Graphical description of unitary transformations on hypergraph states
Abstract
Hypergraph states form a family of multiparticle quantum states that generalizes cluster states and graph states. We study the action and graphical representation of nonlocal unitary transformations between hypergraph states. This leads to a generalization of local complementation and graphical rules for various gates, such as the CNOT gate and the Toffoli gate. As an application, we show that already for five qubits local Pauli operations are not sufficient to check local equivalence of hypergraph states. Furthermore, we use our rules to construct entanglement witnesses for threeuniform hypergraph states.
1 Introduction
Due to its possible applications in quantum information processing, multiparticle entanglement is under intensive research. One of the problems in this field is the identification of families of states which are useful in applications, but nevertheless can be described by a simple formalism. An interesting class of multiqubit quantum states are graph states [1]. Mathematically, these states are described by graphs, where the vertices correspond to particles and the edges represent twobody interactions in a possible generation process. A generalization of these states are hypergraph states [2, 4, 3, 5]. In a hypergraph, an edge can connect more than two vertices, so hypergraph states can be generated with multiqubit interactions. Hypergraph states have turned out to violate local realism in a robust manner [6], they play a role in quantum algorithms [7] and are central for novel schemes of measurementbased quantum computation [8].
A general feature of graph and hypergraph states is that different graphs may lead to quantum states with the same entanglement properties. It is therefore important to study the action of local and nonlocal unitary transformations between these states. For graph states, the socalled local complementation plays an outstanding role [9]: This graphical transformation corresponds to socalled local Clifford operations and these operations represent all possible local unitary transformations between graph states for up to eight qubits [10], only for large qubit numbers other transformations play a role [11, 12].
In this paper we derive graphical rules to represent various unitary transformations between hypergraph states. First, we introduce a generalization of local complementation to hypergraphs and the corresponding unitary transformations. Then we consider different quantum gates, such as the CNOT and Toffoli gate and their graphical representation. In general, the considered unitary transformations are nonlocal, but in some cases they can be combined to give effectively local transformations. With that, we find pairs of fivequbit hypergraph states, which are equivalent under local unitary transformations, but they are not equivalent under local Pauli operations. These are the first examples of this kind, up to four qubits all locally equivalent hypergraph states could be transformed into each other by application of Pauli matrices only [5, 13]. As a second application, we construct entanglement witnesses for hypergraph states which contain only threeedges. This will be useful for characterizing entanglement in these states experimentally.
2 Local complementation of hypergraph states
2.1 Basic definitions and local complementation of graphs
Let us start by defining hypergraph states, a detailed discussion of their properties can be found in Ref. [5]. A hypergraph consists of a set of vertices and a set of hyperedges , with being the power set of , some examples of hypergraphs can be found in Fig. 1. While for graphs edges connect exactly two vertices, hyperedges can connect more than two vertices, or contain just a single vertex. For any hypergraph the corresponding hypergraph state is defined as the qubit state
(1) 
where are the initial singlequbit states, is a hyperedge and is a multiqubit phase gate , acting on the Hilbert space associated with the vertices . Since all these phase gates commute, the order in the product does not matter. It is useful to note that hypergraph states are exactly the states that can be written as
(2) 
with being some binary function. From this representation, one recognizes that hypergraph states are special cases of locally maximally entanglable (LME) states. LME states are wider classes of states as they allow arbitrary equal complex phases (corresponding to an arbitrary ) in the full computational basis. The name LME is due to the fact that they are maximally entangleable to auxiliary systems using only local operations [2].
A very important subclass of hypergraph states are graph states. Their properties and applications have been studied extensively. Graph states correspond to graphs and therefore, only twobody controlled phase gates are required for their generation. They are local stabiliser states and prominent examples of them are GreenbergerHorneZeilinger states and cluster states. For a review on graphs states we direct the reader to Ref. [1].
Once it is established that graph states are important classes of multiqubit states, it is crucial to learn which of these states are equivalent under local actions of each party. As local actions one considers here local unitary transformations. Here the discrete subclass of local Clifford operations play an outstanding role. By definition, local Clifford operations leave the set of Pauli matrices invariant. It has been shown that a graph state can be transformed to another graph state by means of local Clifford action on some parties, if the graph can be obtained from the graph by a series of local complementations [9]. The local complementation of a graph works as follows: One picks a vertex and complements then subgraph in the neighbourhood of , defined as the set containing all adjacent vertices to . The complementation means that vertices in the neighbourhood become disconnected, if they were connected before, and they become connected, if they were disconnected before. Originally, the rule of local complementation has been conjectured to be necessary and sufficient for local unitary equivalence of graph states [9], however, it was later disproved by counterexamples [11, 12].
In order to physically achieve the local unitary transformation corresponding local complementation, the following unitary transform is considered:
(3) 
where and denote the Pauli matrices, and
(4) 
In the next section, we will generalize this to hypergraph states.
2.2 Local complementation of hypergraphs
Now we extend the term local complementation and its action to all hypergraph states. We denote the set of vertices of a hypergraph to be and the set of edges to be . First we introduce a term which can be regarded as a generalization of the term neighbourhood known in graph theory. We call it adjacency of a vertex and denote it by . The elements of are sets of vertices which are adjacent to via some hyperedge. To give an an example, the adjacency of the vertex from the hypergraph in the top image of Fig. 1 (a) is given by . Similarly we can define the adjacency for some set of vertices as .
For formulating our main result, we introduce the concept of a local edgepair complementation in hypergraphs around a vertex . Let us define first the set of adjacency pairs of vertex to be the set of all distinct pairs in the adjacency set. Considering again the top image from the Fig. 1 (a), the set of adjacency pairs for vertex is given by the set . Finally, the local edgepair complementation around a vertex complements the edges in the multiset . Notice that is a multiset and only the edges appearing with odd multiplicity will be affected. We again consider the top image from Fig. 1 (a) for which the multiset is . Complementation of the edges in this multiset means that they are deleted from the hypergraph, if they were already present, and the are added, if they were not present.
In the following theorem we show that a local edgepair complementation transforming a hypergraph state to a hypergraph state around a vertex can be achieved by the the following nonlocal operation:
(5) 
where is the square root of controlled phase gate applied to qubits in edge . It is a diagonal operator with every eigenvalue being one, except for the eigenvalue corresponding to state , which is . In the following, we sometimes write in order to indicate the sign of Note that for a usual graph Eq. (5) corresponds to Eq. (3) and local edgepair complementation corresponds to the local complementation. We can formulate:
Theorem 1.
For any hypergraph state the transformation around a vertex performs a local edgepair complementation on its corresponding hypergraph.
Proof.
Without loss of generality we assume that is the last qubit. Then, we need to fix the following notation. First, let or be an element of the computational basis on the first qubits, can be seen as a string of and . We say that an edge acts on if has the entries on the qubits belonging to . This means that the phase gate changes the sign of We define as the number edges acting on . Since is defined on qubits only, we also define as the number of edges that act on . This means that the edges act on the basis element, where a 1 is appended as the state of the last qubit.
Using this notation, the hypergraph state can be rewritten as: and we can compute:
(6) 
Eq. shows that the sign flip of is defined by . This is nothing but the number of pairs of edges in that act on . This sign flip is equivalently described, if we apply the for all the edges to the hypergraph. As , this means that edges in multiset get complemented. ∎
Some examples for the application of this rule are given in Fig. 1. Note that the map is not always local, since it contains gates that are nonlocal whenever the vertex is contained in at least one edge of cardinality three or more. Thus, this map can change the entanglement properties of the state it is applied to. However, in particular structures of hypergraphs, the map can be chosen to be applied to multiple vertices in a way that the nonlocal gates cancel each other out. Whenever the nonlocal gates cancel each other out we can perform the complementation operation without applying those canceling gates at all. Thus the resulting hypergraph will be obtained by using local operators only.
Fig. 1 (b) and (c) display two examples where a sequence of local complementations can be implemented using only local operators. These are the first examples that demonstrate that two hypergraphs, with edges containing more than two qubits, can be equivalent under local unitary operators but not under local Pauli operators. Finally, it should be noted that our rule of local complementation can also be derived from the general theory given recently in Ref. [12], but our proof is significantly simpler.
3 Permutation unitaries and their applications
In the previous section we considered the extension of local complementation for hypergraph states. In this section we investigate a different family of unitary transformations, we call them permutation unitaries. These transformations permute the vectors of the computational basis. Such permutations are obviously unitary and from Eq. (2) it is clear that they map hypergraph states to hypergraph states, so there must be a graphical description.
The simplest example of such a permutation unitary is PauliX (or NOT) gate, whose action on a hypergraph state was studied before [4, 5], see also Fig. 2 for an example. A nonlocal example of a permutation unitary in two dimensions is a CNOT gate, CNOT. An extension to threequbit is the Toffoli gate, CCNOT. Clearly, is is not necessary to consider all permutations, as for instance any permutation can be viewed as a sequence of transpositions [14]. Another possibility is to look for extensions of CNOT gates. For twoqubit permutations considering one can easily see that NOT and CNOT are enough to cover all possible permutations. Additionally, it is known that every permutation on can be realized by means of a reversible circuit using the NOT, CNOT and CCNOT basis and at most one ancilla bit [15]. It is possible to derive a graphical rule of how such maps transform hypergraph states. Here we give rules explicitly only for the twoqubit CNOT and its multiqubit extensions, but the methodology can be applied to derive any arbitrary permutation unitary if the exact graphical transformation is needed.
Lemma 2.
Applying the CNOT gate on hypergraph state, where is the control qubit and is the target one, introduces/deletes the edges of the form .
Proof.
Without loss of generality we assume that CNOT acts on the first two qubits. We write a hypergraph state as follows:
(7)  
(8)  
(9)  
(10) 
The CNOT gate swaps and , or alternatively Eq. (9) and Eq. (10), but leaves the other parts invariant. Therefore we obtain the following:
(11)  
(12)  
(13)  
(14) 
Equations (1114) show that only the edges containing the control qubit can appear or disappear. More precisely, Eq. (13) shows that the new edges that are added/deleted are of the form . ∎
An example of this rule is shown in Fig. 2. We can directly generalize this rule to extended CNOT gates, such as the Toffoli gate, the proof is essentially the same.
Corollary 3.
Applying the extended CNOT gate on a hypergraph state, where a set of control qubits controls the target qubit , introduces or deletes the set of edges .
Moreover, as mentioned above every permutation can be constructed using NOT, CNOT, and CCNOT and at most one ancilla qubit. An ancilla qubit is necessary to construct the multiqubit gate set, [16] and the set is enough to realize any permutation on indices. As exactly consists of the gates with graphical rules from above, we can state:
Corollary 4.
Every permutation unitary maps a hypergraph state to a hypergraph state and its graphical action can be seen as a composition of rules from graphical rules.
It is interesting to note how the different rules change the cardinality of edges. If is the cardinality of the largest edge in the hypergraph, the NOT gate can only create/erase edges with a cardinality strictly smaller then . The CNOT gate can create/erase edges with cardinality smaller or equal to , but the CCNOT can create edges with cardinality higher then .
Finally, we demonstrate that the rule for the CNOT gate has a direct application: Consider a complete threeuniform hypergraph states, that is, the hypergraph contains all possible threeedges, but nothing else. These states can be thought as generalizations of GHZ states and violate Bell inequalities in a robust manner [6]. If one considers a possible bipartition of the particles, one may ask how the hypergraph can be simplified using unitaries that are local with respect to this bipartition. The following Lemma provides an answer, and we will use this later for the construction of witnesses.
Lemma 5.
Consider an qubit complete threeuniform hypergraph state and a bipartition . Then, using only local actions with respect to this bipartition the hypergraph can be reduced to the form shown in Fig. 3. We call this form a normal form of complete threeuniform hypergraph state respecting the bipartition .
Proof.
The proof consist of an application of a sequence of CNOT gates on both sides of the bipartition. Details are given in the Appendix. ∎
4 Construction of witnesses
In this section we consider the construction of entanglement witnesses as an application of the results derived so far. More specifically, we construct tight witnesses for fullyconnected threeuniform hypergraph states. These states are of special interest, as it has been shown that they violate Bell inequalities with an exponentially increasing amount and the violation is robust against particle loss. The Bell inequalities can be used to prove that there is some entanglement in the state, but in this section we will focus on entanglement witnesses for genuine multiparticle entanglement.
An entanglement witness is an observable which has a nonnegative expectation value for all separable states, thus, a negative expectation value signals the presence of entanglement. There are many ways to construct entanglement witnesses, see Ref. [17] for an overview. One possible way to design a witness for a general state is to consider the following observable
(15) 
where is the maximal overlap between the state and the pure biseparable states. This can be computed by the maximal squared Schmidt coefficient occurring when computing the Schmidt decomposition with respect to all bipartitions,
(16) 
For usual graph states the witness can be determined in the following way [18]: First, for any bipartition one can generate a Bell pair between the two parties by making only local operations with respect to this partition. During local operations, however, the maximal Schmidt coefficient can only increase. This proves directly that for any bipartition , so is a witness. Using a similar construction, we can estimate and write down a witness for threeuniform hypergraph states. Note that this scheme of constructing witnesses has recently been extended to other hypergraph states [19].
Theorem 6.
For any threeuniform hypergraph state the operator
(17) 
is an entanglement witness detecting this state.
Proof.
The proof is similar to the one for graph states, but in this case the aim is to share a the threequbit hypergraph state between the bipartition. For this state the maximal squared Schmidt coefficient is . Given an qubit threeuniform state we consider a bipartition . One can get rid of any edge which entirely belongs to either side of the bipartition. Since the graph is assumed to be connected at least one threeedge remains shared between the two parts. Without loss of generality we can assume that this edge is . Now by making measurements in the PauliZ basis on every qubit except these three in we can disentangle all the qubits from the main hypergraph except . For all possible measurement results, i.e. with probability one the resulting state is, up to local unitarians, a threequbit hypergraph state consisting only of the edge . ∎
The previous witness can be used for any connected threeuniform hypergraph state, but is it no necessarily tight. For the special case of complete threeuniform states, where any possible threeedge is present, we derive a better witness in the following. Since this state is symmetric, the Schmidt coefficients depend only on the size of the partitions.
Lemma 7.
Consider an qubit complete threeuniform hypergraph state. Then, the maximal squared Schmidt coefficient with respect to the bipartition vs. qubits, is
(18) 
For the vs. partition it is . For the vs. partition it is given by if and for one has .
Proof.
The proof is done by tracing out the parties and calculating the Schmidt coefficients as eigenvalues of the reduced states. Details can be found in the Appendix. ∎
Theorem 8.
An improved witness for the qubit complete threeuniform hypergraph state is given by
(19) 
where .
Proof.
We have to show that in general it is sufficient to consider the vs. and vs. partitions. First, the vs. give only smaller Schmidt coefficients, as can be seen from the Lemma 7. For any other bipartition with we use the normal form in Fig. 3. If a resulting hypergraph is reduced either to Fig. 3 (b) or (c) [without the dashed edge], then on qubits the measurements in the PauliZ basis can be made. As a result, the hypergraph state is obtained. We know from the Lemma 7 that the vs. partition has a largest squared Schmidt coefficient less than (unless ). Keeping in mind that measurements can never decrease the squared maximal Schmidt coefficient, we reach the conclusion that the bipartition cannot contribute to the maximal Schmidt coefficient when . If in the normal form in Fig. 3 (c) the dashed edge is present, one can make measurements on both sides of the partition to reduce the state to a Bell state between qubits and . This clearly gives a squared Schmidt coefficient
The final case is the state with a normal form in Fig. 3 (a). Here the strategy is as follows: PauliZ measurements are made on every qubit but eight of them, namely the qubits remain untouched. This leaves us with the state given in Fig. 4, where the qubits have been relabeled. Then, a Pauli Z measurement is made on qubit 1. With probability (in case of outcome 0), the edge is introduced and qubit is disentangled. With probability (outcome ) both qubits and are disentangled. For the first case ( outcome), we again make a PauliZ measurement on qubit , denoted by . This itself gives two possible outcomes with halfhalf probabilities, the outcome gives the edge and disentangles qubit and the outcome disentangles qubits and . Putting all measurement outcomes together with corresponding probabilities yields as a bound on the Schmidt coefficient
(20) 
Note that in this estimation it was used that one minus the largest squared Schmidt coefficient can be viewed as the geometric measure of entanglement for this partition, and this measure decreases under local operations even for mixed states. ∎
5 Conclusions
In summary, we have extended the local complementation rule from graph states to hypergraph states. We also described the action of different gates on hypergraph states with graphical rules. Already for five qubits we showed with a simple example that local Pauli operations only are not enough to exhaust all local unitary equivalence classes of hypergraph states. Based on the rule for the CNOT gate, we developed a normal form for bipartitions of complete threeuniform hypergraph states. Based on this, we derived entanglement witnesses for these states.
There are several directions in which our work can be extended. First, it would be highly desirable to develop a general theory for entanglement witnesses for hypergraph states, similar to the existing theory for graph states [17]. Here, notions of the coulourability of a hypergraph may be developed to characterize how many measurements are needed to estimate the fidelity of a state. All this can help to observe hypergraph states experimentally. Another interesting question concerns the extent to which hypergraph states and their correlations can be simulated classically in an efficient manner. Our findings show that certain unitary operations have a graphical interpretation. This may be useful to decide whether their classical simulation is feasible. For graph states, the GottesmanKnill theorem characterizes a set of operations that can be simulated efficiently and it would be highly desirable to identify similar operations for hypergraph states.
We thank Cornelia Spee for discussions. This work has been supported by the DFG and the ERC (Consolidator Grant 683107/TempoQ). Additionally, MG would like to acknowledge funding from the Gesellschaft der Freunde und Förderer der Universität Siegen.
6 Appendix
6.1 Reduction of three uniform hypergraph states to the normal form in Lemma 5
We prove Lemma 5 by considering first simple bipartitions, where the strategy of the proof is easier to explain. The proof for the general case then follows the same lines.
Lemma 9.
Consider the bipartition for an qubit complete threeuniform hypergraph state. Then this state is locally (for the given bipartition) equivalent to the threeuniform hypergraph state where every vertex is contained in only one edge and edges are of the form: And only if , an additional cardinality two edge appears, which is .
Proof.
Fig. 5 (a) represents the goal hypergraph state respecting a bipartition . The algorithm to achieve this state is as follows:

Erase all the edges which only contain subsets of vertices . This operation is local with respect to the bipartition.
All the remaining edges are . 
Apply , where .
To give an example, we start with the gate. The adjacency of is , where The edges introduced by the gate are and therefore, this action removes all the edges where is contained except the edge and adds the cardinality two edge .
At this step the remaining edges are . Since , it is clear that consecutive gates presented in this step do not modify edges containing . erases all the edges where is presented except already established and the gate where vertices and are presented together . It also adds the cardinality two edge . Repeating this procedure:
All the remaining edges are of the form and cardinality two edges . 
Apply , where
The adjacency of mod right before applying the gate is . The gate, therefore, erases/creates edges . This means that the adjacency of mod is only , and the gate can only erase . See Fig. 5 (b).
Here we have to consider several cases:
is odd: All the remaining edges are of the form , for even and also for unless mod . It is easy to see that all two edges can be removed by action of PauliX’s.
is even: If mod , then the the last edge is erased and the edge cannot be created. Therefore, the last qubit is completely disentangled in this case. See Fig.5 (a).
is even: If mod , then the the last edge is erased and the edge is created. See Fig.5 (b) for the exact procedure.
∎
To sum up the previous theorem, there are three possibilities for the final hypergraph and it only depends on the number of parties in the hypergraph. If odd, then every vertex is exactly in one hyperedge. If , then the final hypergraph corresponds to the one in Fig. 5 (a) including the dashed line. Note that this is in line with the fact that the maximal Schmidt coefficient for this case is (see Lemma 7), as there is a Bell pair shared across the bipartition [18]. In case , the dashed line is missing, therefore, the last qubit can be removed and the result for the maximal Schmidt coefficient matches with case.
Lemma 10.
Considering the bipartition for an qubit fullyconnected threeuniform hypergraph state. Then, this state is locally equivalent to the threeuniform hypergraph state already derived from the bipartition and in addition has the edge .
Proof.
The steps are very similar to the case:

From a hypergraph we remove all the edges which do not contain either party , or .
All the remaining edges are 
Apply .
The adjacency of is for . Augmented by the control qubit , action of removes and creates , .
All the remaining edges are 
Apply .
All the elements in the adjacency for are added as edges to the hypergraph. Thus, all the edges of the type cancel out and can be directly removed.
All the remaining edges are
∎
Corollary 11.
This result is obtained by applying the algorithm from Lemma 9 first to and then to . This ends the proof of Lemma 5.
6.2 Proof of Lemma 7:
Proof.
First we consider vs. bipartition. To calculate the maximal Schmidt coefficient we compute the reduced density matrix. As the state is symmetric, we only have to take the bipartition . We have
(21) 
The diagonal elements follow directly from the representation of the hypergraph state in Eq. (2) and do not depend on the structure of the hypergraph. For computing the offdiagonal entries, we write the hypergraph state as
(22) 
with Since we deal with threeuniform complete hypergraph states, we have and , where is the weight (i.e., the number of “1” entries) of . We can then write
(23) 
The values of and do only depend on . Instead of summing over , we can also sum over all possible in Eq. (23) and distinguish the cases of The value for a given is then up to the sign given by the numbers of possible with the same We have:
(24) 
To give the final result, we have to consider several cases in Eq. (24): If , then , therefore, is maximally mixed and If or , then . Then, it follows that . Similarly, for , and therefore . This ends the computation of .
Second, we look at the vs. bipartitions. The idea of the proof very much resembles the previous case. First, we take the bipartition and trace out the second part:
(25) 
For computing the entries, we express a hypergraph state in the following way:
(26) 
with The diagonal elements of are, as before, easy to determine. This is also the case for the two antidiagonal terms as is always even. The next term, , is derived as Eqs. (23, 24): For the term , is even if is even and is odd if is odd. Therefore . For the last term we find .
Putting all these terms together in the matrix, one can calculate the maximal eigenvalue of :
(27) 
Finally, we have to consider the bipartition and write down the reduced density matrix:
(28) 
where and .
From this we can be derive all possible values of maximal Schmidt coefficient . If , then . If , then . If , then and finally, if , then . It can be easily seen that is decreasing with and it is only greater than when . ∎
References
References
 M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, and H.J. Briegel, Entanglement in Graph States and its Applications, in Quantum Computers, Algorithms and Chaos, edited by G. Casati, D.L. Shepelyansky, P. Zoller, and G. Benenti (IOS Press, Amsterdam, 2006), quantph/0602096.
 C. Kruszynska and B. Kraus, Phys. Rev. A 79, 052304 (2009).
 M. Rossi, M. Huber, D. Bruß, and C. Macchiavello, New J. Phys. 15, 113022 (2013).
 R. Qu, J. Wang, Z. Li, and Y. Bao, Phys. Rev. A 87, 022311 (2013).
 O. Gühne, M. Cuquet, F. E. S. Steinhoff, T. Moroder, M. Rossi, D. Bruß, B. Kraus, and C. Macchiavello, J. Phys. A: Math. Theor. 47, 335303 (2014).
 M. Gachechiladze, C. Budroni, and O. Gühne, Phys. Rev. Lett. 116, 070401 (2016).
 M. Rossi, D. Bruß, and C. Macchiavello, Phys. Scr. T160, 014036 (2014).
 J. Miller and A Miyake, npj Quantum Information 2, 16036 (2016).
 M. van den Nest, J. Dehaene and B. De Moor, Phys. Rev. A 69, 022316 (2004).
 A. Cabello, A.J. LopezTarrida, P. Moreno, and J.R. Portillo, Phys. Rev. A 80, 012102 (2009).
 Z. Ji, J. Chen, Z. Wei, and M. Ying, Quantum Inf. Comp. 10, 97 (2010).
 N. Tsimakuridze and O. Gühne, J. Phys. A: Math. Theor. 50, 195302 (2017).
 X.Y. Chen and L. Wang, J. Phys. A: Math. Theor. 47, 415304 (2014).
 G.W. Leibniz, private notes, Niedersächsische Landesbibliothek Hannover, LH XXXV 4.8 f.12 (1678).
 T. Toffoli, ”Reversible computing,” in Lecture Notes in Computer Science, Vol. 84 (Springer, 1980), pp. 632644.
 S. Xu, arxiv:1506.03777.
 O. Gühne and G. Tóth, Phys. Rep. 474, 1 (2009).
 G. Tóth and O. Gühne, Phys. Rev. Lett. 94, 060501 (2005).
 M. Ghio, D. Malpetti, M. Rossi, D. Bruß, and C. Macchiavello, arXiv:1703.00429.