

Two graph states |G〉 and |G'〉 are LC-equivalent if their corresponding graphs are related by a sequence of local complementations G' = τ_a1○···○τ_an(G). Furthermore, the local Clifford unitary relating two equivalent graph states can be efficiently found [28].

Finding a candidate separable state that can be used for quantifying the entanglement is often a very difficult task. However, proving that this state achieves the minimum value of relative entropy or the maximum overlap is equally difficult. Fortunately, the situation for many classes of graph states is such that the lower and upper bounds on the entanglement measures coincide; therefore the exact value of entanglement is known. This makes direct evaluation of entanglement measures easier because it is enough to find separable state ω and product state |ϕ〉 that yield this value for E_R and E_G, respectively. In the case of the Schmidt measure E_S, knowing its value tells us the minimum number of terms that the linear decomposition of |G〉 into product states needs to contain.

In [12] the authors showed how the upper and lower bounds for REE and geometric measure can be found. The upper bound is obtained by considering perfect discrimination using local operations and classical communication (LOCC) of a subset of the complete orthogonal graph state basis {|G_k1...kN〉}. A lower bound on the number of these states that can be discriminated is given by maximizing the number of stabilizer generators {g_i} that can be determined in a single setting of LOCC measurements. This bound can be achieved by identifying the maximum independent set α(G) of the graph G and measuring X on these qubits and Z on their neighbours. This can, in fact, be used to find an upper bound on the REE and the geometric measure N − |α(G)| ⩾ E_R = E_G [12].

The lower bound for the entanglement can be obtained in the usual way of relaxing the condition of full separability and maximizing the entanglement E_bi between all the bipartitions of the graph. This can be viewed as creating the maximum number of Bell pairs m_p between the bipartitions by CZ and local Clifford operations applied within the bipartitions. Summarizing, the lower and upper bounds for the REE and the geometric measure are [12]

$$\begin{eqnarray*} &&N-|\alpha(G)|\geqslant E_{\rm R}=E_{\rm G}\geqslant E_{\rm bi}=m_{\mathrm{p}}. \end{eqnarray*} \noindent$$

Classes of graph states for which these bounds are equal include d-dimensional cluster states, Greenberger–Horne–Zeilinger (GHZ) states and ring states with an even number of qubits.

The bounds for the Schmidt measure can be found in a similar fashion [16, 26]. The lower bound is given by the maximal Schmidt rank ${\rm SR}_{\max }$, which for graph states is equal to the entropy of entanglement ${\rm SR}_{\max }=-{\rm Tr}[\rho ^A\log \rho ^A]$, where A⊆V is a subset of the vertices. The upper bound is given by the minimal number of local Pauli measurements needed to completely disentangle the graph state. This quantity is referred to as the Pauli persistency, PP(G), and is bounded from above by the size of the minimum vertex cover |β(G)| [24]. So the Schmidt measure is bounded as follows:

$$\begin{eqnarray*} &&|\beta(G)|\geqslant{\rm PP}(G)\geqslant E_{\rm S}\geqslant {\rm SR}_{\max}. \end{eqnarray*} \noindent$$

States up to 7 and 8 qubits for which these bounds are equal are categorized in [26] and [17], respectively.

To understand when the lower and upper bounds are equal, we first demonstrate how the lower bound for all three entanglement measures can be found in terms of a single quantity from graph theory.

A matching M of a graph G is a set of independent edges. Two edges are independent when they do not have any common vertices. A subset of vertices U⊆V is called matched if every vertex a∈U is incident to an edge in M. If the entire vertex set V is matched by M, then it is called a perfect matching.

The notion of matching is very useful because the cardinality of maximum matching |M_max| is equal to the maximum number of Bell pairs that can be created for a bipartition of G as well as the maximal Schmidt rank ${\rm SR}_{\max }$.

At this point we would like to stress one crucial point that has been implicitly covered in section 2.3. We have explained that two graph states |G〉 and |G'〉 which are related by a local Clifford transformation have underlying graphs G and G' related by a series of local complementation operations. However, the properties of the two graphs may be very different. In particular, their maximum matching and minimum vertex cover may have different cardinalities. One such example is the GHZ state pictured in figure 2. This graph state corresponds to a star graph which can be transformed to a complete graph using local complementation. So the graph states corresponding to a star graph and a complete graph are LC-equivalent and therefore have the same entanglement of 1. However, for the star graph we have |M_max| = 1 and |β(G)| = 1 whereas the complete graph has $|M_{\mathrm { max}}|=\left \lfloor \frac {N}{2}\right \rfloor$ and |β(G)| = N − 1.

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To overcome this apparent contradiction, it is necessary to consider the entire LC-equivalency class of a graph G and compute the size of M_max and β(G) for all graphs in the equivalency class to find the minimum. Such canonical forms for all LC-equivalency classes have been identified up to 7 qubits in [16, 26], up to 8 qubits in [17] and up to 12 qubits in [29]. For a graph state up to 12 qubits, it is enough to find to which canonical form it is equivalent. However, for a general graph state of N qubits, one has to first generate the entire LC orbit of the underlying graph. Therefore, when computing the lower and upper bounds for the entanglement of a graph state, it is not enough to consider only the graph itself. The entire LC orbit needs to be taken into account.

The overall complexity of such a search is not so obvious to determine. The first obstacle lies in the fact that it is not known whether the LC orbit can be generated efficiently. Furthermore, we are interested in non-isomorphic connected graphs within the LC orbit. So even if we generate the LC orbit we have to determine which graphs are non-isomorphic. The graph isomorphism problem is one of the few computational problems whose complexity is still unresolved. It is known that it is contained in the NP class; however, it is not known whether it has an efficient solution or is NP-complete, see [30]. Finally, we have to compute the maximum matching M_max and the maximum independent set α(G) for each non-isomorphic graph in the LC orbit. A polynomial algorithm exists for calculating the M_max [31]; however, it is known that the maximum independent set problem is NP-complete. For some special cases, such as bipartite graphs, α(G) can be found efficiently but it is unlikely that this is true in the general case. In section 3.3 we consider various non-bipartite lattices and evaluate α(G) exactly by exploiting symmetries in the lattice.

In the rest of this paper we will assume that the canonical form of the underlying graph with the smallest M_max and β(G) across the LC orbit is known.

Consider a graph G with a maximum matching M_max. Select either endvertex from each independent edge in M_max to form one partition A. The other partition is then given by all the other vertices in V , that is, B = V −A. Apply control-Z gates locally within the partitions to remove any edges that do not cross the bipartition boundary. As a result of this, only edges that contribute towards entanglement across the bipartition are kept. In certain cases this transformation leaves all the vertices in the partitions with either degree 0 or degree 1. This corresponds to the case of Bell pairs being created across the bipartition. However, in the general case, some vertices will have degree higher than 1. Then we need to apply a series of local complementation transformations along with further control-Z gates to either decrease their degree to 1 or completely disconnect them from the rest of the graph. A crucial observation here is that none of the above transformations delete an edge from the maximum matching M_max. Therefore, this procedure produces |M_max| Bell pairs across the bipartition. This method of transforming the graph state has been introduced in [12] and demonstrated for a selection of graph states with underlying bipartite graphs. Figure 3 displays the procedure for a particular example of a non-bipartite graph.

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Having established that the lower bound for all three measures is given by the size of the maximum matching |M_max| and the upper bound by the cardinality of the minimum vertex cover |β(G)|, we will now investigate when these bounds are equal and when not. To do this we will partition the graph states into the set of bipartite and non-bipartite states, and consider them separately.

Bipartite graph states. For all bipartite graph states the lower and upper bounds coincide. This can be seen immediately using König's Theorem [21] which states that for bipartite graphs the sizes of maximum matching and of minimum vertex cover are equal, |M_max| = |β(G)|.

Non-bipartite graph states. The situation for non-bipartite graph states is more complicated as König's Theorem does not hold anymore and in general |M_max| ≠ |β(G)|. However, there are still numerous cases when the two bounds are equal. In fact, the crucial quantity that determines whether the bounds are equal is the size of the maximum independent set |α(G)|:

• 1.  
  If $|\alpha (G)|<\frac {N}{2}$, then |M_max| ≠ |β(G)|.
• 2.  
  If $|\alpha (G)|>\frac {N}{2}$, then |M_max| = |β(G)|.
• 3.  
  If $|\alpha (G)|=\frac {N}{2}$, then we have the following two scenarios:

Statement 1 can be proven easily. For a graph state of N qubits, the maximum number of Bell pairs that can be created is $|M_{\mathrm { max}}|\leqslant \left \lfloor \frac {N}{2}\right \rfloor$. However, the upper bound for entanglement is given by $N-|\alpha (G)|>\frac {N}{2}$ from our assumption. Therefore the bounds are not equal, |M_max| ≠ |β(G)|.

Statement 2 becomes clear when one considers a bipartition of the graph G with partition A given by the vertices in the maximum independent set A: = {v|v∈α(G)} and partition B given by all the other vertices, that is, the minimum vertex cover B: = {v|v∈β(G)}. It is clear that all vertices v∈A are not connected. Now consider the reduced neighbourhood of a vertex a∈B defined as N^red_a: = {v∈A|(a,v)∈E}. So N^red_a is just the usual neighbourhood with vertices in B removed. When looking for the maximum matching of graph G, we can consider each N^red_a separately. The crucial point to note here is that since all v∈N^red_a are incident on a single vertex a∈B, we can only pick one edge that contributes to the maximum matching. The fact that |M_max| = |β(G)| follows straightaway. Including edges between vertices in B in the matching can only decrease the matching's cardinality, so they have been omitted from the argument. This is illustrated in figure 4.

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Statement 3 will become clear if one realizes what it means to have a perfect matching. The upper bound for entanglement in both the cases (a) and (b) is given by $|\beta (G)|=N-|\alpha (G)|=\frac {N}{2}$. If the matching M_max is perfect, then each vertex in the graph is matched. Since a matching is a set of independent edges, this implies that $|M_{\mathrm { max}}|=\frac {N}{2}$. Therefore the entanglement bounds coincide, |M_max| = |β(G)|. On the other hand if M_max is not a perfect matching, then there exists at least one vertex which is not matched by M_max. Hence the number of vertices that are matched by M_max is upper bounded by N − 1. We know that N is even, so the size of the maximum matching is upper bounded by $|M_{\mathrm { max}}|\leqslant \frac {N-2}{2}$. This shows that if $|\alpha (G)|=\frac {N}{2}$ and the maximum matching is not perfect, then the bounds for entanglement do not coincide.

As mentioned in section 2.3 the stabilizer $\mathcal {S}$ of an N-qubit graph state generated by N generators g_i given by equation (2) stabilizes a unique entangled state. Consider a scenario where we discard a set of k generators. The new stabilizer is given by $\mathcal {S}_{N-k}=\langle g_1,\ldots ,g_{N-k}\rangle$ where we have relabelled the remaining generators from $\mathcal {S}$ for convenience. Because we no longer have a full set of N generators, $\mathcal {S}_{N-k}$ does not stabilize a unique state. Rather it stabilizes a set of states {|ψ₁〉,...,|ψ_D〉} that span a D-dimensional subspace. The dimensionality of this subspace depends on the particular structure of the generators $g_i\in \mathcal {S}_{N-k}$ and the states {|ψ_i〉} may be entangled or product states. We are interested in the case when $\mathcal {S}_{N-k}$ stabilizes a set of product states.

Observation 1 $\mathcal {S}_{N-k}$ stabilizes entangled states if it contains at least two generators $g_a,g_b\in \mathcal {S}_{N-k}$ where g_b contains Z_a.

Such a scenario happens when the generators g_a and g_b correspond to adjacent qubits, that is, (a,b)∈E. Generator g_a stabilizes some subspace $\mathcal {H}_a$ spanned by states {|ψ^(a)_i〉} and g_b stabilizes a subspace $\mathcal {H}_b$ spanned by states {|ψ^(b)_j〉}. Since qubits a and b are adjacent, the action of g_a on the states {|ψ^(b)_j〉} is to permute them and the same is true for g_b acting on {|ψ^(a)_i〉}. In order for g_a to stabilize the states {|ψ^(b)_j〉} we have to take superpositions of these states according to how they are permuted by g_a. This finally yields states that are stabilized by $\mathcal {S}_{N-k}$; however, due to the superpositions the stabilized states are entangled. We illustrate this in the following example.

Example 1. Consider a three-qubit open linear graph state given by the stabilizer

$$\begin{eqnarray*} &&\mathcal{S}=\langle \underbrace{XZI}_{g_{1}}, \underbrace{ZXZ}_{g_{2}}, \underbrace{IZX}_{g_{3}}\rangle. \end{eqnarray*}$$

Say, we discard the third generator so the new stabilizer is given by $\mathcal {S}_{2}=\langle XZI, ZXZ\rangle$. Each generator stabilizes a set of states

$$\begin{eqnarray*} g_1 & = & XZI:\{|+0.\rangle,|-1.\rangle\}, \\ g_2 & = & ZXZ:\{|0+0\rangle,|1+1\rangle,|0-1\rangle,|1-0\rangle\}, \end{eqnarray*}\noindent$$

where the states of the form | + 0.〉 mean that the generator does not fix the last qubit. We can quickly check that the action of g₁ on the states stabilized by g₂ is to permute them, that is, g₁|0 + 0〉 = |1 − 0〉 and g₁|1 + 1〉 = |0 − 1〉. Requiring that g₁ stabilizes all the above states we are forced to take superpositions of the states, which finally yields the states that are stabilized by $\mathcal {S}_{2}$ to be

$$\begin{eqnarray*} &&\left\{\frac{1}{\sqrt{2}}(|0+\rangle+|1-\rangle)\otimes|0\rangle,\frac{1}{\sqrt{2}}(|0-\rangle+|1+\rangle)\otimes|1\rangle\right\}.\end{eqnarray*}\noindent$$

The stabilized subspace is two dimensional and because of the form of the generators of $\mathcal {S}_2$ it is spanned by entangled states. On the other hand, we can quickly check that discarding g₂ from the original $\mathcal {S}$ produces a stabilized subspace spanned by product states {| + 0 + 〉,| − 1 − 〉}.

The above observation tells us exactly when $\mathcal {S}_{N-k}$ stabilizes a set of product states. The desired structure of the generators $g_i\in \mathcal {S}_{N-k}$ is achieved when the generators correspond to non-adjacent qubits. As a vertex colouring of the underlying graph G partitions the set of qubits into subsets of independent qubits, the possible ways of discarding generators from the original stabilizer $\mathcal {S}$ are given by all the possible vertex colourings that G admits. More specifically, if the generators corresponding to qubits of the same colour are the only generators not discarded, then $\mathcal {S}_{N-k}$ stabilizes a set of product states. Note that the converse is not true as it is possible for $\mathcal {S}_{N-k}$ to contain generators corresponding to different colours and still stabilize a set of product states. This is illustrated in figure 5.

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Knowing how to discard generators from $\mathcal {S}$ to produce a product basis, we can focus on the important question of doing this optimally. Ideally, we would like to discard as few generators as possible. In light of the above discussion, it is straightforward to see that the generators that need to be kept correspond to qubits in the maximum independent set α(G), or equivalently, the generators that need discarding correspond to the minimum vertex cover β(G). A similar approach has been used to compute upper bounds for the three entanglement measures in [12, 16, 26] although our logic for deriving this result is complementary to the approach in these references.

We will now show how the entanglement measures can be evaluated directly. We refer to the stabilizer of generators corresponding to qubits in the maximum independent set α(G) as $\mathcal {S}_{\alpha }$, the stabilized subspace as $\mathcal {H}_{\alpha }$ and product states spanning this subspace as {|ψ^α_i〉}.

Theorem 1 Given a graph state |G〉, its minimal linear decomposition into product states is a superposition of states {|ψ^α_i〉}

$$\begin{equation} |G\rangle=\frac{1}{\sqrt{D_{\alpha}}}\sum_{i=1}^{D_{\alpha}}(-1)^{f_i(\mathcal{S})}|\psi^{\alpha}_i\rangle, \end{equation} \tag{ 4 }$$

where $f_i(\mathcal {S})$ is a binary-valued function and its value depends on the action of the original stabilizer $\mathcal {S}$ on the states |ψ^α_i〉. This ensures that the form of |G〉 in equation (4) is stabilized by the whole $\mathcal {S}$. D_α is the dimension of the subspace $\mathcal {H}_{\alpha }$ and depends on the size of the minimum vertex cover as D_α = 2^|β(G)|.

Proof. The proof consists of two parts. The first part shows that the decomposition in equation (4) actually gives the graph state |G〉. The second part shows that the decomposition achieves the minimum bound of the Schmidt measure E_S(|G〉).

From observation 1 we can be sure that the states |ψ^α_i〉 are product states. Furthermore, since these states are stabilized by $\mathcal {S}_{\alpha }$ we have g_j|ψ^α_i〉 = |ψ^α_i〉 for all $j\in \mathcal {S}_{\alpha }$ and i∈{1,...,D_α}. All we have to determine is the action of generators in the minimum vertex cover β(G) on the states |ψ^α_i〉. Consider the action of one of the generators in the minimum vertex set g_k, k∈β(G). Due to the structure of the correlation operators this permutes the elements in the stabilized set g_k|ψ^α_j〉 = |ψ^α_i〉. If the qubit k is adjacent to another qubit in the minimum vertex cover, then the action of g_k may be to also introduce a negative sign, g_k|ψ^α_j〉 = −|ψ^α_i〉. By taking superpositions of states |ψ^α_i〉 that are permuted with each other and including the negative amplitudes, we can construct a new set of states |ψ'_i〉 that are stabilized by the generators g_i∈α(G) as well as the new generator g_k∈β(G). The new states will be of the form

$$\begin{eqnarray*} &&|\psi'_i\rangle=|\psi^{\alpha}_i\rangle+g_k|\psi^{\alpha}_i\rangle. \end{eqnarray*}$$

Repeating this process with a new generator g_l where l∈β(G) acting on the states |ψ'_i〉, we obtain a new set of states stabilized by $\mathcal {S}_{\alpha }$ as well as g_k,g_l. Extending this procedure to all of the generators in β(G) we finally arrive at the state in equation (4).

Showing that this decomposition is also minimal becomes now trivial for graph states for which |M_max| = |β(G)| as logD_α reaches the lower bound for the Schmidt measure known from [16, 26]. Therefore the entanglement as evaluated by the Schmidt measure is given by

$$\begin{eqnarray*} &&E_{\rm S}(|G\rangle)=\log D_{\alpha}, \end{eqnarray*}$$

which concludes the proof.   □

Observation 2. Bipartite graph states for which the maximum independent set α(G) is also given by a two-colouring of G can be written as $|G\rangle =\frac {1}{\sqrt {D_{\alpha }}}\sum _{i=1}^{D_{\alpha }}|\psi ^{\alpha }_i\rangle$. Because any qubit in the minimum vertex cover β(G) has all its neighbours in the maximum independent set α(G), the action of any generator $g_k\in \mathcal {S}_{\beta }$ is to only permute the states stabilized by $\mathcal {S}_{\alpha }$, that is, g_k|ψ^α_j〉 =  |ψ^α_i〉 for i,j∈α(G). Therefore all the amplitudes in the linear decomposition of equation (4) are positive.

Using the decomposition in equation (4) we can now compute the other two measures.

Theorem 2 The CSS ω to a given graph state |G〉 is given by

$$\begin{equation} \omega=\frac{1}{D_{\alpha}}\sum_{i=1}^{D_{\alpha}}|\psi^{\alpha}_i\rangle\langle\psi^{\alpha}_i|. \end{equation} \tag{ 5 }$$

So the CSS is given by an equal mixture of the states stabilized by $\mathcal {S}_{\alpha }$.

Proof. States of the form given by equation (5) are clearly separable because they are a mixture of product states. Computing the relative entropy between ρ = |G〉〈G| and ω we obtain

$$\begin{eqnarray*} S(\rho||\omega) & = & -{\rm Tr}[\rho\log\omega] \\ & = & -\frac{1}{D_{\alpha}}\log\frac{1}{D_{\alpha}}\sum_{ijk}^{D_{\alpha}}(-1)^{f_{i}(\mathcal{S})+f_{j}(\mathcal{S})}\delta_{jk}\delta_{ki} \nonumber \\ & = & \log D_{\alpha}. \end{eqnarray*}$$

For graph states that have |M_max| = |β(G)|, this is the lower bound found in [12]. Therefore ω is the CSS and the REE is given by

$$\begin{eqnarray*} &&E_{\rm R}(|G\rangle)=\log D_{\alpha}. \end{eqnarray*}$$

This concludes the proof.   □

The CSS ω can be expressed in a similar form to equation (3) as the sum of all the elements in $\mathcal {S}_{\alpha }$.

Observation 3. We can write the CSS as

$$\begin{equation} \omega=\frac{1}{2^{N}}\sum_{\sigma\in\mathcal{S}_{\alpha}}\sigma. \end{equation} \tag{ 6 }$$

To see this, we consider the action of equations (5) and  (6) on the graph-state basis $|G_{\vec {k}}\rangle =Z^{\vec {k}}|G_{00\ldots 0}\rangle$. Here $\vec {k}$ is a bit-string of length N, $Z^{\vec {k}}=Z^{k_{1}}_1\otimes \cdots \otimes Z^{k_{N}}_N$ and |G_00...0〉 is the usual graph state in equation (4). For clarity we distinguish the two forms of the CSS by using $\omega _1=\frac {1}{D_{\alpha }}\sum _{i=1}^{D_{\alpha }}|\psi ^{\alpha }_i\rangle \langle \psi ^{\alpha }_i|$ and $\omega _2=\frac {1}{2^N}\sum _{\sigma \in \mathcal {S}_{\alpha }}\sigma$.

Let us first consider $\omega _1|G_{\vec {k}}\rangle$. The structure of |ψ^α_i〉 is such that qubit a is expressed in the Z-basis if vertex a∈β(G) and in the X-basis if a∈α(G). This implies the following:

$$\begin{eqnarray*} \omega_1|G_{\vec{k}}\rangle = \left\{ \begin{array}{@{}ll@{}} 2^{-|\beta(G)|}|G_{\vec{k}}\rangle & {\rm if }\ k_a=0\ {\rm for\ all }\ a\in\alpha(G), \\ 0 & {\rm otherwise}. \end{array} \right. \end{eqnarray*}$$

Therefore ω₁ has the following form in the graph-state basis:

$$\begin{eqnarray*} &&\langle G_{\vec{k'}}|\omega_1|G_{\vec{k}}\rangle=2^{-|\beta(G)|}\delta_{\vec{k'}\vec{k}}, \end{eqnarray*}$$

where $\vec {k}$ has a particular form such that $Z^{\vec {k}}$ acts trivially on all vertices a∈α(G). If $Z^{\vec {k}}$ acts non-trivially on any a∈α(G), then the matrix element vanishes.

Now we consider the action of ω₂ on the graph-state basis $|G_{\vec {k}}\rangle$ which depends on whether $Z^{\vec {k}}$ commutes or anticommutes with ω₂. If $[Z^{\vec {k}},\omega _2]=0$, then we have $\omega _2Z^{\vec {k}}|G_{0\ldots 0}\rangle =Z^{\vec {k}}(2^{-N}\sum _{\sigma \in \mathcal {S}_{\alpha }}\sigma )|G_{0\ldots 0}\rangle =2^{-|\beta (G)|}|G_{\vec {k}}\rangle$, where we used $|\mathcal {S}_{\alpha }|=2^{|\alpha (G)|}$, |α(G)| + |β(G)| = N and σ|G_0...0〉 = |G_0...0〉 for all $\sigma \in \mathcal {S}_{\alpha }$. Again from the construction of $\sigma \in \mathcal {S}_{\alpha }$ we see that $Z^{\vec {k}}$ commutes with ω₂ only if $Z^{\vec {k}}$ acts trivially on all qubits in the maximum independent set α(G). If $\{Z^{\vec {k}},\omega _2\}=0$, then $Z^{\vec {k}}$ anticommutes with exactly half of the terms in ω₂, which leads to $\omega _2|G_{\vec {k}}\rangle =0$. Therefore we have

$$\begin{eqnarray*} &&\langle G_{\vec{k'}}|\omega_2|G_{\vec{k}}\rangle=2^{-|\beta(G)|}\delta_{\vec{k'}\vec{k}}, \end{eqnarray*} \noindent$$

where again $Z^{\vec {k}}$ acts trivially on all $a\in \mathcal {S}_{\alpha }$. This established equivalence between the two forms of ω in equations (5) and  (6).

Theorem 3 The closest product state |ϕ〉 is given by any state from the stabilized set {|ψ^α_i〉},

$$\begin{equation} |\phi\rangle=|\psi^{\alpha}_i\rangle,\quad\forall i\in\{1,\ldots,D_{\alpha}\}. \end{equation} \tag{ 7 }$$

Proof. Using the minimum linear decomposition into product states in equation (4) and substituting |ϕ〉 = |ψ^α_i〉, we find that the geometric measure is

$$\begin{eqnarray*} E_{\rm G}(|G\rangle) & = & -\log\left|\left\langle\psi^{\alpha}_i\left|\frac{1}{\sqrt{D_{\alpha}}}\sum_{j=1}^{D_{\alpha}}(-1)^{f_{j}(\mathcal{S})}\right|\psi^{\alpha}_j\right\rangle\right|^2 \\ & = & \log D_{\alpha}. \end{eqnarray*} \noindent$$

This is the lower bound found in [12] and therefore concludes the proof.   □

We demonstrate how to quantify the entanglement measures in the following example of a 6-qubit graph state.

Example 2. Consider a graph state |G〉 of 6 qubits with an underlying graph G pictured in figure 6.

The graph state |G〉 is stabilized by

$$\begin{eqnarray*} \mathcal{S} & = & \langle \underbrace{XIIIIZ}_{g_1},\underbrace{IXIIIZ}_{g_2},\underbrace{IIXIZI}_{g_3}, \\ & & \underbrace{IIIXZI}_{g_4},\underbrace{IIZZXZ}_{g_5},\underbrace{ZZIIZX}_{g_6}\rangle. \end{eqnarray*}$$

The maximum independent set can be quickly obtained by a three-colouring and is α(G) = {1,2,3,4}. Therefore, the corresponding stabilizer is $\mathcal {S}_{\alpha }=\langle g_1,g_2,g_3,g_4\rangle$. |α(G)| > 3, which means that the lower and upper bounds coincide. The stabilized states given by $\mathcal {S}_{\alpha }$ are

$$\begin{eqnarray*} \{|\psi^{\alpha}_i\rangle\}=\{|++++00\rangle,|&ndash;++01\rangle,|++&ndash;10\rangle,|----11\rangle\}. \end{eqnarray*}$$

The action of the stabilizer $\mathcal {S}_{\beta }$ associated with the minimum vertex cover is the following:

$$\begin{eqnarray*} g_5|\psi^{\alpha}_1\rangle & = & |\psi^{\alpha}_3\rangle \quad {\rm and}\quad g_5|\psi^{\alpha}_2\rangle=-|\psi^{\alpha}_4\rangle, \\ g_6|\psi^{\alpha}_1\rangle & = & |\psi^{\alpha}_2\rangle \quad {\rm and}\quad g_6|\psi^{\alpha}_3\rangle=-|\psi^{\alpha}_4\rangle, \end{eqnarray*}$$

which means that the graph state can be written in the following form:

$$\begin{eqnarray*} &&|G\rangle=\textstyle\frac{1}{2}[|\psi^{\alpha}_1\rangle+|\psi^{\alpha}_2\rangle+|\psi^{\alpha}_3\rangle-|\psi^{\alpha}_4\rangle]. \end{eqnarray*}$$

Using the CSS ω in equation (5) and the closest product state in equation (7), we can finally show that for this graph state |G〉

$$\begin{eqnarray*} &&E_{\rm S}(|G\rangle)=E_{\rm R}(|G\rangle)=E_{\rm G}(|G\rangle)=2. \end{eqnarray*}$$

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Knowing how to find the CSS ω to a given graph state |G〉, it is now possible to determine the form of the CSSs to all of the graph states in the orbit generated by local Clifford operations. Two graph states are LC-equivalent, |G'〉 = U^LC|G〉, if they are related by a local unitary U^LC. Using equation (4), |G'〉 can be expressed as a superposition of states U^LC|ψ^α_i〉. Applying the unitary we see that the CSS has the form $\omega '=\frac {1}{D_{\alpha }}\sum U^{\mathrm { LC}}|\psi ^{\alpha }_i\rangle \langle \psi ^{\alpha }_i|U^{\mathrm {LC}\dagger }$.

An immediate consequence of this approach to analysing entanglement is that we can identify which graph states are maximally entangled in many cases. By maximally entangled we mean with respect to the three measures we are considering. For any graph state that is maximally entangled it must be true that E_R = E_G ⩾ |M_max| and E_S ⩾ |M_max|. For bipartite graph states, these measures cannot be larger than this value due to König's Theorem [21]. Therefore any bipartite graph state |G〉 whose underlying graph G has the property that $|M_{\mathrm { max}}|=\left \lfloor \frac {N}{2}\right \rfloor$ is maximally entangled. Examples of such states include linear graph states with open boundaries, ring states with even N and cluster states in d dimensions. This holds true also for non-bipartite graph states for which the bounds are equal. An example of such a state is pictured in figure 4.

The situation is quite different for the case of graph states with unequal entanglement bounds. Again any graph state that is maximally entangled must have $|M_{\mathrm { max}}|=\left \lfloor \frac {N}{2}\right \rfloor$. However, the true value of entanglement is now unclear. For general graph states it is not even possible to calculate this bound. Therefore, we limit ourselves to various types of regular two-dimensional lattices for the rest of this section.

The lattices that we consider are pictured in figure 7. They are the triangular, kagome, hexa-triangular and hexagonal lattices. The triangular, kagome and hexagonal lattices have been shown to be universal resources for measurement-based quantum computation [32]. We are interested in the scaling of the gap between upper and lower bounds Δ = |β(G)| − |M_max|. For the hexagonal lattice, this gap is trivial since the lattice is bipartite and therefore Δ_hexagonal = 0. For the other three lattices that gap is

$$\begin{eqnarray*} &&\Delta_{\rm tri} = \left\lceil\frac{N}{2}\right\rceil-\sqrt{N}-2\sum_{j=1}^{\left\lfloor\frac{N-1}{3}\right\rfloor}(\sqrt{N}-3j),\\ &&\Delta_{\rm hex\mbox{-}tri} = \frac{1}{18}(12N-3\sqrt{9+12N}+9)-\left\lfloor\frac{N}{2}\right\rfloor, \\ &&\Delta_{\rm kag} = \frac{1}{9}(6N-\sqrt{13+3N}-11)-\left\lfloor\frac{N}{2}\right\rfloor, \end{eqnarray*} \noindent$$

where Δ_tri is valid for L > 3, with L² = N being the number of vertices on one side of the triangular lattice. The particular form of Δ depends on the boundary conditions. However, the general behaviour of the scaling remains unchanged for different boundary conditions. Interestingly, the lower bound for all four lattices is the same, $|M_{\mathrm { max}}|=\left \lfloor \frac {N}{2}\right \rfloor$. The scaling of Δ is pictured in figure 8.

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We briefly highlight the PEPs description of entangled states [18]. Consider a graph state |G〉 with N qubits. A PEPs $|\Psi \rangle \in \mathbb {C}^{2^{N}}$ is constructed by replacing a physical qubit a by |N_a| virtual qubits where |N_a| denotes the degree of vertex a. Each physical edge (a,b) is then replaced by the maximally entangled state of the corresponding two virtual qubits $|G_2\rangle =\frac {1}{\sqrt {2}}(|0+\rangle +|1-\rangle )$. The original graph state |G〉 can then be obtained by applying a projector P_a: = |0〉〈0|^a1 ...〈0|^a|N_a|  + |1〉〈1|^a1 ...〈1|^a|N_a| at each physical site.

The above approach can be adapted to describe separable mixed states with few changes. Instead of maximally entangled pairs of virtual qubits the basic building blocks are maximally correlated separable pairs of qubits ω₂. Maximally correlated in this sense means that the relative entropy between the separable states and the tensor product of its subsystems is unity, $S(\omega _2||\omega _2^{(1)}\otimes \omega _2^{(2)})=S(\omega _2||\frac {1}{4}I\otimes I)=1$ where ω⁽¹⁾₂ = Tr₂[ω₂] and similarly for ω⁽²⁾₂. In fact, it will be necessary to use two various separable states ω₂. The projectors being applied to physical sites will also have a different structure compared to the case of pure entangled states.

Define two 2-qubit maximally correlated separable states of virtual qubits i' and j'

$$\begin{eqnarray} &&\omega_{i'j'}^A:=|+0\rangle\langle+0|+|-1\rangle\langle-1|,\nonumber\\ &&\omega_{i'j'}^B:=|0+\rangle\langle0+|+|1-\rangle\langle1-|. \end{eqnarray} \tag{ 8 }$$

Here and in the rest of this subsection we omit the normalization constants. Virtual qubits are denoted by primed letters a' and physical sites by a. These states are, in fact, both CSSs to a 2-qubit graph state. It is also useful to give these states a graphical representation depicted in figure 9. Using the separable states in equation (8) we construct a separable state of virtual qubits by placing either ω^A_i'j' or ω^B_i'j' on the edges of the graph state. The 2-qubit separable states are picked in such a way that all virtual qubits at a physical site a∈α(G) are orange. In the case of bipartite graph states this means that all virtual qubits at physical sites b∈β(G) will be of the same colour, blue. However, this is not true anymore in the case of non-bipartite graph states where virtual qubits at a physical site b∈β(G) will be of both colours. This is illustrated in figure 8. Finally, in analogy to the usual PEPs construction, the virtual qubits at physical sites are projected using the following maps:

$$\begin{eqnarray*} P_a^A & := & |0\rangle\langle0\ldots0|+|1\rangle\langle1\ldots1|, \\ P_a^B & := & |+\rangle\langle\tilde{+}|+|-\rangle\langle\tilde{-}|. \end{eqnarray*}$$

The projector P^A_a is applied if the physical site a∈β(G) contains at least one virtual qubit of blue colour and P^B_a is applied if a∈α(G), which means that all its virtual qubits are orange. $|\tilde {+}\rangle$ is an equal superposition of all tensor product states {| + 〉,| − 〉} which are +1 eigenstates of X⊗···⊗X. Similarly, $|\tilde {-}\rangle$ is an equal superposition of all −1 eigenstates of X⊗···⊗X.

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How this construction works is most easily seen when illustrated with an explicit example. Consider an open linear graph state of 4 qubits as in figure 8. Following the two-colouring of the graph the tensor product of six virtual qubits takes the following form:

$$\begin{eqnarray*} &&\omega_6'=\omega_{1'2'}^A\otimes\omega_{3'4'}^B\otimes\omega_{5'6'}^A. \end{eqnarray*}$$

The desired 4-qubit CSS is then obtained by applying the projector

$$\begin{eqnarray*} &&\omega_4=(P_2^A\otimes P_3^B)\omega_6'(P_2^A\otimes P_3^B), \end{eqnarray*}$$

where P^A₂ = |0〉〈00| + |1〉〈11| acts on physical site 2 and P^B₃ = | + 〉〈 + +| + | + 〉〈 − −| + | − 〉〈 + −| + | − 〉〈 − + | acts on physical site 3.

In this subsection we focus on the third approach of constructing the CSS ω. The common feature of the two previous approaches is the necessity of identifying either the maximum independent set α(G) or the minimum vertex cover β(G). This remains true for the approach presented here as well. However, this time we ask whether there is a simple way of obtaining ω by introducing noise to the pure state rather than resorting to methods based on stabilizer generators or pairs of maximally correlated separable states.

This new approach relies on introducing distinct relative phases to certain qubits and then averaging over them to obtain ω. The minimum number of the relative phases is equal to the cardinality of the minimum vertex cover |β(G)|.

The graph state vector can be written in the following form [24]:

$$\begin{equation} |G\rangle=\frac{1}{2^{N/2}}\bigotimes_{j=1}^{N}(|0\rangle_j+|1\rangle_j Z_{N'_j}), \end{equation} \tag{ 9 }$$

where the Pauli Z matrix is applied to a subset of qubit j's neighbourhood N'_j: = {i∈N_j|i > j}. Now let us define a new vector |Φ〉 that differs from the graph state in equation (9) in that it contains the above-mentioned relative phases ϕ_j

$$\begin{eqnarray*} &&|\Phi\rangle:=\frac{1}{2^{N/2}}\bigotimes_{j=1}^N(|0\rangle_j+\mathrm{e}^{\mathrm{i}\phi_jm(j)}|1\rangle_j Z_{N'_j}), \end{eqnarray*}$$

where m(j):V →{0,1} is an indicator function from the set of all vertices V given by

$$\begin{eqnarray*} m(j) := \left\{ \begin{array}{@{}ll@{}} 0 & {\rm if }\ j\in\alpha(G),\\ 1 & {\rm if }\ j\in\beta(G). \end{array} \right. \end{eqnarray*}$$

So the relative phase ϕ_j is applied only to qubits that correspond to vertices in the minimum vertex cover. For example, a 3-qubit open linear graph state with a relative phase is $|\Phi \rangle =\frac {1}{2\sqrt {2}}(|0\rangle _1+|1\rangle _1Z_2)\otimes (|0\rangle _2+\mathrm {e}^{\mathrm {i}\phi _{2}}|1\rangle _2Z_3)\otimes (|0\rangle _3+|1\rangle _3)$. Finally, the CSS is given by averaging over these phases

$$\begin{eqnarray*} &&\omega=\frac{1}{(2\pi)^{|\beta(G)|}}\int_{0}^{2\pi}\mathrm{d}\phi|\Phi\rangle\langle\Phi|, \end{eqnarray*}$$

where dϕ = Π_{{j|j∈β(G)}} dϕ_j.