Unextendible Sets of Mutually Unbiased Basis Obtained from Complete Subgraphs

: We propose a method, based on the search and identiﬁcation of complete subgraphs of a regular graph, to obtain sets of Pauli operators whose eigenstates form unextendible complete sets of mutually unbiased bases of n -qubit systems. With this method we can obtain results for complete and inextensible sets of mubs for 2, 3, 4 and 5 qubits.


Introduction
In quantum information sciences, the generation, manipulation, and classification of the basis of n-qubits are of great importance, since those techniques have a close relationship with the reconstruction processes [1,2], together with the classification and quantification of entanglement of quantum states [3,4].
In this regard, mutually unbiased bases (MUBs) are of special interest because the amount of information obtained in a measurement performed in these bases about the state of a certain system contains the minimum amount of redundant information in comparison to a measurement performed on another basis. Mutually unbiased bases satisfy for two elements belonging to two different basis, |A, a and |B, b , with a, b = 1, 2, . . . , 2 n , the following condition: Among some of the applications of MUBs are the optimal reconstruction of quantum states [1,2], quantum error correction coding [5,6], and discrete Wigner function construction [7][8][9].On the other hand, several papers have approached the problem of generating complete sets of MUBs of n-qubit systems (for which the maximum number of MUBs cannot exceed 2 n + 1), by following different methodologies [10][11][12][13].
Moreover, in addition to the generation of complete sets of MUBs, other structures of special interest, both in the theoretical and applied context, are the unextendible sets of MUBs [3,[14][15][16][17][18]: a set {B 1 , B 2 , . . . , B m } of MUBs in C d is said to be weakly unextendible (WUS), if there is no other unbiased basis in C d with respect to the basis B j , j = 1, . . . , m.
On the other hand, if no vector of C d satisfies the condition (1) with all the basis elements B j , then the set is designated as strongly unextendible (SUS).
On this matter, some of the most important results regarding WUSs are: (a) in [19], it was shown that in dimension d = 6, the eigenstates of the Pauli operators σ x , σ z , σ x σ z form a WUS of size three. Moreover, it has been conjectured that the eigenstates of the Pauli operators σ x , σ z , σ x σ z form a WUS of dimension d = 2m (the conjecture has been verified for d ≤ 12 [14]); (b) in [17], a procedure to generate sets of WUSs from complementary subalgebras was shown; although this was exclusive for dimensions d = p 2 with p ≡ 3 mod 4; (c) in [15], explicit sets of WUSs were obtained for d = 4 and d = 8; (d) in [16], a methodology to construct explicit examples of unextendible sets up to dimension 16 was given.
This paper is organized as follows: in Section 2, we begin with the basic definitions to introduce the problem of obtaining a WUS and complete sets of MUBs; then, in Section 3, we proceed to pose the problem formally, in terms of graphs; afterwards, in Section 4, we show the results obtained for dimensions 4, 8, 16, and 32; and finally, in Section 5, we give a summary of our results.

Preliminaries
We define the correspondence: where σ z , σ x , σ y , I constitute (with the exception of a global phase) the Pauli group P 1 . Within this correspondence, any element of P n = P 1 ⊗ P 1 ⊗ · · · P 1 can be represented by a 1 × 2n matrix: a b = a n · · · a 1 b 1 · · · b n , with a j , b j ∈ Z 2 representing the monomials: and a factor of the tensorial products in the form σ z σ x = iσ y is for simplicity considered as σ y (omitting the global phase). Moreover, σ a z σ b x commutes with σ a z σ b x if and only if ab + ba = 0. From this commutative condition, it follows that [Z a X b , Z a X b ] = 0 if and only if ∑ n i=1 a i b i + b i a i = 0, or in matrix form: where J is the anti-diagonal matrix of size 2n × 2n given by: Notice that the order employed for the subscripts in (4) allows one to write the matrix condition (6). The group of matrices preserving the matrix J and, hence, the commutative condition (6) is the symplectic group Sp 2n (Z 2 ) = P ∈ GL 2n (Z 2 ) : P T JP = J . With the purpose of simplifying some of the calculations and the results, it is possible to represent only the n generators of each commutative set, for which we take only the first n columns of each P ∈ Sp 2n (Z 2 ) as: they generate a set of 2 n − 1 (omitting the identity operator) commutative Pauli operators, since by construction, (8) is of rank n, and satisfy the commutativity condition (6): Considering that the symplectic matrices preserve (7), all sets of commutative operators can be generated from an initial set that satisfies (6). For example, considering the initial set Y a0 = Z a (which is encoded as a 0 ), then any other commutative set is of the form: The number of commutative sets is smaller than the order of the symplectic group [20]: consequently, in order to select symplectic matrices producing different commutative sets, according to (10); an adequate manipulation of the symplectic group is necessary, which in turn requires establishing the Bruhat decomposition theorem, as well as certain additional definitions, which are as follows: The order of the Borel group is: Definition 2. The Weyl group W 2n is a subgroup of permutations of 2n objects such that if the objects are labeled by the indexes 1, 2, . . . , 2n, the group can be realized as the group of permutations π satisfying: π(2n The generators of the Weyl group W n are: where s n is a transposition that permutes n with n + 1 and s i is the permutation given by s i = (i, i + 1)(2n + 1 − i, 2n − i), for i = 1, 2, . . . , n − 1. Additionally, the length l(π) of a permutation π ∈ W 2n is equal to the minimum number of generators (13) needed to generate π, while the order of the Weyl group is given by: For every permutation π ∈ W 2n , one can identify π with the matrix: For every matrix [π], we construct a symplectic parameterized matrix: where [A π ] is obtained by inserting the free parameters (i.e., any x ∈ Z 2 ) of [π], such that the following condition is satisfied: every zero in the matrix [π] that is located, simultaneously, below and to the left of the ones is replaced by a free parameter (see [20] for more details).

Theorem 1.
The symplectic group Sp 2n (Z 2 ) can be decomposed into matrices of the form A π B + 2n , π ∈ W 2n . The number of free parameters in each A π is n 2 − l(π) [20,21].
When generating commutative sets from (10), using Theorem 1, it can be concluded that the only matrices that generate commutative sets, different from the initial one, are the matrices A π , since the matrices of the Borel group are upper triangular and the initial commutative set is encoded in a vector with zeros in its last n lines; therefore: corresponds (according to (5)) to the set initial set of commutative operators. Among the different matrices A π , some of them generate the same commutative sets when applied as in (10). We refer to such matrices (e.g., A π k and A π m ) as equivalents (i.e., A π k ∼ A π m ).

Proposition 1.
Consider two elements of the Weyl group π k and π m , such that the first n columns of their respective matrix representations [π k ] and [π m ] coincide up to one permutation, then: Proof. Suppose, without loss of generality, that l(π m ) > l(π k ) and that s ∈ S n (S n being the group of permutations of n objects) is a permutation that organizes the first n columns of [π m ] in the same order as [π k ] while r ∈ S n is a permutation that organizes the last n columns of [π m ] in the same order as [π k ], i.e., where: where Ξ n is an anti-diagonal matrix of size n × n with ones in the anti-diagonal entries. Furthermore, since l(π m ) > l(π k ), then the number of free parameters in A π k is greater than in A π m [20]. Therefore, by considering that a representation is obtained by the intersection of the free parameters in [π] only at the left and under the ones, we built [s] such that the free parameters does not appear at the right of or over a one. In this way, in the product A π m D s , the free parameters are again only at the left and under the ones (since [s] permutes the columns of [π m ]). By choosing a suitable set of free parameters in D s , we obtain: Since the action of D s over the initial vector is trivial, The construction of the representation [s] of the permutation s is completely equivalent to the construction of the matrix A s of the Bruhat decompositions of GL n (Z 2 ) [20] and GL n (Z 2 ) = ∪A s B + , where s ∈ S n and B + is the group of the non-degenerated upper triangular matrices of size n × n. In particular, it is easy to calculate the order of S ={[s] : s ∈ S n }, i.e., According to Proposition 1, we can divide all the matrices A π k : k = 1, . . . , 2 n n! into equivalence classes, such that for each of these classes, A π k ∼ A π m . This implies that for generating all the different commutative sets, we can take only one representative of each class, containing a minimum number of free parameters. This can be performed already at the level of the corresponding permutations π ∈ B + 2n (Z 2 ); in fact, π k ∼ π m (hence, they generate A π k ∼ A π m ) if the first n columns of [π k ] and [π m ] coincide up to one permutation.
By considering that a single permutation π allows the intersection of free parameters in the corresponding A π only at positions (i, j), i.e., and that any π ∈ B + 2n (Z 2 ) is defined unequivocally by its first n elements, we can arrange such elements in a strictly increasing form, l 1 > l 2 > . . . > l n , such that a single permutation π * , defined as: π * (1) = l 1 , . . . , π * (n) = l n , is of maximum length in the corresponding equivalence class; hence, it generates A π * with a minimum number of free parameters (Theorem 1).
Since the cardinality of each equivalence class is the same as the number of permutations of n objects, i.e., n!, then the number of matrices A π required for the generation of all different commutative sets is B + 2n /n! = 2 n . In this regard, by using (21) and (12), we can calculate the total number of commutative sets for n qubits (i.e., the number of distinct bases with an underlying factorization structure defined in the Hilbert space 2 n ):

Operators Producing MUBs
In this section, we describe the sets of operators producing MUBs, together with the exclusion criteria of these operators allowing one to pose the problem of obtaining a WUS and complete sets of MUBs.
Consider sets of operators Y k j , where k = 1, 2, . . . labels the different sets (9) while j = 1, . . . , 2 n − 1 labels an element of the set. In this regard, hence, the exclusion condition is represented as: It was shown in [10] (see also [11] for the specific case of monomials with the dimension of powers of prime numbers) that the eigenstates of two of such sets form MUBs. In this context, the maximum number of commuting sets that are mutually exclusive is 2 n + 1 [3]; consequently, we call the corresponding operators Y k j maximal. Moreover, since each set of commutative operators (9) is isomorphic to a linear space generated by the first n columns of a symplectic matrix P (8), it is possible to establish simple criteria for the exclusion of two sets.
Suppose that the sets Y 1 and Y 2 are generated respectively by the matrices P 1 and P 2 , then the sets Y 1 and Y 2 are disjoint sets if the 2n × 2n matrix formed with the first n columns of P 1 and the first n columns of P 2 is non-singular, implying that the subspaces generated by the first n column vectors of P 1 and P 2 have no intersection.
The criteria described in the last paragraph allow formally demonstrating the following Proposition (see [12] for the proof): Proposition 2. Let Y be any set of 2 n commutative Pauli operators. Thus, there are exactly 2 n(n+1) 2 sets of commutative Pauli operators that are disjoint with Y.

Proposition 2 implies that each basis formed by the eigenstates of a set of operators
Y is mutually complementary with another 2 n(n+1) 2 basis, therefore within the idea of the proposition together with the fact that the number of Pauli commutative sets in 2 n dimension is (2), if we can formulate the problem of finding sets of operators that generate a WUS and complete sets of MUBs in terms of graphs.
In other words, we constructed a graph with the vertex (23), associating with each set of operators (9) a single vertex. Furthermore, the complementarity relations among the corresponding basis (represented within the vertex) are established through the exclusion criteria described before in the following way: two vertices in the graph are connected by an edge only if the sets of corresponding operators are disjoint, i.e., if the matrix formed by the first n columns of the two symplectic matrices that generate these sets of operators (8) is non-singular.
Once the graph is constructed and according to Proposition 2, we obtained for n qubits an undirected graph of the vertex (23), which is regular (i.e., the number of edges in each vertex is 2 n(n+1) 2 ) and simple (there are no parallel edges, nor loops). Furthermore, to determine the vertex of the graph, the symplectic matrices (23) were selected as described in [12] for which the exclusion criteria described above were applied to determine which vertex was connected through an edge.
In this regard, if there exists a subgraph of m vertices lying inside the graph formed as described above, in which every single pair of distinct vertices is connected by an edge, this graph represents a set of m MUBs. Such subgraphs are known as complete graphs (cliques) and are denoted by K m [22].
Moreover, any clique K m is called maximal, if m is the highest integer such that K m is a clique, i.e., there is no other vertex in the graph that is connected to each one of the m vertices of K m . For a maximal K m with m = 2 n + 1, K m represents a complete set of MUBs, while for m < 2 n + 1, a maximal K m represents a weakly unextendible set of MUBs.
To determine the maximal cliques, we first constructed the adjacency matrix of the regular graph, and afterwards, we employed the Bron-Kerbosch [23] and Östergard [24] search algorithms.

Results
We began with two qubits; the graph formed in the way described earlier was constituted by 15 vertices where each vertex was of degree eight. The complete sets of MUBs correspond to the maximal cliques K 5 , i.e., subgraphs with five vertices for which each one of the vertices is connected to the other four vertices of the subgraph. On the other hand, the WUS, corresponds to the maximal cliques K m with m < 5. Therefore, over the adjacency matrix of the 15-vertex graph, we implemented an exhaustive search of cliques of size ≤5, finding the following results: there were only maximal cliques of sizes three and five, i.e., there was only a WUS of size three and complete sets of MUBs. In Figure 1, we depict the graph corresponding to two qubits in which each subgraph with edges of the same color represents a maximal clique (only the maximal cliques are highlighted). Table 1 shows the number of complete sets of MUBs and the WUS found for two qubits.  For three qubits, the corresponding graph was constituted by 135 vertices in which each vertex was of degree 64.
The complete sets of MUBs corresponded to the maximal cliques K 9 , i.e., subgraphs with nine vertices in which each of these vertices was connected to the other eight vertices of the subgraph. On the other hand, the WUS corresponded to maximal cliques K m , with m < 9; thus, over the adjacency matrix of the graph with 135 vertices, we again performed the search for the cliques of size ≤9, finding the following results: there were only maximal cliques of sizes five and nine, i.e., there was only a WUS of size five and complete sets of MUBs. In Table 2, the number of sets of WUSs and complete sets of MUBs found for three qubits are shown.

No. of Cliques Size
Maximal 960 m = 9 unextendibles 24,192 m = 5 Therefore, for two and three qubits, a list of all cliques was obtained, verifying that in both cases, there were only unextendible sets of one single size (m = 3 for two qubits and m = 5 for three qubits) [14], and cliques of maximal size (m = 5 for two qubits and m = 9 for three qubits), i.e., a list of all the maximal cliques K m was obtained (maximal and unextendibles).
In the case of four qubits, the possible existence of a WUS of size nine was conjectured in [15].
With our approach, an adjacency matrix of size 2295 × 2295 was constructed, which we took advantage of based on the symmetry of the graph by performing the search over a single fixed vertex, reducing in this way its complexity. The search was performed employing the Östergard algorithm [24], looking for the maximal cliques of a specific size, i.e., each specific execution of the algorithm looks for maximal cliques of a given size m, m ≤ 17 (unlike the Bron-Kerbosch algorithm, which covers all the graph looking simultaneously at all maximal cliques of any size). The results obtained in this way showed that there were WUSs of sizes 5,8,9,11, and 13 and, of course, complete sets of MUBs (of size 17). In Tables 3-7, we show explicit examples of sets of operators that generate WUSs, for 5,8,9,11, and 13 (for simplicity, the symbol ⊗ was omitted). The search for five qubits was carried out in the same way as for the case of four qubits, obtaining WUSs of size 9, 11, 13, 15, and 17 and, of course, complete sets of MUBs of size 33 (in Appendix A, we include Table A1, showing an example found for this case). Table 3. Commutative sets (generators) whose eigenstates form a WUS of degree 5.

N.
Commutative Set Iσ x I I σ x Iσ x σ x σ z Iσ z I σ z I Iσ y 4 I Iσ x σ x σ x I Iσ x Iσ y I I σ z Iσ z σ y 5 I Iσ x I σ y σ x Iσ x σ x σ z I I σ x I Iσ y Table 6. Commutative sets (generators) whose eigenstates form a WUS of degree 11.

N. Commutative Set
1 σ z Iσ x σ x Iσ z σ x I σ x σ x σ y σ x σ x Iσ x σ y 2 σ y Iσ x σ x Iσ y σ x σ x σ x σ x σ y I σ x σ x Iσ y 3 σ x Iσ x I Iσ z I I σ z Iσ z σ x I Iσ x σ z 4 σ x σ x σ x σ x σ z σ y I I σ z Iσ z I σ z I Iσ z 5 σ x Iσ x σ x Iσ y I I σ z Iσ y σ x σ z Iσ x σ z 6 σ x σ x σ x I σ z σ y σ x σ x σ z σ x σ y σ x Iσ x σ x σ y 7 Iσ x σ x σ y σ z I Iσ x Iσ z σ z σ x σ x Iσ x σ z 8 Iσ x σ x σ x σ y I I I Iσ z σ y I Iσ z Iσ z 9 I Iσ x I σ z σ x I I σ x σ y I I I I Iσ y 10 I Iσ x σ x σ y σ x Iσ x σ x σ z I I σ x Iσ z σ z 11 I I Iσ x σ x σ x I I σ z σ z σ x I Iσ x σ y I Table 7. Commutative sets (generators) whose eigenstates form a WUS of degree 13.

N. Commutative Set
1 σ y Iσ x I Iσ z σ x I σ x σ x σ y I I I Iσ z 2 σ y Iσ x σ x Iσ z I I σ x Iσ y σ x σ x Iσ x σ y 3 σ y σ x σ x σ x σ x σ y I I σ x Iσ y I σ x I Iσ z 4 σ y σ x σ x I σ x σ y σ x σ x σ x σ x σ y σ x Iσ x σ x σ y 5 σ x σ x σ x I σ z σ z I I σ z Iσ y σ x I Iσ x σ z 6 σ x σ x σ x σ x σ z σ z σ x I σ z σ x σ y σ x σ z Iσ x σ z 7 σ x Iσ x I Iσ y σ x σ x σ z σ x σ y I Iσ x Iσ z 8 σ x Iσ x σ x Iσ y Iσ x σ z Iσ y I σ z σ x Iσ z 9 Iσ x σ x σ x σ z Iσ x σ x σ x σ z σ z σ x σ x σ z σ x σ z 10 Iσ x σ x I σ z I Iσ x Iσ z σ y I σ x I Iσ y 11 I Iσ x σ x σ z σ x Iσ x σ x σ z Iσ x σ x σ x σ z σ z 12 I Iσ x I σ z I I I Iσ y I I I I Iσ y 13 I I Iσ x Iσ x I I σ x I I I I Iσ z I

Summary
We found a method that allowed us to obtain, simultaneously, complete and unextendible sets of mutually unbiased bases. The method optimized the search for MUBs from sets of commutative Pauli operators and the employment of complete subgraphs within a regular graph. With this method, we obtained complete and unextendible sets for two, three, four, and five qubits.