Joint measurability structures realizable with qubit measurements: incompatibility via marginal surgery

A given set of measurements can exhibit a variety of incompatibility relations and an intuitive way to represent its joint measurability structure is via hypergraphs. The vertices of such a hypergraph denote measurements and the hyperedges denote (all and only) subsets of compatible measurements. Quantum measurements represented by positive operator-valued measures (POVMs) can realize arbitrary joint measurability structures. This latter fact opens up the possibility of studying generalized contextuality a la Spekkens on joint measurability structures not realizable with projective measurements. Here we explore the scope of joint measurability structures that are realizable with POVMs on a qubit. We develop a technique that we term marginal surgery to obtain nontrivial joint measurability structures starting from a set of measurements that are all compatible. We show explicit examples of marginal surgery on a special set of qubit POVMs -- which we term planar symmetric POVMs -- to construct joint measurability structures such as $N$-cycle and $N$-Specker scenarios for any integer $N\geq 3$. We also show the realizability of various joint measurability structures with $N\in\{4,5,6\}$ vertices. In particular, for the case of $N=4$ vertices, we show that all possible joint measurability structures are realizable with qubit POVMs. We conjecture that all joint measurability structures are realizable with qubit POVMs. This is in contrast to the arbitarily large dimension required by the construction in R. Kunjwal et al., Phys. Rev. A 89, 052126 (2014). Incidentally, our realization of all $N$-Specker scenarios on a qubit also renders this previous construction maximally efficient in terms of the dimensions required for realizing arbitary joint measurability structures with POVMs. Along the way, we also prove some new sufficiency conditions for joint measurability of qubit POVMs.

Measurements in quantum theory exhibit incompatibility, i.e., it is impossible to simulate certain sets of measurements by coarse-graining a single measurement [1]. This lack of joint measurability is crucial to the demonstration of nonclassicality in quantum theory, e.g., both Bell inequality violations [2][3][4] and Kochen-Specker (KS) contextuality [5] are impossible in the absence of incompatible measurements. While the joint measurability (or compatibility) of a set of projective measurements is a binary property, characterized entirely by their pairwise commutativity, the joint measurability of general quantum measurements given by positive operatorvalued measures (POVMs) is, in general, not characterized by pairwise commutativity. That is, it is possible to have nonprojective measurements that are noncommuting but they nonetheless admit a joint POVM that can be coarse-grained to obtain their statistics. Recent years have seen a steady increase in research activity geared towards a better understanding of the joint measurability of POVMs, its connection to steering, state discrimination, Bell nonlocality, as well as the general area of determining conditions for joint measurability of sets of POVMs [6][7][8][9][10][11][12].
In the sphere of contextuality research, the use of nonprojective POVMs in proofs of KS-contextuality has been controversial [13]. However, within the generalized approach to contextualityà la Spekkens, arbitrary POVMs can be accommodated [14]. The joint measurability relations among a set of POVMs can be represented by a hypergraph that we will refer to as the joint measurability structure of the set, following Ref. [8]. 1 That is, a joint measurability structure is a collection of (all and only) jointly measurable subsets of a given set of POVMs. Since any set of projective measurements is jointly measurable if and only if they commute pairwise [1], its joint measurability structure corresponds to a graph, i.e., a 1 We will recall the definition of joint measurability structure more formally later in this paper but for now it suffices to note that it specifies whether any subset of POVMs in a given set is compatible or incompatible. Note also that we use the terms 'joint measurability' and 'compatibility' interchangeably in this paper and 'incompatibility' denotes the lack of joint measurability.
collection of jointly measurable subsets, each of size two. Ref. [15] showed that all joint measurability structures corresponding to graphs can be realized by projective measurements. Later, Ref. [8] realized arbitrary joint measurability structures (going beyond the ones that are graphs) using POVMs. The result of Ref. [8] indicates that there is a whole zoo of joint measurability structures that remains to be explored from the point of view of contextuality, where only nonprojective POVMs can provide any evidence of nonclassicality, projective measurements being useless for this purpose. The simplest example of such a joint measurability structure is Specker's scenario (Fig. 2) which was shown to be realizable on a qubit [16] well before Ref. [8] showed the quantum realizability of arbitrary joint measurability structures. In Specker's scenario, at least under the assumption that the operational theory is quantum theory, one can witness contextuality [17,18]. It remains to extend such a demonstration of contextuality to arbitrary scenarios in general operational theories [19][20][21]. Before such a project can be undertaken, however, one needs a better understanding of the scope of joint measurability structures that can be realized by quantum systems of limited Hilbert space dimension, particularly if one is keen to build experimental tests and quantum information protocols based on such joint measurability structures. Such an understanding can also potentially aid the elucidation of facts about joint measurability that are particular to quantum theory in the broader landscape of general probabilistic theories [22][23][24][25][26]. Already, in Ref. [26], it was shown that considering the joint measurability structure corresponding to a complete graph on four vertices is enough to separate almost quantum correlations (generalized to include single laboratory situations) [27,28] from those correlations that can be realized with projective measurements in quantum theory [28]. This result was taken as a failure of Specker's principle [26,29] -that any set of pairwise compatible measurements is globally compatible -for any measurements underlying almost quantum correlations. However, Specker's principle also fails to hold within quantum theory if the measurements are not projective, so one can always ask: what if quantum measurements are allowed to be arbitary POVMs rather than just projective measurements? Is there still, in any sense, a qualitative difference between measurements in an almost quantum theory and those in quantum theory if we allow arbitrary POVMs in the latter? Presumably, an answer to this question would require a framework for addressing nonclassicality of quantum correlations arising from arbitrary POVMs and, hence, the ability to deal with arbitary joint measurability structures. Such a framework would be a natural extension of the one proposed in Ref. [30]. All these considerations motivate the present study as a first step towards addressing general features of joint measurability in quantum theory for systems of limited Hilbert space dimension.
Coming back to the construction of Ref. [8], we note that it requires a steadily growing Hilbert space dimen-sion to realize joint measurability structures corresponding to ever larger sets of measurements, rendering it quite inefficient in this sense. This raises the natural question of whether it is possible to realize joint measurability structures for arbitrarily large sets of measurements using the smallest possible Hilbert space dimension, i.e., using qubit POVMs. We show that this can be done for at least two families of joint measurability structures: N -cycle scenarios and N -Specker scenarios (Examples 1 and 4) for any finite number, N ≥ 3, of measurements. To do this, we introduce a technique that we term marginal surgery, which requires tweaking a joint POVM of a set of compatible POVMs to realize new joint measurability structures. We show the realizability of many joint measurability structures with N vertices, N ∈ {4, 5, 6}. In particular, for N = 4, we show that all conceivable joint measurability structures can be realized with qubit POVMs. This is also obviously true for N = 3 vertices, where the realizability of 3-Specker scenario is enough to make this claim, the realizability of other scenarios requiring only incompatibility of pairs of POVMs. Motivated by the fact that all conceivable joint measurability structures for N ∈ {3, 4} vertices are thus realizable with qubit POVMs, we conjecture that arbitrary joint measurability structures can be realized with qubit POVMs. Although we do not have a concrete proposal, we expect that a general recipe for realizing arbitrary joint measurability structures with qubit POVMs could potentially be obtained by some clever application of the method of marginal surgery. On the other hand, we also mention a potential counter-example to the conjecture, namely, a joint measurability structure which can perhaps be shown to be not realizable with qubit POVMs. This potential counter-example is of the type {E 1 , E k }, k ∈ {2, . . . , N } , for some N > 4. We now outline the structure of this paper: • In Section II, we start with a review of definitions and some known facts about joint measurability of POVMs that will be used in this paper. Sections II B 2, II B 3, and II B 8 contain (to our knowledge) new material including definitions, theorems, and corollaries that we use in this paper. In particular, they define the notion of geometrically equivalent sets of qubit POVMs.
• In Section III, we introduce marginal surgery and apply it, firstly, to the case of pairs of POVMs out of a set of qubit POVMs (Sec. III A), realizing N -cycle scenarios as an example, and then to the case of arbitrary subsets of a set of qubit POVMs (Sec. III B), realizing N -Specker scenarios as an example.
• In Section IV, we obtain a sufficient condition for joint measurability of a set of coplanar and unbiased binary qubit POVMs with the same purity.
• In Section V, we study the qubit realizability of various joint measurability structures "in between" N -cycle and N -Specker for N = 4, 5, 6 vertices (cf. Sec. V A). In Sections V B and V C, we show the qubit realizability of arbitrary joint measurability structures on N = 4 vertices.
• We conclude with discussion and some open questions in Section VI.

II. A REVIEW OF JOINT MEASURABILITY OF BINARY QUBIT MEASUREMENTS
Here we review the joint measurability conditionsnecessary and/or sufficient -for binary qubit POVMs that are known from previous work in this area. We also make some general observations about joint measurability that will be useful in later sections. We begin with basic definitions below.
A. POVMs and their joint measurability Definition 1 (POVMs and binary qubit POVMs). Given a non-empty set X and a σ-algebra 2 F of subsets of X, a positive operator-valued measure (POVM) E on F is defined as the map E : Here B + (H) denotes the set of positive semidefinite operators on the Hilbert space H and I is the identity operator on H. E becomes a projection-valued measure (PVM) under the additional constraint that E(Y ) 2 When the Hilbert space is two-dimensional, i.e., H ∼ = C 2 , and we let X = {x (1) (1) , x (2) }}, we have the notion of a binary qubit POVM as any POVM E : F → B + (C 2 ). Definition 1 implies that for binary qubit POVMs it is enough to specify E({x (1) (1) and x (2) are outcomes of the measurement and we will henceforth label them x (1) = 1 and x (2) = −1 in keeping with the convention for spin-1/2 POVMs in quantum theory. 3 For simplicity, we write is a positive semidefinite operator bounded above by the identity, i.e., an effect. We can write it in the usual Pauli operator basis of B(C 2 ). Then every binary qubit POVM E can be parametrized with four real parameters α and a = (a 1 , a 2 , a 3 ) (see Sec. 3.1 in Ref. [1]): The POVM is a projection-valued measure (PVM) when α = a = 1. We will call the vector a the Bloch vector of the POVM E. Three or more POVMs will be called coplanar if their Bloch vectors are coplanar.
Following Ref. [8], we also need the notion of a joint measurability structure: Definition 3 (Joint measurability structure). A joint measurability structure is a hypergraph with a set of vertices, V , and a family of (finite) subsets (called hyperedges) of V , denoted E ⊆ {e|e ⊆ V }. Each vertex denotes a measurement in a set of measurements (indexed by V ) and each hyperedge denotes a subset of compatible (or jointly measurable) measurements. Any subset of vertices that do not share a common hyperedge represents an incompatible subset of the given set of measurements. To model the requirement that every subset of a set of compatible measurements is also compatible, a joint measurability structure must also satisfy e ∈ E, e ⊆ e ⇒ e ∈ E. (3) We will call a joint measurability structure quantumrealizable, or that it admits a quantum representation, if its vertices can be represented by quantum measurements, i.e., POVMs, satisfying the (in)compatibility relations dictated by it.
We now mention some important examples of joint measurability structures that we will realize with binary qubit POVMs: Example 1 (N -cycle scenario). Let s = {E 1 , . . . , E N } be a set of N vertices (representing measurements). (In)compatibility relations on s of the form are said to form a joint measurability structure called the N -cycle scenario (Fig. 1). Trivially, an (N, 1)-compatible set is just a set of N pairwise incompatible observables, while (N, N )compatible set is just a set of N compatible measurements. We now follow with two more special cases of (N, M )-compatibility scenarios for M = 2 and M = N − 1.
Example 3 (N -complete scenario). Let s = {E 1 , . . . , E N } be a set of N vertices (representing measurements). A joint measurability structure on s where each pair of measurements is compatible but no three measurements are (i.e., s is an (N, 2)-compatible set) is called an N -complete scenario. The hypergraph representing this structure is a complete graph with N vertices.
Example 4 (N -Specker scenario). Let s = {E 1 , . . . , E N } be the set of N vertices (representing measurements). If each (N − 1)-element subset of s is compatible while s itself is incompatible (i.e. s is an (N, N − 1)compatible set) then the joint measurability structure of s is called an N -Specker scenario. This is the generalization of the notion of Specker's scenario which is the simplest non-trivial N -Specker scenario for N = 3. We provide more examples of joint measurability structures in Section V along with their quantum realizations with binary qubit POVMs.

B. Conditions for joint measurability of qubit POVMs
Previous research has uncovered many analytical criteria for the joint measurability of POVMs, particularly for qubits. We now collect some known results on joint measurability of qubit POVMs that will be of interest to us.

Two binary qubit POVMs
Yu et al. [31] proved a necessary and sufficient condition for the joint measurability of two binary qubit POVMs. We state this condition below. Theorem 1. Two binary qubit POVMs E 1 (1) = 1 2 (α 1 I + a 1 · σ) and E 2 (1) = 1 2 (α 2 I + a 2 · σ) are jointly measurable if and only if where we have Of particular interest to us will be a class of POVMs called unbiased binary qubit POVMs. We define them below: Definition 4 (Unbiased binary qubit POVMs). Binary qubit POVMs specified by where x = ±1, n = || n|| = 1 and 0 ≤ η ≤ 1 are called unbiased. Otherwise, they are biased. The parameter η is usually referred to as the purity or sharpness parameter, it's upper bound η = 1 corresponding to the case of a sharp (projective) measurement.
In the light of the Definition 4, we define the bias b associated with an outcome x = +1 of a binary qubit POVM E (simply, "the bias of E") by rewriting Eq. (1) as An unbiased qubit POVM has zero bias.
Theorem 2. Two unbiased binary qubit POVMs specified by E 1 (x 1 ) = 1 2 (I + x 1 η 1 n 1 · σ) and E 2 (x 2 ) = 1 2 (I + x 2 η 2 n 2 · σ) are jointly measurable if and only if The proof of this theorem follows directly from Eq. (5), although it was independently proven earlier [32]. The reader may consult Ref. [1] for a guide to previous literature on necessary/sufficient conditions for joint measurability.

Joint measurability and relabelling of outcomes
We now note two observations on joint measurability of POVMs that we will use later on: . . E m }, each POVM with outcome set O, exhibits a particular joint measurability structurerepresenting the (in)compatibility relations between the POVMs -and if we define POVM F such that F (Perm(o)) = E k (o) for all o ∈ O under some relabelling (or permutation of labels) denoted by Perm for some k ∈ {1, 2, . . . , m}, then the same joint measurability structure is realized by the POVMs Proof. Note that F is physically the same POVM as E k in the sense that, for any quantum state, it yields a probability distribution over outcomes in O that is a permutation of the probability distribution yielded by E k over O, hence the joint measurability structure will remain unaltered and the only thing that will change is the labelling of outcomes of the joint POVMs that can be coarse-grained to yield E k so that they can now be coarse-grained to yield F . More precisely, we show the following: . We can then simply define a joint POVM for F and The same argument works in the other direction, i.e., given the compatibility of F with M k , we can infer the compatibility of E k with M k and this is true for any set of compatible POVMs M k ⊂ M.
Hence the (in)compatibility relations of E k with all the POVMs in the set M\{E k } will be the same as that of F with M\{E k }.
That is, the joint measurability structure realized by Remark 1. Intepreted for the case of unbiased binary qubit POVMs, Theorem 3 says that their joint measurability is dependent only on the lines on which their Bloch vectors lie and not the particular orientation of a Bloch vector along a line. Hence, the choice of which outcome is labelled +1 and which is labelled −1 does not affect their joint measurability. Corollary 2. Let E 1 , E 2 and E 3 be three POVMs, each with outcome set O. If E 1 is jointly measurable with E 2 and E 3 is just a relabelling of the operators of E 2 , i.e., Then we have E 3 (Perm(o)) = E 2 (o) = o1∈O G 12 (o 1 , o) which we can use to define the joint POVM of E 1 and E 3 as The joint POVM of E 2 and E 3 , related to each other by relabelling of outcomes given by Perm, is just the POVM

Joint measurability and geometrically equivalent sets of POVMs
We will now prove some general relations between sets of POVMs such that they exhibit the same joint measurability structure. In particular, we will define a notion of geometrically equivalent sets of POVMs.
If f k (x k ) and e k (x k ) are related by some transformation we will formally write this as and we have that 1. the two sets exhibit the same joint measurability structure, 2. if the s 1 ⊆ s 1 is a compatible subset with joint POVM G s 1 , then a joint POVM of the corresponding subset in s 2 , i.e., s 2 = Os 1 ⊆ s 2 , is given by Proof. This follows from the fact that e k , f k ∈ R 3 and O(3) is the group of isometries that fix the origin in R 3 . If we passively act on the chosen axes in R 3 with O −1 we would get the new orthogonal coordinate axes such that s 2 looks the same way in the new coordinate system as s 1 looks in the old one and vice-versa. Since joint measurability is a notion independent of choice of the axes in R 3 , s 1 and s 2 must exhibit the same joint measurability structure. To see this more rigorously, we show that for each subset s 1 ⊆ s 1 that is (in)compatible, its corresponding subset s 2 ⊆ s 2 is also (in)compatible. For convenience, we assume the POVMs in s 1 are labelled such that those in s 1 are given by {E l } |s 1 | l=1 and, therefore, those in s 2 are given by {F l } |s 2 | l=1 , where |s 1 | = |s 2 |. Suppose s 1 is compatible. Then s 1 has a joint POVM where for each E k ∈ s 1 we have . , x k , . . . , x |s 1 | ).
(15) This yields, more explicitly, Consider a set of operators Since O is an orthogonal transformation, we have so that all of the operators {G s 2 ( x)} x∈O s 2 are positive semidefinite. Now, recalling that |s 1 | = |s 2 | and marginalizing, we have so that G s 2 is a joint POVM for s 2 . Hence, the compatibility of s 1 implies the compatibility of s 2 . Now suppose s 1 is incompatible and its corresponding s 2 is compatible. Then s 2 would have some joint POVM G s 2 . By a similar analysis as above, we would then have that G s 1 = O −1 G s 2 is a joint POVM for s 1 , contradicting the assumption that s 1 is incompatible. Hence, the incompatibility of s 1 implies the incompatibility of s 2 .
We have thus shown that any subset of s 1 is incompatible if and only if its corresponding subset in s 2 is also incompatible. This means that s 1 and s 2 exhibit the same joint measurability structure.
If there exist relabellings of outcomes for each of the N POVMs in either set (s 1 or s 2 ), given by Perm = (Perm 1 , . . . , Perm N ), and an orthogonal transformation then we formally write this as and we have 1. s 1 and s 2 exhibit the same joint measurability structure, 2. if s 1 ⊆ s 1 is compatible with its joint POVM being G s 1 then its corresponding set s 2 = OPerm s 1 ⊆ s 2 admits a joint POVM G s 2 = OPerm G s 1 , where Perm denotes those permutations Perm k ∈ Perm that refer to the respective POVMs E k ∈ s 1 .
Proof. This follows directly from Theorems 3 and 4.
then we say that s 1 and s 2 are geometrically equivalent sets of POVMs. As shown by Theorem 5, two geometrically equivalent sets of POVMs exhibit the same joint measurability structure. If a joint measurement for the compatible set s 1 ⊆ s 1 is G s 1 , then the corresponding joint measurement for its corresponding set s 2 = OPerms 1 ⊆ s 2 is given by G s 2 = OPermG s 1 , where Perm denotes the relabelling of outcomes, if necessary, on the set s 1 .
Proof. This follows from Definition 5 (of geometrically equivalent sets of POVMs), Theorem 5 and Remark 1 following Theorem 3.

Multiple binary qubit POVMs
We first note a necessary condition for the joint measurability of three binary qubit POVMs (that may be biased) obtained by Pal and Ghosh [33]: Theorem 6. A necessary condition for the joint measurability of three binary qubit POVMs -denoted E k (x k ) = where v FT is the Fermat-Toricelli (FT) point of the following four points in R 3 : For the case of unbiased qubit POVMs where n 1 , n 2 , n 3 are mutually orthogonal directions, this condition (Eq. (24)) reduces to the sufficient condition for joint measurability proved earlier in Ref. [32], namely, Note that v FT = (0, 0, 0) in this case. In Ref. [35], the sufficiency of Eq. (24) for the case of any three unbiased qubit POVMs (that need not be mutually orthogonal) was shown. Hence, Eq. (24) is necessary and sufficient for joint measurability of three unbiased binary qubit POVMs. Furthermore, if a set of three unbiased binary qubit POVMs is also coplanar, then the four vectors { v i } 3 i=0 are also coplanar and their FT vector can be explicitly computed and is given by the following recipe: if the four coplanar vectors { v i } 3 i=0 make a convex quadrilateral then their FT point is located at the intersection of the diagonals of that quadrilateral; on the other hand, if one of the points is inside the triangle determined by the convex hull of the other three (i.e., it is a convex combination of the other three points), then the FT point is the point inside the triangle (see Ref [36]). Using this property in Ref. [35] (their Example 1), the necessary and sufficient conditions were derived for this coplanar case and we mention them below.
when a 2 is not a convex combination of a 1 and a 3 , and when a 2 is a convex combination of a 1 and a 3 .
Proof. In Example 1 of Ref [35], it was shown that in the case that a 2 is not a convex combination of a 1 and a 3 then v 2 = a 1 − a 2 + a 3 from Eq (24) lies inside the triangle determined by the convex hull of v 0 , v 1 and v 3 and thus v FT = v 2 . In the case that a 2 is a convex combination of a 1 and a 3 , the vectors { v j } 3 j=0 make a convex quadrilateral and we have that v FT points from the origin to the intersection of the diagonals of that quadrilateral. Substituting these values for v FT into Eq (24), we obtain the joint measurability conditions of Corollary 4.
Notice that in the case where a 2 is a convex combination of a 1 and a 3 (and, thus, E 2 is a convex combination of E 1 and E 3 due to unbiasedness), then the 3-way joint measurability condition reduces to the joint measurability condition for two POVMs E 1 and E 3 , cf. Theorem 2. In other words, as intuitively expected, if two unbiased binary qubit POVMs are compatible, then any convex combination of theirs is also compatible with them.
Corollary 5. Three unbiased binary coplanar POVMs, E k , k ∈ {1, 2, 3}, with the same purity η are jointly measurable if and only if where φ 1 , φ 2 ≤ π/2 are angles between the lines determined by the Bloch vectors of E 1 , E 2 and E 3 with the arrangement shown in Fig. 3.
Proof. This is a special case of the Corollary 4 where the observables have the same purity. Then the norm of Bloch vectors of all three POVMs is equal to η and, hence, none of them can be a convex combination of other two. It is then easy to verify that inequality given in Eq. (24) reduces to Eq. (28). So far, we have a necessary and sufficient condition for compatibility of two binary qubit POVMs (Theorem 1) [31] and a condition that is necessary and sufficient for compatibility of three unbiased binary qubit POVMs (Theorem 6) [33,35]. We will now state one necessary and one sufficient condition for joint measurability of a set of N unbiased binary qubit POVMs with same purity η.

Specker's scenario with unbiased binary qubit POVMs
Among the most important immediate consequences of Corollaries 1, 5 and 6 is the existence of Specker's scenario on a qubit, i.e., the setting with three incompatible qubit POVMs that are pairwise compatible, which was introduced in Example 4. Note that such a joint measurability structure is impossible with projective measurements [8,16]. There are at least two standard ways to construct Specker's scenario using unbiased binary qubit POVMs [16]: , k ∈ {1, 2, 3} be three orthogonal unbiased binary qubit POVMs (see Fig. 4, left), i.e., e 1 . e 2 = e 2 . e 3 = e 3 . e 1 = 0. By Corollary 6 we have that each pair of POVMs is jointly measurable 2 that yields pairwise joint measurability but no triplewise joint measurability.

Example 6. (Specker's scenario with trine spin axes)
Consider three POVMs in an equatorial plane of the Bloch ball equiangularly separated (see Fig. 4, right): . These are the so-called trine spin POVMs.
Corollary 1 combined with Corollary 5 says that each pair of these POVMs is jointly measurable if and only if η ≤ 1 sin π 6 + cos π while all three of them are compatible if and only if which means that Specker's scenario is realized for η ∈ 2 3 , √ 3 − 1 .

Adaptive strategy for constructing joint POVMs
Here we recall the general adaptive strategy outlined for joint measurements in Ref. [10] for the case of unbiased binary qubit POVMs. The form of the qubit POVM is the following: We seek a joint measurement of where n i ∈ R 3 and || n i || = 1 for all i ∈ {1, 2, . . . , N }. We denote the joint POVM as G and its POVM elements Following the recipe in Ref. [10], we want to obtain these POVM elements as the post-processing of a set of projective measurements, {Π j } N j=1 : where {µ j } N j=1 is a probability distribution according to which the PVMs {Π j } N j=1 are implemented, each PVM given by PVM elements Π j (β j ) = 1 2 I + β j n j . σ . By "Π j = β j ", we denote the fact that the effect Π j (β j ) (or the outcome β j ) was registered in a measurement of Π j . Here we have n j ∈ R 3 , || n j || = 1, and β j ∈ {±1} for all j ∈ {1, 2, . . . , N }.
The vectors { n j } N j=1 are chosen such that n i . n j = 0 for all i ∈ [N ] and j ∈ [N ]. The conditional probability distribution p(α 1 , α 2 , . . . , α N |Π j = β j ) is given by where sgn( n i . n j ) = +1, if n i . n j > 0, and −1, if n i . n j < 0. The goal is to choose an appropriate set of parameters for G such that for all i ∈ [N ], so that G is a valid joint POVM for

Planar symmetric POVMs
We now define a family of qubit POVMs that will be of particular interest in this paper: Definition 6 (Planar symmetric POVMs). A set of planar symmetric POVMs is any set of N coplanar and unbiased binary qubit POVMs with the same sharpness whose Bloch vectors define lines that equiangularly dissect the plane, i.e., the angle between successive lines is φ 0 = π N , where N is a positive integer. Note that, by Theorem 3, the joint measurability structure of such a set of POVMs is independent of how we label the outcomes, i.e., the orientation along the line of a particular Bloch vector does not matter. However, unless specified otherwise, we will always consider the case where the POVMs are in an equatorial plane of the Bloch ball and the "+1" outcomes are assigned to vectors in the upper half of the plane and "−1" to the opposite vectors in the lower half, cf. Fig. 5. With this convention, these POVMs are: Note that trine spin POVMs are just the special case when N = 3. The following theorem, proven in Ref. [10], is of important for many constructions in this paper: Theorem 8. N planar symmetric POVMs are jointly measurable if and only if their sharpness η satisfies The effects of a joint POVM that reproduces the whole range η ∈ 0, are given by Here, outcome x is associated uniquely to the unit vector e( x) by the relation and the null (or zero) effect is assigned to any outcome x not obtainable as x = sgn n 1 · e( x) , . . . , sgn n N · e( x) .

Remark 2.
Note that the set { e( x)} x : • coincides with the set {± n k } for odd N ; • contains vectors that bisect the angles between the successive {± n k } for even N (see Fig. 6) Remark 3. Note that all of the outcomes corresponding to non-zero effects of the joint POVM correspond to strings of length N arranged in a cycle of consecutive '+1's concatenated with consecutive '−1's ( Fig. 7). In other words, the outcome strings associated with nonzero effects are of the form

Geometric equivalence of planar symmetric POVMs
We discussed the geometric equivalence of general sets of qubit POVMs in Section II B 3. One of the results presented there is Corollary 3 which states that two sets of binary qubit POVMs are geometrically equivalent (and therefore exhibit the same joint measurability structure) if and only if the lines determined by their Bloch vectors are related by some orthogonal transformation O ∈ O(3). In this section we explore the consequences of this result for the joint measurability properties of planar symmetric POVMs. Considering their Bloch lines, we see that certain subsets are certainly geometrically equivalent. For example, if we act with an anti-clockwise rotation of π N on the set {E 1 , E 2 }, we obtain the set {E 2 , E 3 } (see Fig. 8), and these two subsets are thus geometrically equivalent. A general question then arises: what is the symmetry group of the Bloch lines of planar symmetric POVMs? Since they are coplanar, we expect this to be a discrete group that is a subgroup of linear isometries of the plane, namely, O(2). 6 To see what the symmetry group is, consider a regular 2N -gon overlaid on the Bloch lines as in Fig. 9. This makes it clear that the group we are searching for is the group of symmetries of the diagonals of this 2N -gon. If the opposite ends of the 2N -gon are labelled differently, this group is just the dihedral group D 2N 7 of order 4N (see Refs. [37] and [38]). However, our Bloch lines do not have a sense of orientation since the orientation of the particular Bloch vector along them does not matter. Therefore the actual symmetry group is the symmetry group of a regular 2N -gon with opposite vertices identified which we will denote by S N . This group is a quotient group of D 2N and the group H 2N of symmetries of the regular 2N -gon that sends every vertex either to FIG. 6. Green arrows represent the set { e( x)} x for N = 6, a representative example for even N . The first 'green arrow' above the (positive) x axis is assigned to the outcome x = (+1, +1, +1, +1, −1, −1). Going counter-clockwise, the next one is assigned to x = (+1, +1, +1, +1, +1, −1) and so on. For odd N , the correspondence between e( x) and x works in a similar way.

FIG. 7.
For example, for N = 6 we have a cycle of 6 pluses concatenated to 6 minuses. In this figure, the red loop identifies an outcome that has a non-zero effect associated with it. Reading counter-clockwise, this outcome is x = (+1, +1, +1, +1, −1, −1). All the other outcomes associated to non-zero effects are obtained in a similar way, i.e., by identifying, counter-clockwise, a string of six ±1 in the figure.
itself or to its opposite vertex i.e., We now recall of some basic definitions from group theory, namely those of a normal subgroup and semidirect

product.
Definition 7 (Normal Subgroup). Let G be a group with the identity element id and N be its subgroup. We say that N is a normal or invariant subgroup of G, which is denoted by N G, if for each g ∈ G and for each n ∈ N the product gng −1 (which is called the conjugation of n by g) belongs to N .
Theorem 9 (Equivalent definitions of Semidirect product). Let G be a group with the identity element id, H its subgroup and N its normal subgroup. The following statements are equivalent: 1. for each g ∈ G, there exist n ∈ N and h ∈ H such that g = nh, where N ∩ H = {id}; 2. for each g ∈ G there exist unique n ∈ N and h ∈ H such that g = nh; 3. for each g ∈ G there exist unique n ∈ N and h ∈ H such that g = hn; 4. G is a semidirect product of N and H denoted by For more properties of the semidirect product see Ref. [39], specifically Definition 7.14 and Lemma 7.15.
where C 2 is the second order cyclic group generated by the positive rotation by π, denoted C 2 8 , and σ x is the second order cyclic group generated by the reflection on the x axis, defined by two vertices of the polygon, here denoted by σ x ; 2. for N > 2 it holds Proof. The symmetry transformation that preserves three vertices or two non-opposite vertices i.e. that sends each of them to itself, must preserve all vertices so it is just the identity transformation, id (cf. Theorem 2.2 in Ref. [40]) 9 . The symmetry transformation that preserves two opposite vertices (i.e. sends each of them to itself) is a reflection about the diagonal connecting them. It is easy to see that this transformation satisfies the properties required by H 2N (namely, that it sends every vertex either to itself or to its opposite vertex) only for N = 2 (see Fig. 10). The symmetry transformation that preserves no vertices and satisfies the requirements of H 2N is the one that sends each vertex to its opposite one i.e. the central symmetry on the origin, and in the planar case this is the same as the positive (or counter-clockewise) rotation by π, namely the operation C 2 . So, only in the case N = 2 we have that (42) and in the case N > 2 we have Theorem 10. Group S N has the following properties Proof. Remember that the dihedral group D N is the semidirect product of C N and σ x (we always take x axis to be defined by two vertices of the polygon). Therefore, for N = 2, For N > 2 we have Here we have used the following facts: that each cyclic subgroup is always a normal subgroup; that two cyclic groups of the same order are isomorphic to each other; that a cyclic group of the order n generated by r has cyclic subgroups of the order p generated by r n/p if p divides n and we applied the second and the third isomorphism Theorem (cf. Theorems 4.22 and 4.23 in Ref. [38]). However, if we identify each pair of opposite vertices, we have that the reflections about orthogonal lines give the same result and therefore they are identified. Also, the rotation C p 2N (counter-clockwise, by an angle pπ/N ), and C N +p 2N (counter-clockwise, by an angle π + pπ/N ) produce the same result so they have to be identified as well. Hence, the group S N consists of the following rotations: C 2N , C 2 2N , . . . , C N 2N = id 10 and the reflections in D 2N , where reflections about any pair of orthogonal lines are identified. This group is of order 2N (since it is isomorphic to D N ).
Corollary 7. Let s be a set of N planar symmetric POVMs. For any two subsets of s, say s 1 and s 2 , that are related by a transformation O ∈ S N (i.e., s 2 = OPerms 1 ), where and σ φ is a reflection about the line making the angle of φ with the positive x-axis (cf. Fig 11), we have that s 1 and s 2 are geometrically equivalent and hence exhibit the same joint measurability structure. If a joint POVM for s 1 is G (s1) then the corresponding joint POVM for s 2 is given by G (s2) = OPermG (s1) , where Perm denotes the relabelling of outcomes on s 1 that may be necessary for the orientations of the Bloch vectors of s 1 , after being acted upon by O, to match those of s 2 .
Proof. This follows from the fact that the discrete group S N is the symmetry group of planar symmetric POVMs and Corollary 3.
where σ denotes the reflections on the lines shown in Fig.  11 FIG. 11. Bloch lines of N = 4 planar symmetric POVMs with the reflection symmetry lines denoted explicitly. 11 Notice that there is still some ambiguity in this notation. For example, take N = 6. The sets {E 1 , E 2 , E 5 } and {E 1 , E 3 , E 4 } are geometrically equivalent, so we may denote their equivalence class by either of them. Given some joint POVM with the effects E (s1) ( x) for some set s 1 of N POVMs, we can obtain a joint POVM for any subset s 1 (with M < N POVMs, say) of s 1 by marginalizing the effects E (s1) ( x) over the outcomes of POVMs from s 1 \s 1 . In what follows, we will introduce a technique that tweaks the marginalized joint POVM on s 1 and uses that to obtain an incompatible set s 2 (of same cardinality as s 1 ) and its compatible subset s 2 (of same cardinality as s 1 ). Thus, using this technique, we can go from a set of N compatible POVMs to a set of N POVMs that are incompatible but such that a subset of M of them is still compatible.
As an example, consider a set s 1 of unbiased qubit POVMs with the same purity η that are compatible if and only if η ∈ (0, η max ] and let {E (s1) ( x)} x∈{±1} N be a joint POVM for them, also valid for any η ∈ (0, η max ]. Then we choose the set s 1 ⊂ s 1 of M < N POVMs and marginalize to obtain {E (s 1 ) ( x )} x ∈{±1} M , a joint POVM for s 1 . This joint POVM for s 1 is thus valid for η ∈ (0, η max ]. Now we tweak the effects E (s 1 ) ( x ) to get new effects E (s 2 ) ( x ) which still constitute a joint measurement on s 1 but this new M -wise joint POVM allows for a broader range of purity η ∈ (0, η MAX ], where η MAX > η max , and consequently allows us to obtain a set s 2 (similar to set s 1 but with η / ∈ (0, η max ]) of M compatible POVMs (by marginalization) with η ∈ (η max , η MAX ]. It thus also allows us to obtain an incompatible set s 2 (of N POVMs) similar to the compatible set s 1 but with η ∈ (η max , η MAX ] and containing the compatible subset s 2 . Thus, by choosing η ∈ (η max , η MAX ] we can realize a joint measurability structure where the N POVMs in s 2 are incompatible but the M POVMs in its subset s 2 are compatible. This technique can be applied to obtain many new joint measurability structures from a set of compatible POVMs, as we will show below. Since the technique proceeds by tweaking marginal joint POVMs obtained by marginalizing a bigger joint POVM, we term this technique marginal surgery. We now proceed to illustrate how this method works in practice by obtaining Ncycle and N -Specker compatibility scenarios using qubit POVMs. As a consequence of this, we will also be able to construct arbitrary joint measurability structures using these qubit POVMs, improving upon the (rather weak) dimension bounds for realizing arbitrary joint measurability structures obtained in Ref. [8].
A. Marginal surgery on any pair of compatible POVMs: constructing N -cycle scenarios on a qubit We will now apply marginal surgery on the joint POVM of any pair of POVMs out of N planar symmetric compatible POVMs, {E 1 , E 2 , . . . , E N }, and use this to construct N -cycle scenarios on a qubit.
Consider two of the POVMs, E 1 and E k , where 1 < k ≤ N . We will also denote p ≡ k −1, so that the angular separation between the lines defined by the considered POVMs is φ 1,k ≡ (k−1)π N = pπ N . Now consider the joint POVM effects G( x), x ∈ {±1} N , for all N POVMs from Eq. (38). To get the pairwise joint measurement E (1,k) we marginalize G over all of the outcomes of POVMs E i , i / ∈ {1, k}: where we have chosen to indicate the outcomes that are held fixed (while all the others are summed over) on top of the summation sign.
We will now exploit the fact that a pairwise joint POVM here is completely determined by knowing just one effect, say E (1,k) (+1, +1), with the others given by so we only have to focus on finding E (1,k) (+1, +1): Paying attention to Remark 3, consider the outcomes x = (x 1 , . . . , x N ) to which non-zero effects of the joint POVM {G( x)} x∈{±1} N are assigned. Of these, those which have +1 at the first index and the k-th index (x 1 = x k = +1) must be of the type and there are exactly N − k + 1 = N − p of them. This lets us write for odd N , while for even N which arise from the way of assigning effects to outcomes described in Theorem 8. In both cases, we get (see Appendix A, Lemma 5) This enables us to write which in combination with results of Appendix A, Corollary 8 of Lemma 6, Eq. (A20) gives where is a unit vector. This completely specifies E (1,k) (+1, +1) which combined with Eq. (50) yields (recalling that p = k − 1) where t is found to be This motivates the following ansatz for the general form of the joint POVM of E 1 and E k : Requiring correct marginalization for, say, E 1 (+1) = G (1,k) (+1, +1) + G (1,k) (+1, −1), gives Requiring positivity of G (1,k) we have that Eliminating everything but η from the linear system consisting of Eqs. (66) and (67) we get This means that for every η ∈ 0, 1 sin pπ 2N + cos pπ

2N
there are suitable α, β and µ such that G (1,k) is a valid joint POVM for E 1 (x 1 ) = 1 2 (I ± x 1 η n 1 · σ) and E k (x k ) = 1 2 (I ± x k η n k · σ). We propose one particular choice that is consistent with Eqs. (66) and (67): which gives Note that this construction is such that when η saturates the upper bound of Eq. (68), G (1,k) is a rank one POVM. From Corollary 1, we see that the sufficient condition proven by this construction for the joint measurability of E 1 and E k is also necessary, so the joint POVM G (1,k) that we have constructed is optimal. That is, it applies to the whole range of η that is necessary and sufficient for joint measurability of E 1 and E k . Vectors s and t are situated so that s is the unit vector bisecting the angle between the lines determined by n 1 and n k , while the vector t is the unit vector perpendicular to s that makes an acute angle with n 1 (see Fig. 12). For convenience we have chosen the set {E 1 , E k }. Acting on this set by the symmetry operation C k1−1 2N , i.e., by the positive rotation through (k1−1)π N , we can obtain every other two element set, {E k1 , E k1+p mod N }, k 1 > 1, that has the angular separation of pπ N between the Bloch lines of its POVMs. By Corollary 7, all of these sets are geometrically equivalent to {E 1 , E k } and, thus, compatible if and only if the set {E 1 , E k } is compatible. Therefore, the condition of Eq. (68) is sufficient (and necessary by Corollary 1) for joint measurability of every pair E k1 and E k2 , where k 2 = k 1 + p mod N . According to Corollary 7, the joint POVM for this pair is given by G (k1,k2) = C k1−1 2N G (1,k) . In the particular case analysed in this section this reduces to rotating the vectors s and t through (k1−1)π N in the counter-clockwise direction to preserve the geometrical situation in Fig. 12. Using Eqs. (70) we immediately This gives us the necessary and sufficient condition for all the compatibility relations required in an N -cycle scenario, i.e., every POVM is compatible with its immediate neighbours if and only if the purity η satisfies Eq. (73). What remains is to obtain the necessary and sufficient condition for the incompatibility of all the remaining pairs of POVMs so that the set of N POVMs realizes an N -cycle scenario.
and E l are incompatible if and only if η > 1 sin x 2 +cos x 2 . Since we need this to hold for all pairs {E j+1 , E l } such that x ≥ 2π N , the necessary and sufficient condition for obtaining all the incompatibility relations required in an N -cycle scenario is which follows from properties of function f (x) that we discussed and which are also illustrated in Fig. 13.

B. Marginal surgery on arbitrary M-tuples of POVMs: N-Specker scenarios on a qubit
In this section we will extend the marginal surgery described for pairs of POVMs to an arbitrary M -element subset of the set of N planar symmetric POVMs, s ≡ {E 1 , E 2 , . . . , E N } (cf. Definition 6). Let us select s ≡ {E k1 , E k2 , E k3 , . . . , E k M }, where we assign the labels {k i } M i=1 such that k 1 = 1 < k 2 < k 3 < · · · < k M and denote by E (s ) the joint measurement obtained by marginalization of G: where (x k1 , x k2 . . . , x k M ) ∈ {±1} M are held fixed and the summation is carried out over the remaining entries of x ∈ {±1} N .
Now the summation becomes and where we used the fact that there are k p+1 − k p non-zero effects assigned to outcomes that start with +1 in the first k p slots and end with −1 starting from the k p+1 -th slot (and the same is true when the first k p slots are −1 and from k p+1 they are all +1), and also that e(− x) = − e( x). There are exactly M non-zero effects of E (s ) assigned to outcomes that start with at least one +1, i.e., outcomes labelled by M -element strings in  (79) show that we can focus on specifying those effects that start with at least one +1 and then those that start with at least one −1 can be found just by inverting the sign of the corresponding geometric parts (i.e., the σ-dependent parts of the POVM elements). In order to obtain the effects of E (s ) , we need to evaluate the sum and we will do so by converting it into the sum over some index i as we did in the previous section for the case of pairs of POVMs (see Appendix A, Lemma 7 for how the limits on i in the summation are obtained): and x1,..., where we have assumed that p < M . Using results from Appendix A, Lemma 6, we get in both cases The case p = M corresponds to the outcome (x 1 , x k2 , . . . , x k M ) = (±1, ±1, . . . , ±1) (all signs are the same), so that we have: Now we have a complete specification of the effects of the joint POVM E (s ) : and Note that t(k p , k p+1 ) is the unit vector perpendicular to the direction that bisects the angle between n kp and n kp+1 and oriented such that it makes an acute angle with n kp , while s(k M ) is a unit vector that bisects the angle between n k M and n 1 . 13 The POVM E (s ) is valid on s for η ∈ (0, 1/(N sin(π/2N ))] since it is a marginal of the joint POVM G. Now let us apply marginal surgery on E (s ) to obtain a joint POVM for s that is valid for a broader range of the purity parameter η. Motivated by the form of E (s ) we consider the following ansatz for the joint POVM of s : Requiring the correct marginalization for E kr where r ∈ 13 This follows from the following facts: unit vector that bisects the angle between (cos x, sin x) and (cos y, sin y) is (cos x+y 2 , sin x+y 2 ). A vector perpendicular to (a, b) is (b, −a) so we have that the vector perpendicular to the bisecting one is (sin x+y 2 , − cos x+y 2 ). If x < y, it makes an acute angle with the vector (cos x, sin x) because (sin x+y 2 , − cos x+y 2 ) · (cos x, sin x) = sin y−x 2 > 0.
{1, 2, . . . , M } we get which together with Eq. (89) yields Requiring the positivity of each effect of G (s ) , according to Eq. (89) we have the following constraints: Eliminating everything except η in Eqs. (91) and (92) we obtain an upper bound on η for which we can find suitable α(k p , k p+1 ) and β(k M ) such that G (s ) is a valid joint measurement on s for values of η constrained by Geometrically, (k p+1 − k p )π/N and (k M − 1)π/N are respectively the angles between n kp+1 and n kp , and n k M and n 1 , and we see that the upper bound on η is an explicit function of half of those angles. In case of M = 2, i.e., when we choose only two POVMs, Eq. (93) reduces to Eq. (68) and in this case the condition of Eq. (93) is also necessary (cf. Corollary 1). It is readily seen from Corollary 5 that this condition is necessary for M = 3 as well. For M = N , i.e., when we consider all N POVMs, Eq. (93) reduces to η ≤ 1/(N sin π/2N ) which is also known to be necessary from Theorem 8. So for M ∈ {2, 3, N } the given condition is both necessary and sufficient for joint measurability. However, for every other M , while it is certainly sufficient, we cannot say if it is necessary for joint measurability. We conjecture its necessity (see Conjecture 1). 14 It remains to make a choice for α(k p , k p+1 ) and β(k M ) for each η. We do this in the following manner, subject to constraints from Eqs. (91) and (92): which gives for G (s ) : When η takes the value of its upper bound, G (s ) forms a rank-1 POVM. Note that so far we have taken E k1 = E 1 for calculational convenience. However, our procedure can be applied to any arbitrary choice of M out of N planar symmetric POVMs. Namely if we take s = {E k1 , E k2 , . . . , E k M } such that 1 < k 1 < k 2 < · · · < k M , then the set s = {E 1 , E k2−k1+1 , . . . , E k M −k1+1 } is geometrically equivalent to the set s because it is related to s by a counter-clockwise rotation of (k1−1)π N in an equatorial plane (XY plane) of the Bloch ball and thus it is jointly measurable if and only if s is jointly measurable (cf. Corollary 7) and the joint POVM on s is given by This together with Eq. (95) yields: All of this serves as a proof for the following sufficient condition for joint measurability: . . , N }} be the set of planar symmetric POVMs. A sufficient condition for its subset s = {E k1 , E k2 , . . . , E k M } to be jointly measurable is given by . (99) As we already noted, we conjecture this condition is necessary as well, i.e., (100) We will split the proof into two cases: Case 1: k 1 , . . . , k N −1 are successive integers which geometrically corresponds to n k1 , . . . , n k N −1 being as closely grouped as possible, i.e., the angular separation between the successive vectors is π/N and the total angular span is ( which completes the proof. the joint measurability structure of s forms an N -Specker scenario (cf. Example 4).
Proof. We have that s is compatible if and only if leaving an open gap for which s is incompatible but every (N − 1)-element subset of s is compatible, thus realizing an N -Specker scenario.
Theorem 12 shows that we can realize any N -Specker scenario on a qubit with planar symmetric POVMs. Ref. [8] provided a constructive proof that all conceivable joint measurability structures for any finite set of measurements are realizable with POVMs in quantum theory. Crucial to the construction in Ref. [8] are the socalled minimal incompatible sets of POVMs (of which, N -Specker scenarios are examples) 15 constructed for subhypergraphs in the hypergraph representing the joint measurability structure to be realized. The construction in 15 A minimal incompatible set of vertices in a joint measurability structure is such that every proper subset of this set is compatible, i.e., shares a hyperedge. N -Specker (N ≥ 3) scenarios constitute all examples of minimal incompatible sets, except one: a pair of incompatible measurements. Such a pair forms a minimal incompatible set since each POVM in the pair is trivially compatible with itself but the two POVMs are incompatible. One could even call it a "2-Specker" scenario.
Ref. [8] proceeds by decomposing a joint measurability structure into minimal incompatible sets. Each minimal incompatible set is then realized on some Hilbert space H i and the dimensionality of the overall Hilbert space, H = ⊕ i H i , required to realize a joint measurability structure based on this recipe is dim H = i dim H i . Our construction of all N -Specker scenarios on a qubit here can therefore be used to make the construction of all joint measurability structures in Ref. [8] maximally efficient given the recipe adopted there, i.e., we have dim H i = 2 for all minimal incompatible sets indexed by i.

IV. SUFFICIENT CONDITION FOR JOINT MEASURABILITY OF BINARY QUBIT POVMs FROM MARGINAL SURGERY
In the previous sections, we started with the setting of planar symmetric POVMs (cf. Definition 6 and Figure 5) and then applied marginal surgery to obtain joint POVMs for arbitrary M -tuples of them. It turns out that geometric insights from Eq. (95), which is the result of the aforementioned procedure, can be used to construct the joint POVM for an arbitrary set of N coplanar unbiased binary qubit POVMs with the same purity η bounded above by some η max that depends on the particular setting.
Theorem 13. Let s = {E 1 , E 2 , . . . , E N } be a set of coplanar unbiased binary qubit POVMs with the same purity defined by and let α k , k ∈ {1, 2, . . . , N − 1}, be the angles defined by the rays in the upper half plane of the lines determined by n k (where the X-axis is chosen to be along n 1 )(see Fig.  14). Then a sufficient condition for the compatibility of s is and a joint POVM satisfying this constraint is where s = cos setting all other effects of G s to 0.
FIG. 14. Lines defined by n k Proof. We will prove this by construction of a joint POVM for s using basic geometric insights gained in the previous subsection. Without loss of generality, we can choose that all of the Bloch vectors are in the upper half plane. With this convention, α k−1 is the angle between n k and the x-axis i.e., n k = (cos α k−1 , sin α k−1 , 0).
• t(k p , k p+1 ) is the unit vector that is orthogonal to the line that bisects the angle between n kp and n kp+1 and makes an acute angle with respect to n kp ; • s(k M ) is a unit vector that bisects the angle between n 1 and n k M and is oriented towards the upper half plane.
This suggests searching for a joint POVM for s of the following form (cf. Eq. (94)): setting all other effects of G s to 0. The positivity constraint on the effects implies that each G s ( x) is a valid effect for (cf. Eq. (93)) Now we have to show that the marginalization is correct: The geometric part becomes: where e z = (0, 0, 1) and we have used the fact that which can easily be derived by linear decomposition ( n p+1 + n p )× e z = a n p +b n p+1 , and finding a and b. There are three distinct cases: k = 1, k = N and 1 < k < N , but in all of them by trivial algebraic manipulations of Eq. (111) we obtain and this in combination with Eq. (110) yields This means that G s marginalizes to the POVMs in s correctly and the proof is complete.
The sufficient condition given by this theorem is known to be necessary as well for N = 2 and N = 3 (cf. Corollary 1 and Corollary 5), as well as for arbitrary N planar symmetric POVMs (cf. Theorem 8). Therefore, we make the following conjecture: Conjecture 2. The sufficient condition for joint measurability of coplanar unbiased binary qubit POVMs with same purity given by Theorem 13 is also necessary.
The truth of this conjecture would imply the truth of Conjecture 1.

FIG. 15. An example with 5 POVMs
Notice that we can rewrite: which suggest further generalizations of the result given by Theorem 13. We pursue this suggestion below for the case of N , possibly biased, binary qubit POVMs.
Theorem 14. Let s = {E 1 , . . . , E N } be the set of some binary qubit POVMs (possibly biased) specified with The set s is compatible if Adopting the convention that the outcomes of E k are labelled such that positive bias is associated with the outcome '+1', i.e. all b k ≥ 0, and that the POVMs themselves are labelled by non-decreasing bias (b k ≥ b p for k > p), a joint POVM that satisfies the given joint mea-surability condition on s is given by where with all other effects of G s set to 0.
Proof. We will prove this by construction. Consider the following candidate for a joint POVM of s: (120) As for the marginalization, it is easy to see that the geometric part of the joint POVM marginalizes correctly: Requiring the correct marginalization of the remaining part of G s , we get From Eqs. (120) and (122) we get Substituting the expressions for S and T p we get If we label {E p } N p=1 such that b p increases with p and further label outcomes such that b p is positive, we can choose Hence, we have constructed a valid joint POVM for the set s.
The sufficient condition derived here is known to be necessary in the case of N = 2 unbiased POVMs (cf. Theorem 2) and in the case of N = 3 coplanar unbiased POVMs if a 2 is not a convex combination of a 1 and a 3 (cf. Corollary 4). Motivated by this, we make the following conjecture: Conjecture 3. The sufficient condition given in Theorem 14 is also necessary in the case of unbiased coplanar POVMs where 1) the Bloch vectors a k , in order from k = 1 to k = N , lie on the lines that make the total angular span less than π (as in Fig. 14), 2) the outcomes are labelled such that they all point towards the upper half of the plane, and 3) none of the Bloch vectors is a convex combination of any other pair of Bloch vectors.
The truth of Conjecture 3 would imply the truth of Conjecture 2 and, hence, the truth of Conjecture 1.

V. MISCELLANEOUS JOINT MEASURABILITY STRUCTURES ON A QUBIT
In this section we give explicit construction of the joint measurability structures realizable on a qubit, many of which are not realizable with PVMs on any quantum system, using previous results.

A. Progression from N-Cycle to N-Specker scenarios
We start with planar symmetric POVMs (cf. Definition 6): We will examine how the joint measurability structure changes when we change η for sets of planar symmetric POVMs, up to N = 6. All sets geometrically equivalent (cf. , √ 2 2 , we have the joint measurability structure where every pair of POVMs is jointly measurable (but no larger sets are), illustrated in Figure 17. The joint measurability structure is a complete graph with four vertices.  This joint measurability structure therefore corresponds to a complete graph with five vertices (Figure 20).
we have the joint measurability structure given by  Example 14. 5-Specker scenario is realized for η ∈ 1 5 , according to Theorem 12. Remember that this is only a sufficient condition -the upper bound is only known to be sufficient for 4-way compatibility (cf. Lemma 2 and Conjecture 1), even though the lower bound is necessary and sufficient for 5-way incompatibility (Theorem 8).   We could continue enumerating examples for N = 7, 8, . . .. By continuously decreasing η (i.e., by making the measurements more noisy) from 1 to 0, we go from N incompatible POVMs to N compatible ones, passing through various joint measurability structures in between. In a sense, we get more and more compatibility as we add noise. For N = 4, we know the exact compatibility structure for each η ∈ [0, 1], which we can verify passing through examples for 4 POVMs in the next subsection. This follows from the fact that Conjecture 1 holds for N = 4. However, passing through the examples for N = 5, we see that we cannot infer the exact compatibility relation for 4 Proof. We provide an example of a joint measurability structure that does not admit such a realization. The hypergraph for this example is shown in Fig. 29). This hypergraph is the same as the one in Fig. 28 labelled by number 6. Suppose that this hypergraph can be realized with binary, unbiased, planar qubit POVMs with the same purity denoted by {E 1 , E 2 , E 3 , E 4 }. Without loss of generality, we can take the directions of their Bloch vectors as in Fig. 30, i.e., that they are of the form In this case we have that According to Corollary 1, the prescribed pairwise joint measurability relations require that following inequalities hold: as well as Using first inequalities from both Eqs. (170) and (171), we get Rearranging terms in this new inequality we find Applying sum to product rules this becomes Inequality (174) branches into two cases: which reduces to β >π for 2α > β, or (177) β <π for 2α < β.
We take only the possibility that is consistent with the constraint from Eq. (169) (in particular, β < π). Hence, we consider the case 2α < β. In the similar succession of steps combining the second relation in Eq. (171) with the first one in Eq. (170) as well as by combining the third relation from Eq. (171) with the second one from Eq. (170) we obtain 2α < γ and 2β < γ. In total, we obtain that the following constraints must hold: Now if we combine the second relation from Eq. (171) with the third one from Eq. (170) we will get Rearranging this becomes: Applying sum to product rules we get Using Eq. 169 i.e. that α ∈ (0, π) we can cancel the factor of sin α 4 knowing that it is positive so we further obtain Combining the first relation from Eq. 171 with the second one from Eq. 170 and as well as the third relations from both of them, we obtain in a similar succession of steps 2β − α > π and 2γ − β > π. In total this yields Now using the second relations in both Eqs. (179) and (184) we will obtain which is in contradiction with the constraint given by Eq. (169). This means that the system of constraints listed in Eqs. (169), (170) and (171) is inconsistent. Hence, the proposed joint measurability structure ( Fig. 29) cannot be realized with unbiased binary coplanar qubit POVMs with the same purity.
However, the hypergraph in Fig. 29 can be realized with binary qubit POVMs if we relax some of the constraints. We provide two examples with two distinct ways of achieving this: in one of them we give up on the same purity for the POVMs and in the other one we allow one of them to not be coplanar with the others.
Example 20 (Giving up on the same purity). In this example we stay in an equatorial plane of the Bloch ball but we let the POVMs differ in their purity. Here it is enough to choose three POVMs, say E 1 , E 2 and E 3 to have same purity η such that the sets {E 1 , E 2 } and {E 1 , E 3 } are compatible but that the set {E 2 , E 3 } is incompatible, and to assign some other purity µ to E 4 . The strategy is then to arrange the Bloch vectors of E 1 , E 2 and E 3 (using Corollary 1) such that (in)compatibility relations prescribed in Fig. 29 hold. We can then choose E 4 to be a measurement in the same direction as E 1 such that it's a projective measurement (setting µ = 1). This choice ensures that E 4 is incompatible with E 2 and E 3 but compatible with E 1 , which is what we wanted. One of many possible choices is  Example 21 (Allowing non-coplanar POVMs). Here we keep the same purity η of four POVMs {E 1 , E 2 , E 3 , E 4 } but allow for non-coplanar directions. The strategy is to put the Bloch vectors of E 2 and E 3 symmetrically in the equatorial XY plane with respect to the Bloch vector of E 1 such that they make an angle α < π 4 with it. Then we put the Bloch vector of E 4 in XZ plane such that it makes the same angle α with the direction of the Bloch vector of E 1 (see Fig. 32). The Bloch vector of E 4 then makes an angle φ with the Bloch vectors of E 2 and E 3 which satisfies α < φ < 2α. 16 Therefore, the range for φ is a subset of 0, π 2 . Now we search for η such that the prescribed joint measurability relations in Fig. 29 are satisfied. According to Corollary 1, we must have to have pairwise compatibility of E 1 with each of E 2 , E 3 and E 4 , while at the same time we must demand to exclude other possible pairwise compatibility relations in {E 1 , E 2 , E 3 , E 4 }. Because φ < 2α < π 2 , and from the property that the function is decreasing for x ∈ (0, π/2] (see the graph of this function in Fig 13), Eq. (192) is automatically satisfied when Eq. (191) is satisfied. Also from this decreasing property and the fact that φ > α we find that Eqs. (190) and (191) leave an open gap for the purity, namely, such that the joint measurability structure from Fig. 29 is realized.

VI. CONCLUSIONS AND OPEN QUESTIONS
In the preceding sections, we have done a fairly exhaustive study of joint measurability structures realizable on a qubit. One of our key motivations was to answer the following question: Is it possible to realize every conceivable joint measurability structure with qubit POVMs? While we haven't settled this question, the realizability of countably infinite sets of joint measurability structures (e.g., N -Specker and N -cycle scenarios for all N ) makes us conjecture that this is the case: Conjecture 4. All joint measurability structures are realizable with qubit POVMs. cos φ = cos 2 α ensures that cos φ is positive i.e. φ < π 2 . Now using the trigonometric identity cos 2 α = 1+cos 2α 2 we find the following cos φ = 1 + cos 2α 2 ⇒ cos 2α = 2 cos φ − 1.
2. Conjecture 4, namely, the realizability of arbitrary joint measurability structures with qubit POVMs. Potential counter-examples to this conjecture could include, for example, a joint measurability structure with N vertices for some N > 4 such that one vertex is compatible with each of the remaining N − 1 vertices but all the other pairs of vertices are incompatible. This would be a generalization of the joint measurability structure in Fig. 29. Another candidate counter-example is an N -complete joint measurability structure for some N > 5.
3. The study of noise-robust contextuality for joint measurability structures not covered by previous work and implications of this for the project of characterizing the set of quantum correlations from principles [17,19,20,26]. In particular, given a joint measurability structure that is only realizable with non-projective POVMs in quantum theory, is there a meaningful distinction to be made between quantum and post-quantum theories? How should this distinction be made precise? An early hint that this is indeed the case was provided by the work of Ref. [17] and followed up in Ref. [20].
More broadly, the study of incompatibility as a resource for quantum information protocols has been a very active area of research in recent years [12,41,42]. We believe that studying the role that realizability of particular joint measurability structures can play in quantum information is an area of research that hasn't seen a lot of work and could potentially lead to insights on previously under-appreciated aspects of incompatibility that are relevant for fundamental as well as applied reasons. The satisfaction of a particular joint measurability structure by a set of POVMs is usually the first pre-requisite that determines their potential relevance, for example, in generating correlations that can demonstrate nonclassicality, whether in the case of Bell inequality violations [6,43] or of contextuality [16][17][18]. We hope that the results of this paper and the questions they motivate will go a long way towards elucidating the role of joint measurability structures in quantum theory and beyond.

Acknowledgment
We are thankful to Tomáš Gonda who suggested that we find the symmetry point group of planar symmetric POVMs. This research was supported in part by the Perimeter Institute for Theoretical Physics, where N.A. was an undergraduate summer research student when most of this work was carried out. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. R.K. is supported by the Program of Concerted Research Actions (ARC) of the Université libre de Bruxelles. , sin (2i−1)π 2N , 0 for even N. (A1) Proof. We have to sum over all e( x) such that x "starts" with +1 in the first k slots. The boundaries for the summation index i are determined from the conditions n 1 · cos (i − 1)π N , sin (i − 1)π N , 0 > 0, n k · cos (i − 1)π N , sin (i − 1)π N , 0 > 0, for odd N ; (A2) n 1 · cos (2i − 1)π 2N , sin (2i − 1)π 2N , 0 > 0, n k · cos (2i − 1)π 2N , sin (2i − 1)π 2N , 0 > 0, for even N.
(A21) Lemma 7. We derive the bounds for index i.