Unextendible mutually unbiased bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

We consider questions posed in a recent paper of Mandayam, Bandyopadhyay, Grassl and Wootters [10] on the nature of"unextendible mutually unbiased bases."We describe a conceptual framework to study these questions, using a connection proved by the author in [19] between the set of nonidentity generalized Pauli operators on the Hilbert space of $N$ $d$-level quantum systems, $d$ a prime, and the geometry of non-degenerate alternating bilinear forms of rank $N$ over finite fields $\mathbb{F}_d$. We then supply alternative and short proofs of results obtained in [10], as well as new general bounds for the problems considered in loc. cit. In this setting, we also solve Conjecture 1 of [10], and speculate on variations of this conjecture.


INTRODUCTION
Finite-dimensional quantum systems -that is, "multiple qudits" -exhibit many interesting properties like quantum entanglement and quantum non-locality and play, therefore, a crucial role in numerous physical applications like Quantum Cryptography, Quantum Coding, Quantum Cloning/Teleportation and/or Quantum Computing, to mention just a few. As these systems live in finite-dimensional Hilbert spaces, further insights into their behavior require, obviously, a proper understanding of the structure of the associated Hilbert spaces. Within the past few years, a lot of activity in this direction has been devoted to the study of so-called mutually unbiased bases ("MUBs").
Recall that two orthonormal bases B and B ′ of the Hilbert space C ℓ (ℓ ∈ N × ) are mutually unbiased if and only if | φ|ψ | 2 = 1/ℓ for all |φ ∈ B and |ψ ∈ B ′ . It is a fundamental conjecture, with many applications, that the theoretical upper bound ℓ + 1 of a set of mutually unbiased bases can only be reached when ℓ is a prime power.
It has been suspected for a long time that there are deep connections between Quantum (Information) Theory and Finite Geometry -see, for instance, Wootters [22,23]. (See also [5]- [6] and [14]- [16], and references therein.) As a specific example, proving a conjecture of Saniga and Planat [16], the author showed in [19] that the generalized Pauli operators can be identified with the points, and maximum sets of pairwise commuting members of them with the lines (or subspaces of higher dimensions), of a specific finite incidence geometry so that the structure of the operator space can fully be inferred from the properties of the geometry in question. The incidence geometry is the geometry of a non-degenerate alternating bilinear form over a finite field, called symplectic polar space. Using this connection, it is easy to construct maximal sets of MUBs by just translating known results in the theory of symplectic polar spaces.
In a recent paper [10], Mandayam, Bandyopadhyay, Grassl and Wootters introduced unextendible mutually unbiased bases ("UMUBs") (and several variations and related concepts; details can be found in §3) as a natural generalization of maximal sets of mutually unbiased bases. One of the main results of [10] reads as follows.
Theorem 1.1 (Mandayam et al. [10]). Given three Pauli classes C 1 , C 2 , C 3 belonging to a complete set S of classes in d = 4, there exists exactly one more maximal commuting class C of Pauli operators in C 1 ∪ C 2 ∪ C 3 . The class C together with the remaining two classes C 4 and C 5 of S forms an unextendible set of Pauli classes, whose common eigenbases form a weakly UMUB of order 3.
Using the connection with the polar space, we will give a short proof of this result. Moreover, we generalize this result for all dimensions ℓ = prime 2 . (In fact, we present a construction of a new class of maximal partial spreads of the symplectic polar space W 3 (ℓ) for any odd prime power ℓ, which translates to UMUBs in the case ℓ is a prime.) In dimension ℓ = 8, a similar result is obtained in [10].
Motivated by Theorem 1.1 and the result in dimension 8, the following conjecture is then stated in [10]. CONJECTURE 1.2 (Mandayam et al. [10]). Given ℓ/2+1 maximal commuting Pauli classes C 1 , C 2 , . . . , C ℓ/2+1 belonging to a complete set S of classes in ℓ = 2 n =: d n , there exists exactly one more maximal commuting class C of Pauli operators in ∪ 1≤i≤ℓ/2+1 C i . The class C together with the remaining classes of S forms an unextendible set of Pauli classes of size ℓ/2 + 1, whose common eigenbases form a weakly UMUB of order ℓ/2 + 1.
We will show that this conjecture is true if and only if n = 2 or n = 8. In fact, we will consider the conjecture in any characteristic d (i.e., for any prime d), and show that it is true if and only if d = 2 and n = 2 or n = 8.
We then indicate that an alternative version of the conjecture might be true, and describe several new construction techniques to obtain weakly unextendible sets of MUBs.
At the end of the paper, we discuss a special kind of weakly unextendible sets of MUBs, called "Galois MUBs," which attain an optimal bound in relation to being unextendible.
Acknowledgement. The author wishes to thank Marcus Grassl and William K. Wootters for various interesting communications on the subject of this note.

THE GENERAL PAULI GROUP
Let d be an odd prime. Let {|s |s = 0, 1, . . . , d − 1} be a computational base of C d . Define the d 2 (generalized) Pauli operators of C d as where X d and Z d are defined by the following actions The set P of generalized Pauli operators of the N -qudit Hilbert space C d N is the set P of d 2N distinct tensor products of the form where the σ i k run over the set of (generalized) Pauli matrices of C d . Denote P × = P \ {I}. These operators generate a group P = P N (d) -the general Pauli group or discrete Heisenberg-Weyl groupunder ordinary matrix multiplication, which has order d 2N +1 .
For the case of N -qubit Hilbert spaces, we refer the reader to [19] -it is completely similar.

UNEXTENDIBLE SETS OF MUBS AND OPERATOR CLASSES
Let U be a set of d 2 mutually orthogonal unitary operators in C d using the Hilbert-Schmidt norm: operators A and B are orthogonal if tr(AB † ) = 0. Along with the identity operator I, U constitutes a basis for the C-vector space of (d × d)-complex matrices M d×d (C). A standard construction of MUBs outlined in [1] relies on finding classes of commuting operators, with each class containing d−1 mutually orthogonal commuting unitary matrices different from the identity I.

Maximal commuting operator classes. A set of subsets
constitutes a (partial) partitioning of U \ {I} into mutually disjoint maximal commuting classes if the subsets C j are such that (a) the elements of C j commute for all 1 ≤ j ≤ ℓ and (b) C j ∩ C k = ∅ for all j = k. In the rest of the paper, we use the term "Pauli classes" to refer to mutually disjoint maximal commuting classes formed out of the N -qudit Pauli group P N (d) ≤ U d N (C). 1 The correspondence between maximal commuting operator classes and MUBs is stated in the following lemma, originally proved in [1]. The correspondence between MUBs and maximal commuting operator classes gives rise to a weaker notion of unextendibility, based on unextendible sets of such classes.

SYMPLECTIC POLAR SPACES AND THE PAULI GROUP
Consider the projective space PG(2N − 1, d) of dimension 2N − 1, N ≥ 2, over the field F d with d elements, d an odd prime. Let F be a non-degenerate symplectic form of PG (2N − 1, d). For F one can choose the following canonical bilinear form [7]: Then the symplectic polar space W 2N −1 (d) consists of the points of PG(2N − 1, d) together with all totally isotropic spaces of F [7]. Here, a totally isotropic subspace is a linear subspace of PG(2N − 1, d) that vanishes under F .
One can also define this space in the underlying 2N -dimensional vector space V (2N, d) over F d using a non-degenerate alternating bilinear form (which induces a symplectic form on the projective space).
(i) The derived group P ′ = [P, P] equals the center Z(P) of P.
(iv) We have the following short exact sequence of groups: defines a non-degenerate alternating bilinear form on V (2N, d), so defines a symplectic polar space W 2N −1 (d). Here the derived group P ′ is identified with the additive group of F d . Applying this result, one can easily construct sets of MUBs of maximal size ℓ + 1 using the symplectic geometry [19].

UNEXTENDABLE MUTUALLY UNBIASED BASES AND PAULI CLASSES
In this section, we explain in detail the correspondence between Pauli classes and generators of symplectic polar spaces of [19]. It has the same proof as Theorem 4.4, but we make the relation between (unextendible) commuting Pauli classes and the generators more explicit. We indicate the proof in several steps. Let d be any prime and N ∈ N \ {0, 1}. Let S be a partial spread of W 2N −1 (d), i.e., a set of (N − 1)dimensional isotropic subspaces which are two by two disjoint. Let M + 1 be the number of elements in S, and note that M + 1 ≤ d N + 1 (equality holds when S is a spread). Then S corresponds to a set of mutually unbiased bases in the associated d N -dimensional Hilbert space, in the following way.
Step 1 To S corresponds a set of M +1 subgroups H i , i ∈ {0, 1, . . . , M }, of P of size d N +1 which mutually (two by two) intersect (precisely) in Z(P).
which are not contained in Z(P), so that no two such elements are in the same Z(P)-coset.
Step 3 Then U(S) is a set of commuting unitary classes.
Step 4 If S is a complete partial spread of W 2N −1 (d), that is, if S is not strictly contained in another partial spread, then U(S) is unextendible, and the corresponding set of MUBs is weakly unextendible of size M + 1. In particular, this construction applies when U(S) is a set of Pauli operators (each Z(P)-coset contains precisely one Pauli operator).

The bijection
, and let C(C d N ) be the set of commuting classes of Pauli operators (of size d N − 1). Note that from the above, it follows that we have a bijection which sends an element U ∈ C(C d N ) to a generator, following the scheme explained above. It is indeed a bijection: to each generator corresponds a unique maximal abelian subgroup A ≤ P ≤ U d N (C) as above (and conversely), and each Z(P)-coset in this subgroup contains precisely one Pauli operator. Together, the set of (nontrivial) Pauli operators in A form one commuting class of Pauli operators of size d N − 1, that is, one element of C(C d N ).

5.2.
Prime dimension. Now let N = 1 (i.e., prime dimension) and d = 2. Then Proposition 4.2 tells us that P is a group of size d 3 and exponent d, and its center has size d -in other words, P is extra-special.
In P one can now choose subgroups H i j as above, and again [1] applies. If d = 2, the result is well known (but it can also be derived as above).

THE CASE d PRIME, N = 2 -SMALL AND LARGE EXAMPLES
If d = p is a prime and N = 2, the corresponding symplectic polar space is W 3 (p) =: X, with ambient projective space PG(3, p), and it has two types of linear subspaces which are completely contained in X, namely points of PG(3, p), and (projective) lines.

Grids and point-line duals.
Before proceeding, we explore some synthetic properties of W 3 (p) which will makes things easy below.
Let P be the point set of W := W 3 (p), L its line set, and let I be the (symmetric) "incidence relation" on (P × B) ∪ (B ∪ P) which says that xIL (LIx) if and only if the point x is on the line L. Then this point-line geometry is a generalized quadrangle [13], and a very deep and extensive theory exists on these structures. Note that each line contains p + 1 points and that on each point there are p + 1 lines. Also, recall the following defining projection property for generalized quadrangles: if x is a point and X a line not containing x, there is a unique line Y I x which meets X (and then in a unique point).
Now consider the point-line geometry Q with line set B, point set P and the same incidence relation I -the so-called point-line dual of W 3 (p). Then by [13, 3.2.1], Q is isomorphic to the point-line geometry of an orthogonal quadric Q(4, p) in PG(4, p). Moreover, if p = 2, Q and W are isomorphic [13, 3.2.1].
Let "⊥" denote the orthogonality relation in W 3 (p), and let V, W be arbitrary lines which do not meet. Then {V, W } ⊥ consists of p + 1 lines which are mutually disjoint. If p is odd, The set of points on the lines of R 1 := {V, W } ⊥ , which is the same as the set of points on the lines of 6.2. Antiregularity. If ℓ is any odd prime power, we will use the fact that W 3 (ℓ) has no (3 × 3)-grids. This is a corollary of a property called "antiregularity," and can be found in [13, 3.3.1(i), dual].
6.3. The case p = 2. We start with giving an alternative and very short proof of Theorem 1 of [10] using the connection between Pauli classes and partial spreads of symplectic polar spaces. Theorem 6.1 (Mandayam et al. [10]). Given three Pauli classes C 1 , C 2 , C 3 belonging to a complete set S of classes in d = 4, there exists exactly one more maximal commuting class C of Pauli operators in C 1 ∪ C 2 ∪ C 3 . The class C together with the remaining two classes C 4 and C 5 of S forms a unextendible set of Pauli classes, whose common eigenbases form a weakly UMUB of order 3.
Proof. Interpret S in W 3 (2); then by Theorem 5.1 to the C i correspond lines L i of W 3 (2) (i = 1, . . . , 5), and they form a spread. Consider the lines L 1 , L 2 , L 3 . Then either there is precisely one line L of W 3 (2) meeting them all, or there are three such lines. In the latter case the lines L 1 , L 2 , L 3 form a regulus of a (3 × 3)-grid, and as we have seen any line of W 3 (2) meets the point set of such a grid, leading to the fact that L 1 , L 2 , L 3 would not be extendible to a spread, contradiction. So we are in the fomer case, and the class C corresponding to L is the one of the statement. The set C, C 4 , C 5 obviously is unextendible, since extending it with a class C would mean than the corresponding line L be contained in the point set of L 1 ∪ L 2 ∪ L 3 , implying that there would be another line besides L meeting all of L 1 , L 2 , L 3 , contradiction.
We now give a short proof of another result of [10], namely Theorem 5 of that paper. Theorem 6.2 (Mandayam et al. [10]). Given an unextendible set of three Pauli classes C 1 , C 2 , C 3 in d = 4, the nine operators in C 1 ∪ C 2 ∪ C 3 can be partitioned into a different set of three maximal commuting classes C ′ 1 , C ′ 2 , C ′ 3 such that each C ′ i has precisely one operator in common with each C j , i, j ∈ {1, 2, 3}.
Proof. Let L 1 , L 2 , L 3 the lines corresponding to C 1 , C 2 , C 3 in W 3 (2); we have seen that either one or three lines are contained in {L 1 , L 2 , L 3 } ⊥ ; in the latter case, an easy counting argument shows that all lines of W 3 (2) intersect with L 1 ∪ L 2 ∪ L 3 , so suppose we are in the former case, and let {L} := {L 1 , L 2 , L 3 } ⊥ . Then each point on L is incident with precisely one line besides L and not in {L 1 , L 2 , L 3 }. By the projection property of generalized quadrangles, there are six lines different from L 1 , L 2 , l 3 which meet the six points of L 1 ∪ L 2 ∪ L 3 not on L in precisely two points. So the total number of lines meeting L 1 ∪ L 2 ∪ L 3 is 13, and {L 1 , L 2 , L 3 } is indeed extendible (since there are 15 lines in total).
In the next subsection, we will see a general approach for constructing unextendible Pauli classes in C d 2 with d a prime number, starting from a complete set. As a corollary, we will obtain yet another proof for the result of Mandayam et al. Proof. As before, we pass to W 3 (p). Let T be any spread of W 3 (p). Now let U be any line of W 3 (p) which is not contained in T; then there are precisely p + 1 lines in T which hit U (each in exactly one point), due to the fact that the lines of T partition the point set of W 3 (p). Call this line set T U . Now consider the line set Note that |T(U )| = p 2 − p + 1. If it is not a complete partial spread, there is at least one other line R of W 3 (p) not meeting any line of T(U ), and as a point set it clearly must be contained in the point set "of" T U . And then all lines of T U meet both U and R. If yet another line R ′ would exist that extends T U ∪ {R}, R ′ would also be met by the lines of T U -in other words, R ′ ∈ T ⊥ U while T U = U ⊥ ∩ R ⊥ . As we have seen, this is not possible, so only at most one line R can be added.
Translating back to Pauli classes gives the desired result.
It is easy to see that both cases of Theorem 6.7 can occur.
As we have not used the fact that p is prime, we can translate immediately to symplectic polar spaces over any finite field. Corollary 6.4. For each prime power ℓ there exists a complete partial spread S of W 3 (ℓ) of size ℓ 2 − ℓ + 1 or ℓ 2 − ℓ + 2.
Remark 6.5. For ℓ even, we have seen this result in the literature (see, e.g. [3] and the references therein) -it would be safe to attribute this result to folklore though. We presume the odd case is somewhere as well, but the way of proving is needed below, so it is included anyhow for the sake of completeness.
One could apply the technique in the proof of Theorem 6.7 multiple times to obtain examples with less elements. And indeed, this works quite well, as we will demonstrate now. We will work immediately in W 3 (ℓ), and will not restrict ourself only to the prime case. So ℓ is a prime power. We do ask that ℓ is odd -it will be used in the proof.
Proof. Let k be as in the statement, and consider any subset R of {0, 1, . . . , (ℓ − 1)/2} of size k + 1; for simplicity, we consider w.l.o.g. the set {0, 1, . . . , k}. Then define the following set: It is straightforward to see that S R is a partial spread of size ℓ 2 − (k + 1)ℓ + (3k + 1). As for completeness, suppose we could enlarge S R with some line U to another partial spread. As S is a spread, U must be contained in ∪ u∈R S u , and it cannot be contained in S nor S R . By the Pigeon Hole Principle, some S w must have at least three lines meeting U since U has d + 1 points and k + 1 < d+1 2 (in case U ∈ {L, M } ⊥ , one does not need the Pigeon Hole Principle). However, this implies the existence of a (3 × 3)-grid, contradiction.
For each odd prime power ℓ the bounds appear to be new (up to some small coincidences). For fixed ℓ, we obtain complete partial spreads of respective sizes Translating back to Pauli operators, we obtain the next result. The construction has many variations, all using roughly the same ideas, and all giving similar (but not the same) bounds. We will come back to these variations in a forthcoming paper.   [10]). Given ℓ/2+1 maximal commuting Pauli classes C 1 , C 2 , . . . , C ℓ/2+1 belonging to a complete set S of classes in ℓ = 2 N =: d N , N ∈ N \ {0, 1}, there exists exactly one more maximal commuting class C of Pauli operators in ∪ 1≤i≤ℓ/2+1 C i . The class C together with the remaining classes of S forms an unextendible set of Pauli classes of size ℓ/2 + 1, whose common eigenbases form a weakly UMUB of order ℓ/2 + 1.
In this section we will show that this conjecture is true if and only if N = 2 or N = 3. In fact, we will consider the conjecture in any characteristic d (i.e., for any prime d), and show that it is true if and only if d = 2 and N = 2 or N = 3.
Translated to the geometric setting, we obtain: "given 2 N −1 + 1 elements α 1 , α 2 , . . . , α 2 N −1 +1 belonging to a spread S of the polar space W 2N −1 (2), there exists exactly one more generator χ which is completely contained in the union of these elements, such that χ together with the remaining elements of S constitutes a complete partial spread." Note that the situation implies that χ meets each α j , j = 1, 2, . . . , 2 N −1 + 1.
We will replace d = 2 by any prime d, and consider the same situation (immediately in the geometric setting). We will also slightly generalize the statement by replacing "exactly one" by "at least one." So let S be a spread of W 2N −1 (d), d a prime. We assume that the conjecture above is true (in the more general setting).
First suppose that U and U ′ are different subsets of S, both of size d N −1 + 1. Let α be a generator which meets all elements of U and is covered by these elements, and let α ′ be a generator which meets all elements of U ′ and is covered by them. Then α = α ′ .
In the next counting argument, we will use the fact that the number of generators of . Per subset of S of size d N −1 + 1, by the conjecture we have at least one generator meeting all of its elements, and covered by them. Such a generator is never contained in S. So we have that Here, (Note that equality should hold in (5) in the "precisely one statement.") Now (5) is equivalent to or, slightly simplyfied: and that Observe that if for some value N = M , we have then the same inequality holds for all M ′ ≥ M . This is already enough to conclude with a contradiction for d ≥ 5; d = 3 and N ≥ 3; and d = 2 and N ≥ 6. The cases (d, N ) = (3, 2), (2,5), (2,4) all yield a contradiction when substituted in (5); the substitution (d, N ) = (2, 2), which is precisely the case of W 3 (2) which was already studied before, leads to equality in (5), as does the substitution (d, N ) = (2, 3), which is the case of W 5 (2).
In the next section we will formulate and discuss variations on Conjecture 1.2; to that end, we first try to generalize Theorem 6.7.

EXISTENCE OF MAXIMAL PAULI CLASSES
Before proceeding, let us first introduce a simple lemma about complete "partial spreads" of general incidence structures. Let Γ = (E, t, T ) be a triple, with T = {0, 1, . . . , n}, n ∈ N × , and t a surjective function from the set E = ∅ to T . For each i ∈ {0, 1, . . . , n}, put E i := t −1 (i), and call its elements the elements of type i. So (12) E = ∪ i∈T E i , and |E| ≥ |T |.
In particular, we call elements of E 0 "points." We now assume that for i > 0, every element of E i is a subset of E 0 . This is a natural assumption: we see each "i-space" (= element of type i) as a point set.
An i-spread of Γ is a partition of E 0 in elements of type i. Complete i-spreads are introduced naturally as above. Proof. If S \ S χ ∪ {χ} could be completed to an i-spread S ′ of Γ, S ′ must have elements which all are subsets of Ω(S, χ), and which partition Γ.
If ℓ is the maximum number of elements of type i contained in Ω as subsets and which are two by two disjoint, the number of elements in a maximal partial i-spread containing S \ S χ ∪ {χ} is at most |S| − |S χ | + ℓ. (Note that ℓ ≥ 1 as χ itself is in Ω(S, χ).) Remark 8.2 (Back to the prime case). Note that the first part of Theorem 6.7 is an application of the construction method of Proposition 8.1 (with χ = L).
8.1. U-Sets. Motivated by Proposition 8.1, a U-set with carrier χ is a set S χ of mutually disjoint generators of W 2N −1 (d) which all meet some generator χ ∈ S χ such that (14) χ ⊂ ∪ Y ∈Sχ Y, and such that ∪ Y ∈Sχ Y cannot be partitioned by a partial spread P of generators which includes χ.
Note that the number of elements of an U-sets is not uniquely determined by N and d. (One U-set could also have different carriers.) Proof. Let S χ be a U-set. If S χ is not contained in a spread, then we are done, so suppose it is contained in some spread S. Then by Proposition 8.1 we have that S \ S χ ∪ {χ} cannot be completed to a spread.
Note that this proposition can also be applied to general incidence geometries.
For the rest of this section, we suppose d is an odd prime. Before proceeding, recall that a spread S (of generators) of W 2N −1 (d) is regular if the following property is satisfied: if for every three distinct elements α, β, γ in S, L is the set of lines of PG(2N − 1, d) which meet each of α, β, γ, then there are d − 2 further elements of S which meet every line in L. It is well known that every symplectic polar space has regular spreads. Now let R be a regular spread of W 2N −1 (d). Take a generator χ which meets some α ∈ R in a space of dimension N − 2 (and note that this is possible), and let R χ be the set of elements in R which meet χ. Note that |R χ | = d N −2 + 1. Now consider a generator β = χ, α which contains χ ∩ α, and which is disjoint from the elements in R χ \ {α}. (For the existence of such a generator, see Appendix A.) Then because R is a regular spread, one notes that S χ := R χ \ {α} ∪ {β} is a U-set. For, suppose that ∪ Y ∈Sχ Y can be partitioned by a partial spread P of generators which includes χ. Let γ ∈ P \ {χ} contain some point b of β; then γ ∩ β = {b}. Let B be any line in γ containing b; then B meets d + 1 different elements of S χ , one of which is β. As d ≥ 3, the fact that R is a regular spread implies that b ∈ α, contradiction.
In the next theorem, we prove that unextendible sets of Pauli classes of C d N always exist (ignoring possible sizes completely), that is, that complete partial spreads which are not spreads always exist in W 2N −1 (d). This fact is not necessarily true for general incidence geometries which have i-spreads (using the nomenclature of above): consider for instance an incidence geometry for which the elements of type i precisely form one i-spread. So although the existence of complete partial spreads is probably seen as folklore, we see the need to formally write it down. Proof. Translated to W 2N −1 (d), we need to show that the latter geometry always contains complete partial spreads which are not spreads. So take a regular spread R. Consider a generator χ as above, and construct the U-set S χ := R χ \ {α} ∪ {β}. Now apply Proposition 8.3.

Remark 8.5.
Note that if R χ in the proof of the previous theorem is such that there does not exist a generator besides χ which is contained in ∪ α∈Rχ α, then In the special case d = 2, we would end up with an unextendible set of Pauli classes of size 2 N −1 + 1.

8.2.
Reformulation of Conjecture 7.1. We have seen that Conjecture 7.1 is only true when N = 2 or N = 3. On the other hand, there seems to be some evidence that the bound of that conjecture could be attained (see, e.g., the previous remark). So we reformulate the conjecture as follows -we will do it in geometric terms, over all fields F ℓ with ℓ a prime power, but again, for the applications in Quantum Information Theory, one takes ℓ to be prime. When d = 2, one obtains the same bound as in Conjecture 7.1. We hope to come back to this conjecture in the near future.

"GALOIS MUBS"
When d = 2, 3, 5, 7 or 11, there exist extremely exotic examples of unextendible sets of Pauli classes of size d 2 − 1 in C d 2 . (Details, constructions and references can be found in [4].) We propose to call the corresponding sets of MUBs "Galois MUBs," because they are all related to exotic 2-transitive representations of special linear groups, as was first noted by Galois (see also [4]). They are extremely special amongst Pauli classes of C d 2 , d a prime, or even all Hilbert spaces, due to the following result. Theorem 9.1 (See [13], §2.7). Let Γ be a generalized quadrangle of finite thick order (s, s), and let C be a complete partial spread of Γ. If Γ is not contained in a spread of Γ, then (16) |C| ≤ s 2 − 1.
As we have seen that the points and lines of any W 3 (d) form a generalized quadrangle, this result applies to W 3 (d) and hence also to Pauli classes in C d 2 .  1)-bound, and conjecturally they are the only ones. Geometrically, they also satisfy very extreme properties, which rightly translate to Pauli classes. Much more details on the geometric structure of partial spreads of size s 2 − 1 in generalized quadrangles of order (s, s) can be found in the author's paper [18].
The next theorem, taken from the author's paper [18], says that when d = 2, up to isomorphism there is only one complete partial spread of size 3 = 2 2 − 1 in W 3 (2). Theorem 9.4 ([18]). Up to isomorphism there is only one complete partial spread of size 3 in W 3 (2). Corollary 9.5. Up to isomorphism, there is only one unextendible set of Pauli classes of size 3 in C 4 . Remark 9.6 (On isomorphisms). Of course, one needs to specify what isomorphisms between unextendible sets of Pauli classes are. Because of the General Connecting Theorem (and the bijection ρ), we propose to say that unextendible sets of Pauli classes U and U ′ in C d N are isomorphic if there exists an automorphism of W 2N −1 (d) which maps the complete partial spread S(U) corresponding to U, to the complete partial spread S(U ′ ) corresponding to U ′ . This is a natural notion of "isomorphism," since automorphisms of W 2N −1 (d) preserve collinearity of points, so also the commuting of operators at the level of Pauli operators. (One could also define isomorphisms through the general Pauli group. On the other hand, such automorphisms induce automorphisms of W 2N −1 (d) anyhow, while the converse is not true. So one misses (many) maps which should be considered as isomorphisms in this way.)

CONCLUSION
The geometry underlying the space of the generalized Pauli operators/matrices characterizing N dlevel quantum systems, with N ≥ 2 and d any prime, is that of the symplectic polar space of rank N and order d, W 2N −1 (d).
Using this connection, we have derived a short proof of a recent result of [10] on the unextendibility of MUB sets in C 4 (their Theorem 1). Moreover, we generalized this result for all d = square of a prime, and presented a construction of a class of maximal partial spreads of W 3 (ℓ) for any odd prime power ℓ, attaining new bounds in generically every case, which rightly translates to UMUBs in the case ℓ is a prime. We also gave a very short proof of Theorem 5 of [10].
We then considered Conjecture 1 of [10] which conjecturally generalizes the aforementioned result of [10] to any dimension and showed that it is true if and only if N = 2 or N = 3.
We then indicated that an alternative version of the conjecture might be true, and described several new construction techniques to obtain weakly unextendible sets of MUBs.
Finally, we discussed a special kind of weakly unextendible sets of MUBs, called "Galois MUBs," which attain an optimal bound in relation to being unextendible.

APPENDIX A. PROPERTIES OF (SYMPLECTIC) POLAR SPACES
Consider the space W 2N −1 (d), d a prime, N ≥ 2. (We restrict ourselve to the prime case because that's the class which translates to Pauli operators, but everything works when d is a prime power as well.) A.1. Let γ be a generator, and x a point not in γ. Then a well-know property (of general polar spaces) -see e.g. [20, p.137, (c)] -says that there is a unique generator on x which meets γ in an (N − 2)space, γ(x). (If "⊥" is the orthogonality relation coming from the associated alternating form, then γ(x) = x ⊥ ∩ γ.) Now let γ and γ ′ be disjoint generators. Then it is not hard to see that for every (N − 2)space δ in γ ′ , there is precisely one point y ∈ γ such that y, δ is a generator (y = γ ∩ δ ⊥ ). So the map (17) µ is a bijection between the points of γ and the hyperplanes of γ ′ .
A.2. Now let α be an (N − 2)-space contained in W 2N −1 (d); it is well-known that there are d + 1 generators g 0 , . . . , g d containing α. Let β be a generator disjoint from α. By the surjectivity above, it follows that some g i has to intersect β, and then necessarily in one point.
A.3. Structure of spreads. Let S be a spread of W 2N −1 (d), d a prime, N ≥ 2. Let α = g 0 ∈ S, and let τ be an (N − 2)-space in α. Let g 1 , . . . , g d be the other generators containing τ . By the previous paragraph, each element of S \ {α} meets some g i (i = 0) in precisely one point. And conversely, each point of g j \ τ is contained in precisely one spread element. Indeed, #(points of g i \ τ ).

APPENDIX B. SOME MORE PROPERTIES OF W 3 (d)
As in the first appendix, for the applications in Quantum Information Theory considered here, one wants to think of d as being prime, but everything holds when d is a prime power as well. What we do ask is that d is odd.
Let S be a classical spread of W 3 (d); point-line dualize to obtain Q(4, d) -S becomes an elliptic quadric, denoted O. Now let x be a point of Q(4, d) not contained in O. As usual, let "⊥" denote the orthogonality relation associated to the defining quadratic form of Q(4, d) (say, corresponding to the variety with equation X 2 0 + X 1 X 2 + X 3 X 4 = 0). Then x ⊥ ∩ O is a conic section, and since d is odd, there is precisely one other point y ∈ O for which (19) y Note that the latter expression is equal to {x, y} ⊥ . Going back to W 3 (d) (i.e., dualizing again), we obtain that if X is a line of W 3 (d) not in S, and S X is the set of d + 1 lines in S which meet X, then there is precisely one other line Y not in S such that (20) S