Andreas Bluhm

Ion Nechita
email: nechita@irsamc.ups-tlse.fr

Joint measurability of quantum effects and the matrix diamond

In this work, we investigate the joint measurability of quantum effects and connect it to the study of free spectrahedra. Free spectrahedra typically arise as matricial relaxations of linear matrix inequalities. An example of a free spectrahedron is the matrix diamond, which is a matricial relaxation of the 1-ball. We find that joint measurability of binary POVMs is equivalent to the inclusion of the matrix diamond into the free spectrahedron defined by the effects under study. This connection allows us to use results about inclusion constants from free spectrahedra to quantify the degree of incompatibility of quantum measurements. In particular, we completely characterize the case in which the dimension is exponential in the number of measurements. Conversely, we use techniques from quantum information theory to obtain new results on spectrahedral inclusion for the matrix diamond. Contents 1. Introduction 1 2. Main results 2 3. Concepts from Quantum Information Theory 4 4. Free spectrahedra 8 5. Spectrahedral inclusion and joint measurability 10 6. Lower bounds from cloning 13 7. Lower bounds from free spectrahedra 16 7.1. Dimension dependent and symmetric lower bounds 16 7.2. Dimension independent lower bounds 18 8. Upper bounds 19 8.1. Zhu's necessary condition for joint measurability 19 8.2. Pairwise anti-commuting unitary operators and spectrahedral inclusion constants 21 9. Discussion 23 9.1. The shape of the different compatibility regions 23 9.2. The shape of the inclusion sets 26 9.3. Outlook: POVMs with more outcomes 26

Introduction

One of the defining properties of quantum mechanics is the existence of incompatible observables, i.e. measurements that cannot be performed simultaneously [Hei27, Boh28]. A classic example of this behavior are the observables of position and momentum. One of the central notions to capture this property of quantum mechanics is joint measurability. Observables are jointly measurable if they arise as marginals from a common observable. This has practical implications for quantum information tasks [BCP + 14], as only incompatible observables can violate Bell inequalities [Fin82].

It is well-known that incompatible observables can be made compatible by adding a sufficient amount of noise [BHSS13]. Although many works study compatibility questions for concrete observables (see [HMZ16] for a topical review), there has also been an interest in how much incompatibility there is in quantum mechanics and other generalized probabilistic theories [BHSS13, Gud13]. In the present work, we continue this line of research by studying the degree of incompatibility in quantum mechanics in more detail. We will be interested in the compatibility regions for a fixed number of binary measurements in fixed dimension and for different types of noise.

For this, we will use tools from the study of free spectrahedra (see [START_REF] Helton | Free convex algebraic geometry[END_REF] for a general introduction). Concretely, we are interested in the problem of (free) spectrahedral inclusion [HKM13b]. Originally, the inclusion of free spectrahedra has been introduced as a relaxation to study the inclusion of ordinary spectrahedra [BTN02,[START_REF] Helton | Dilations, linear matrix inequalities, the matrix cube problem and beta distributions[END_REF]. In contrast to that, we will be interested in the inclusion constants for their own sake. Often, results on free spectrahedral inclusion work for large classes of spectrahedra, e.g. spectrahedra with symmetries [START_REF] Helton | Dilations, linear matrix inequalities, the matrix cube problem and beta distributions[END_REF][START_REF] Davidson | Dilations, inclusions of matrix convex sets, and completely positive maps[END_REF]. Recently, results have been found which study maximal and minimal free spectrahedra for the p-norm unit balls [START_REF] Passer | Minimal and maximal matrix convex sets[END_REF]. It is especially the latter work which is most useful to us. We would also like to mention that another point of contact between quantum information theory and free analysis is the extension (or interpolation) problem for completely positive maps, see [HJRW12,[START_REF] Ambrozie | An interpolation problem for completely positive maps on matrix algebras: solvability and parametrization[END_REF].

In this work, we establish the connection between free spectrahedral inclusion and joint measurability. The matricial relaxation of the 1 -ball is known as the matrix diamond and plays a central role in our setting. We can then use results on inclusion constants for this free spectrahedron to characterize the degree of incompatibility of quantum effects in different settings. Conversely, we translate techniques to prove upper and lower bounds on quantum incompatibility to study spectrahedral inclusion. Let us note that since the problems of joint measurability and quantum steering are closely related [UBGP15], many of our results can be translated to the steering framework.

Main results

In this section, we will briefly outline the main findings of our work. Its main contribution is to connect the following two seemingly unrelated problems.

One is the problem of joint measurability of binary quantum observables. Given a g-tuple of quantum effects E 1 , . . . E g , we can ask the question of how much noise we have to add to the corresponding measurements fo make them jointly measurable. Joint measurability means that there exists a joint POVM R i 1 ,...,ig from which the binary POVMs we are interested in arise as marginals. Noise can be added in different ways to a measurement. We will mainly consider the case in which we take convex combinations of a quantum measurement with a fair coin, i.e.

E := sE + (1 -s)I/2
for s ∈ [0, 1]. The set of g-tuples s ∈ [0, 1] g which make any g binary POVMs of dimension d compatible will be denoted as Γ (g, d).

The other problem comes from the field of free spectrahedra. A free spectrahedron is a matricial relaxation of an ordinary spectrahedron. The free spectrahedron D A is then the set of self-adjoint matrix g-tuples X of arbitrary dimension which fulfill a given linear matrix inequality g i=1 A i ⊗ X i ≤ I.

If we only consider the scalar elements D A (1) of this set, this is just the ordinary spectrahedron defined by the matrix tuple A. The inclusion problem for free spectrahedra is to find the scaling factors s ∈ R g + such that the implication (1)

D A (1) ⊆ D B (1) ⇒ s • D A ⊆ D B
is true. We will be interested in the case in which D A is the matrix diamond, i.e. the set of matrices X such that

g i=1 i X i ≤ I ∀ ∈ { -1, 1 } g .
The set of all such s which make the implication in Equation (1) true for any B ∈ (M sa d ) g in this case will be written as ∆ (g, d).

The main contribution of our work is to relate these two problems and use this connection to characterize Γ(g, d). In Theorem 5.3, we find the following:

Theorem. Let E ∈ (M sa d ) g and let 2E -I := (2E 1 -I d , . . . , 2E g -I d ). We have (1) D ,g (1) ⊆ D 2E-I (1) if and only if E 1 , . . . , E g are quantum effects.

(2) D ,g ⊆ D 2E-I if and only if E 1 , . . . , E g are jointly measurable quantum effects.

(3) D ,g (k) ⊆ D 2E-I (k) for k ∈ [d] if an only if for any isometry V : C k → C d , the induced compressions V * E 1 V, . . . V * E g V are jointly measurable quantum effects.

This shows that there is a one-to-one correspondence between different levels of the spectrahedral inclusion problem and different degrees of compatibility. Furthermore, we show in Theorem 5.7 that finding spectral inclusion constants corresponds to making POVMs compatible through adding noise:

Theorem. It holds that Γ(g, d) = ∆(g, d).
This result allows us to use results on spectrahedral inclusion in order to characterize the set Γ(g, d). We find that the higher dimensional generalization of the positive quarter of the unit circle plays an important role in this:

QC g := s ∈ R g + : g i=1 s 2 i ≤ 1 .
The adaptation of some results of [START_REF] Passer | Minimal and maximal matrix convex sets[END_REF] allows us to show in Theorem 7.7:

Theorem. Let g, d ∈ N. Then, it holds that QC g ⊆ Γ(g, d). In other words, for any g-tuple E 1 , . . . , E g of quantum effects and any positive vector s ∈ R g + with s 2 ≤ 1, the g-tuple of noisy effects

E i = s i E i + (1 -s i ) I d 2 is jointly measurable.
If the dimension of the effects under study is exponential in the number of measurements, Theorem 8.8 provides us with a converse result. Again, this theorem is based on a result of [START_REF] Passer | Minimal and maximal matrix convex sets[END_REF].

Theorem. Let g ≥ 2, d ≥ 2 (g-1)/2 . Then, Γ(g, d) ⊆ QC g .
Thus, for g ≥ 2, d ≥ 2 (g-1)/2 we infer that Γ(g, d) = QC g ; this equality was known previously only in the case g = 2. However, this can no longer be the case for many measurements in low dimensions, as we point out in Section 9. For other types of noise added to quantum measurements, we can use similar results to give upper and lower bounds on the compatibility regions. The bounds we obtain with our techniques improve greatly on past results in the quantum information literature. As an example, the best lower bound in the symmetric case came from cloning and was of order 1/g for fixed g and large d, see Proposition 6.2 . Our results yield a lower bound of 1/ √ g, which turns out to be exact in the regime g d; we refer the reader to Section 9 for a detailed comparison of these bounds.

Conversely, we can use techniques from quantum information such as asymmetric cloning (Section 6) to give bounds on spectrahedral inclusion in different settings. In particular, we introduce in this work a generalization of the notion of inclusion constants from [START_REF] Helton | Dilations, linear matrix inequalities, the matrix cube problem and beta distributions[END_REF] in two directions: first, by restricting both the size and the number of the matrices appearing in the spectrahedron, and then by allowing asymmetric scalings of the spectrahedra, see Definition 4.1. Our contribution to the inclusion theory of free spectrahedra is going beyond the results from [START_REF] Passer | Minimal and maximal matrix convex sets[END_REF], by studying the asymmetric and size-dependent inclusion constants.

Concepts from Quantum Information Theory

In this section, we will start by reviewing some notions from quantum information theory related to measurements. Subsequently, we will define several versions of incompatibility of quantum measurements and show basic relations between them. For an introduction to the mathematical formalism of quantum mechanics see [HZ11] or [Wat18], for example.

Before we move to the quantum formalism, let us introduce some basic notation. For brevity, we will write [n] := {1, . . . , n} for n ∈ N and R g

+ for { x ∈ R g : x i ≥ 0 ∀i ∈ [g] }, g ∈ N.
Additionally, • : R → Z will be the ceiling function. Furthermore, for n, m ∈ N let M n,m be the set of complex n × m matrices. If m = n, we will just write M n . We will write M sa n for the self-adjoint matrices and U d for the unitary d × d matrices. I n ∈ M n will be the identity matrix. We will often drop the index if the dimension is clear from the context. For A ∈ (M sa d ) g , let OS A be the operator system defined by the g-tuple A, i.e.

OS A := span { I d , A i : i ∈ [g] } .
Moreover, we will often write for such tupels 2A -

I := (2A 1 -I d , . . . , 2A g -I d ) and V * AV := (V * A 1 V, . . . , V * A g V ) for V ∈ M d,k , k ∈ N.
A quantum mechanical system is described by its state ρ ∈ S(H), where H is the Hilbert space associated with the system and

S(H) := { ρ ∈ B(H) : ρ ≥ 0, tr[ρ] = 1 } .
In the present work, all Hilbert spaces will be finite dimensional. To describe transformations between quantum systems, we will use the concept of completely positive maps. Let T : B(H) → B(K) be a linear map with H, K two Hilbert spaces. This map is k-positive if for k ∈ N, the map

T ⊗ Id k : B(H) ⊗ M k → B(K) ⊗ M k is a positive map. A map is called completely positive if it is k-positive for all k ∈ N. If this map is additionally trace preserving, it is called a quantum channel.
Let Eff d be the set of d-dimensional quantum effects, i.e.

Eff d := { E ∈ M sa d : 0 ≤ E ≤ I } .
Effect operators are useful to describe quantum mechanical measurements. In quantum information theory, measurements correspond to positive operator valued measures (POVMs). A POVM is a set

{ E i } i∈Σ , E i ∈ Eff d for all i ∈ Σ, such that i∈Σ E i = I.
Here, Σ is the set of measurement outcomes, which we will assume to be finite for simplicity and equal to [m] for some m ∈ N. For the case of binary POVMs (m = 2), we will identify the POVM { E, I -E } with its effect operator E ∈ Eff d .

If a collection of POVMs can be written as marginals of a common POVM with more outcomes, we will say that they are jointly measurable (see [HMZ16] for an introduction to the topic). Definition 3.1 (Jointly measurable POVMs). We consider a collection of d-dimensional POVMs

E (i) j j∈[m i ]
where m i ∈ N for all i ∈ [g], g ∈ N. These POVMs are jointly measurable or

compatible if there is a d-dimensional POVM R j 1 ,...,jg with j i ∈ [m i ] such that for all u ∈ [g] and v ∈ [m u ], E (u) v = j i ∈[m i ] i∈[g]\{ u } R j 1 ,...,j u-1 ,v,j u+1 ,...,jg .
It is well-known [HMZ16] that not all quantum mechanical measurements are compatible. In concrete situations, the joint measurability of POVMs can be checked using a semidefinite program (SDP) (see e.g. [WPGF09]). Note that the SDP for g binary POVMs has 2 g variables, so when the number of effects g is large, it becomes computationally costly to decide compatibility. However, incompatible measurements can be made compatible by adding noise to the respective measurements. A trivial measurement is a POVM in which all effects are proportional to the identity. Adding noise to a measurement then means taking a convex combination of the original POVM and a trivial measurement. In order to quantify incompatibility of measurements, we can define several sets which differ in the type of noise we allow. We will restrict ourselves to binary POVMs in this work. For our first set, we allow different types of noise for every POVM:

Γ all (g, d) := { s ∈ [0, 1] g : ∀E 1 , . . . E g ∈ Eff d , ∃a ∈ [0, 1] g s.t. s i E i + (1 -s i )a i I are compatible } .
Another possibility is to consider only balanced noise:

Γ(g, d) := s ∈ [0, 1] g : s i E i + 1 -s i 2 I are compatible ∀E 1 , . . . E g ∈ Eff d .
Sometimes, it is inconvenient that the map from the original measurements to the ones with added noise is non-linear in the effect operators. To remedy this, we define

Γ lin (g, d) := s ∈ [0, 1] g : s i E i + (1 -s i ) tr[E i ] d I are compatible ∀E 1 , . . . E g ∈ Eff d .
The restriction of this set to equal weights has appeared before in the context of quantum steering [HKR15, UMG14]. Instead of restricting the type of noise allowed, we can also consider less general POVMs and restrict to those which are unbiased:

Γ 0 (g, d) := s ∈ [0, 1] g : s i E i + 1 -s i 2 I are compatible ∀E 1 , . . . E g ∈ Eff d s.t. tr[E i ] = d 2 .
Finally, let us introduce a set of parameters related to (asymmetric) cloning of quantum states

Γ clone (g, d) := {s ∈ [0, 1] g : ∃T : M ⊗g d → M d unital and completely positive s.t. (2) ∀X ∈ M d , ∀i ∈ [g], T I ⊗(i-1) ⊗ X ⊗ I ⊗(n-i) = s i X + (1 -s i ) tr[X] d I}.
All these sets are convex sets, as the next proposition shows. is again a POVM and it can easily be verified that

i j ∈[2] j∈[g]\{ u } λR i 1 ,...,i u-1 ,1,i u+1 ,...,ig + (1 -λ)R i 1 ,...,i u-1 ,1,i u+1 ,...,ig = [λs u + (1 -λ)t u ]E u + [1 -(λs u + (1 -λ)t u )]I/2.
As the effects were arbitrary, this proves the assertion for Γ(g, d).

Remark 3.3. Using convexity, it can easily be seen that (1/g, . . . , 1/g) ∈ Γ # (g, d), where # ∈ { all, ∅, lin, 0, clone }. It can be seen that the standard basis vector e i is in each of the sets for i ∈ [g]. The above statement then follows by convexity. See also [HMZ16] for an intuitive argument.

In the next proposition, we collect some relations between the different sets we have defined; Proposition 3.4. Let g, d ∈ N. Then the following inclusions are true:

(1) Γ(g,

d) ⊆ Γ all (g, d) (2) Γ lin (g, d) ⊆ Γ all (g, d) (3) Γ(g, d) ⊆ Γ 0 (g, d) (4) Γ lin (g, d) ⊆ Γ 0 (g, d) (5) Γ clone (g, d) ⊆ Γ lin (g, d) (6) Γ 0 (g, 2d) ⊆ Γ(g, d) (7) F Γ all (g, d) ⊆ Γ(g, d),
where

F : [0, 1] g → [0, 1] g , F (s 1 , . . . , s g ) = s 1 2 -s 1 , . . . , s g 2 -s g .
Proof. The first two assertions are true since we restrict the trivial measurements we are mixing with in both cases. The third assertion follows in the same way, but this time compatibility has to hold for less states. The fourth assertion follows since tr [E i ]/d = 1/2 for the effects considered for Γ 0 (g, d). For the fifth claim, let s ∈ Γ clone (g, d) be an arbitrary scaling g-tuple and consider quantum effects E 1 , . . . , E g ∈ Eff d . Define, for every bit-string b of length g

F b := T (E (b 1 ) 1 ⊗ E (b 2 ) 2 ⊗ • • • ⊗ E (bg) g ),
where we set E

(1) i = E i and E (0) i = I -E i , and T is a map as in (2). Since the map T is (completely) positive, we have that all the operators F b are positive semidefinite. Moreover, the marginals can be computed as follows:

b∈{0,1} g , b i =1 F b = T (I ⊗i-1 ⊗ E i ⊗ I ⊗n-i ) = s i E i + (1 -s i ) tr[E i ] d I =: E i ,
which shows that the mixed effects E i are compatible, proving the claim.

For the sixth assertion, let s ∈ Γ 0 (g, 2d). Then, for a g-tuple of arbitrary d × d quantum effects E i , the quantum effects (of size 2d)

s i [E i ⊕ (I d -E i )] + (1 -s i ) d 2d I 2d are unbiased (tr[E i ⊕ (I d -E i )] = d
) and thus compatible. Truncating the above effects to their upper-left corner proves the claim.

Let us now prove the seventh and final claim. It is enough to show that, for any effect E ∈ Eff d and any mixture E = sE + (1 -s)aI with some trivial effect aI (a ∈ [0, 1]), there is a further mixture E = xE +(1-x)bI = yE +(1-y)I/2. Working out the relations between the parameters s, x, y, a, b, we find the following two equations

y = xs and b = 1 -xs -2xa(1 -s) 2(1 -x) . Γ(g, d) Γ all (g, d) Γ clone (g, d) Γ lin (g, d) Γ 0 (g, d) Γ 0 (g, 2d) ⊆ Γ(g, d) F Γ all (g, d) ⊆ Γ(g, d)
Figure 1. The different inclusions for the sets Γ # , # ∈ {∅, lin, all, 0, clone} proven in Proposition 3.4. Full arrows represent inclusions of sets, while dashed arrows represent special conditions.

Asking that, for all values of a ∈ [0, 1], b is also between 0 and 1, we obtain the desired inequality y ≤ s/(2 -s). Let (E 1 , . . . E g ) be the compatible effects corresponding to s. Then E 1 , . . . , E j-1 , b j I, E j+1 , . . . , E g are compatible as well, since we obtain a joint POVM for the effects without E j by summing over the j-th index and b j I is a trivial measurement and as such compatible with all effects. Then, we obtain the element

E = (x 1 E 1 + (1 -x 1 )b 1 I, . . . , x g E g + (1 - x g )b g I) from (E 1 , . . . , E g
) by successively taking convex combinations with elements of the form (E 1 , . . . , E j-1 , b j I, E j+1 , . . . , E g ). As convex combinations of compatible tupels stay compatible (see proof of Proposition 3.2), we infer that E is compatible and the assertion follows.

Remark 3.5. It would be interesting to see if, in general, Γ(g, d ) ⊆ Γ lin (g, d) for some parameter d which depends on d.

Now we show that these sets become smaller when we increase the dimension of the effects considered.

Proposition 3.6. Let g,d ∈ N and # ∈ { all, ∅, lin, 0, clone }. Then

Γ # (g, d + 1) ⊆ Γ # (g, d).
Proof. Let us first show the inclusion for Γ all ; the proofs for Γ and Γ 0 are almost identical. Let E i ∈ Eff d for all i ∈ [g]. We can embed these effects into Eff d+1 by choosing E i = E i ⊕ 0. Let s ∈ Γ all (g, d + 1). Then there exists an a ∈ [0, 1] g such that the effects

s i E i + (1 -s i )a i I d+1 , i ∈ [g], are compatible. Let V : C d → C d+1 be an isometry such that V V * is the projection onto the first d entries. It is easy to check that for a POVM { R i } i∈[g] with R i ∈ Eff d+1 for all i ∈ [g], the set { V * R i V } i∈[g] is again a POVM with elements in Eff d . Furthermore, V * s i E i + (1 -s i )a i I d+1 V = s i E i + (1 -s i )a i I d , which implies that the s i E i + (1 -s i )a i I d are compatible and therefore s ∈ Γ all (g, d).
To prove the claim for Γ lin , we use the same idea as before, but with the following linear embedding of d × d matrices into (d + 1) × (d + 1) matrices

Ψ : M d → M d+1 X → X ⊕ tr[X] d .
As tr[Ψ(X)] = (d + 1)/d tr[X], the claim follows.

Finally, since Ψ can easily be seen to be completely positive and unital, we can use it to together with the embedding V from above to define the cloning map for dimension d through the one for dimension d + 1. With T d+1 and T d the maps appearing in (2) for Γ clone (g, d + 1) and Γ clone (g, d), respectively, we obtain

T d (X) := V * T d+1 (Ψ ⊗g (X))V.
It can then be verified that the map indeed has the desired properties.

Remark 3.7. We would like to point out that the sets Γ(g, d) give rise to compatibility criteria, i.e. sufficient conditions for compatibility, as follows. Let s ∈ Γ(g, d) be such that s i > 0 for all i ∈ [g]. Then, the following implication holds:

∀i ∈ [g], 1 s i E i - 1 -s i 2s i I d ∈ Eff d =⇒ (E 1 , . . . , E g ) compatible.
Indeed, using s ∈ Γ(g, d) and the hypothesis, it follows that the effects

s i 1 s i E i - 1 -s i 2s i I d + (1 -s i ) I d 2 = E i
are compatible. These criteria become useful if the corresponding SDP is intractable.

Free spectrahedra

In this section, we will review some concepts from the study of free spectrahedra which we will need in the rest of the paper. We will start with the definition of free spectrahedra and their inclusion. Then, we will review the link between spectrahedral inclusion and positivity properties of certain maps. All the theory needed in this work can be found in [HKM13b, [START_REF] Helton | Dilations, linear matrix inequalities, the matrix cube problem and beta distributions[END_REF][START_REF] Davidson | Dilations, inclusions of matrix convex sets, and completely positive maps[END_REF].

Let A ∈ (M sa d ) g . The free spectrahedron at level n corresponding to this g-tuple of matrices is the set

D A (n) := X ∈ (M sa n ) g : g i=1 A i ⊗ X i ≤ I nd .
For n = 1, this is a usual spectrahedron defined by a linear matrix inequality. The free spectrahedron is then defined as the (disjoint) union of all these sets, i.e.

D A := n∈N D A (n).
A free spectrahedron which we will very often encounter is the matrix diamond of size g. It is defined as

D ,g (n) = X ∈ (M sa n ) g : g i=1 i X i ≤ I n ∀ ∈ { -1, +1 } g .
To see that this is a free spectrahedron, we can take the direct sum of all these constraints. The matrices defining this free spectrahedron are thus diagonal. At level 1, the matrix diamond is the unit ball of the 1 -norm. Therefore, it is obviously bounded. For free spectrahedra, the inclusion

D A ⊆ D B means D A (n) ⊆ D B (n) for all n ∈ N.
Inclusion at the level of spectrahedra (n = 1) does not guarantee inclusion of the free spectrahedra. That is, the implication

D A (1) ⊆ D B (1) ⇒ D A ⊆ D B
does not hold in general. However, scaling the set D A down, the implication becomes eventually true.

Definition 4.1. Let D A be the free spectrahedron defined by A ∈ (M sa d ) g . The inclusion set ∆ D A (k) is defined as

∆ D A (k) := s ∈ R g + : ∀B ∈ (M sa k ) g D A (1) ⊆ D B (1) ⇒ s • D A ⊆ D B .
For D A = D ,g , we will write ∆(g, k) for brevity. Here, the set

s • D A := { (s 1 X 1 , . . . , s g X g ) : X ∈ D A }
is the (asymetrically) scaled free spectrahedron.

Note that the definition above generalizes the one from [START_REF] Helton | Dilations, linear matrix inequalities, the matrix cube problem and beta distributions[END_REF] by restricting the size of the matrices defining the containing spectrahedra and by allowing non-symmetric scaling; one recovers the definition of inclusion constants from [START_REF] Helton | Dilations, linear matrix inequalities, the matrix cube problem and beta distributions[END_REF] by considering the largest constant s ≥ 0 such that (s, . . . , s g times

) ∈ ∆ D A (k), ∀k ≥ 1.
As in the case of POVMs, we are also interested in the inclusion constant set where we restrict to inclusion into free spectrahedra defined by traceless matrices.

Definition 4.2. Let D A be a spectrahedron defined by A ∈ (M sa d ) g . The traceless-restricted inclu- sion set ∆ 0 D A (d) of D A is defined as ∆ 0 D A (k) := {s ∈ R g + : ∀C ∈ (M sa k ) g s.t. ∀i ∈ [g], tr[C i ] = 0, D A (1) ⊆ D C (1) =⇒ s • D A ⊆ D C }.
For D A = D ,g , we will again write ∆ 0 (g, k) for brevity.

The next proposition shows that both inclusion sets we have defined are convex.

Proposition 4.3. Let A ∈ (M sa d ) g . Both ∆ D A (k) and ∆ 0 D A (k) are convex. Proof. Let B ∈ (M sa k ) g and X ∈ D A . Further, let s, t ∈ ∆ D A (k) and λ ∈ [0, 1]. The assumptions on s, t yield g i=1 B i ⊗ (λs i + (1 -λ)t i )X = λ g i=1 B i ⊗ s i X + (1 -λ) g j=1 B j ⊗ t j X ≤ I.
This proves the first assertion, because B was arbitrary. The second assertion follows in a very similar manner.

The inclusion of spectrahedra can be related to positivity properties of a certain map. Let A ∈ (M sa D ) g and B ∈ (M sa d ) g define the free spectrahedra D A and D B , respectively. Let Φ : OS A → M d be the unital map defined as Φ :

A i → B i ∀i ∈ [g]
. Then, we can find a one-to-one relation between properties of Φ and the inclusion of the free spectrahedra at different levels. This has been proven in [HKM13b, Theorem 3.5] for real spectrahedra and we include a proof in the complex case for convenience. Proof. The "only if" direction is true by the unitality and n-positivity of Φ. For the "if" direction, let Y ∈ M sa n (OS A ). Without loss of generality, we can assume I D , A 1 , . . . A g to be linearly independent. Then

Y = I D ⊗ X 0 - g i=1 A i ⊗ X i for (X 0 , . . . X g ) ∈ M g n .
We claim that X 0 , . . . X g are self-adjoint. Then (I D ⊗e * i )(Y -Y * )(I D ⊗e j ) = 0 for all i, j ∈ [n] and an orthonormal basis { e i } n i=1 of C n if and only if e i , (X l -X * l )e j for all i, j ∈ [n] and for all l ∈ [g] ∪ { 0 }. This proves the claim. If Y ≥ 0, it holds that X 0 ≥ 0. Let us assume that this is not the case. Then there exists an x ∈ C n such that x, X 0 x < 0. Positivity of Y yields

- g i=1 x, X i x A i > 0.
Therefore, λ( x, X 1 x , . . . , x, X g x ) ∈ D A (1) for all λ ≥ 0. This contradicts the assumption that D A (1) is bounded. Let us now assume that Y ≥ 0 and that

X 0 > 0. Then (Φ ⊗ Id n )Y ≥ 0, because X -1/2 0 XX -1/2 0 ∈ D A (n) ⊆ D B (n).
For Y ≥ 0 and X 0 ≥ 0, positivity of (Φ ⊗ Id n )Y follows from exchanging X 0 by X 0 + I n , > 0 and letting go to zero.

Remark 4.5. The complete positivity of Φ can be checked using an SDP [HKM13b, HJRW12]. Therefore, the inclusion problem at the level of free spectrahedra is efficiently solvable. This is not necessarily the case for the usual spectrahedra at level 1, because checking the positivity of a linear map is in general a hard problem (the set of positive maps between matrix algebras is dual to the set of separable states, and deciding weak membership into the latter set is known to be NPhard [Gur03]). Seeing the free spectrahedral inclusion problem as a relaxation of the corresponding problem for (level 1) spectrahedra is a very useful idea in optimization, see [BTN02, HKM13b].

Using the previous lemma, we obtain a useful corollary if we assume that D A (1) is bounded, which is enough for us. Remark 4.7. The result of Corollary 4.6 without the boundedness assumption has appeared before in [HKMS14, Lemma 2.3] for real spectrahedra with a longer proof. Their proof carries over to the complex setting. Therefore, the boundedness assumption is not necessary, but it shortens the proof considerably.

Spectrahedral inclusion and joint measurability

In this section, we establish the link between joint measurability of effects and the inclusion of free spectrahedra. The main result of this work is Theorem 5.3 which we prove at the end of this section. It connects the inclusion of the matrix diamond into a spectrahedron with the joint measurability of the quantum effects defining this spectrahedron. Before we can prove the theorem, we will need two lemmas concerning compressed versions of a (free) spectrahedron.

Lemma 5.1. Let k ∈ N, 1 ≤ k ≤ d and let V : C k → C d be an isometry. For A ∈ (M sa d ) g , it holds that D A ⊆ D V * AV . Proof. Let X ∈ D A (n). Then from the definition g i=1 A i ⊗ X i ≤ I dn .
Multiplying the equation by V ⊗ I n from the right and by its adjoint from the left, it follows that

g i=1 V * A i V ⊗ X i ≤ I kn .
Here, we have used that V is an isometry and that the map

Y → W * Y W for matrices Y , W of appropriate dimensions is completely positive. Lemma 5.2. Let k ∈ N, 1 ≤ k ≤ d and let V : C k → C d be an isometry. For A ∈ (M sa d ) g , it holds that (3) D A (k) = V :C k →C d isometry D V * AV (k).
Proof. Let us denote the right hand side of Eq. (3) by C. From Lemma 5.1, it follows that D A (k) ⊆ C. For the reverse inclusion, let X ∈ C. This implies especially X ∈ (M sa k ) g . We write

Y := g i=1 A i ⊗ X i -I dk .
To prove the assertion, we need to show that Y ≥ 0. Let y ∈ C d ⊗ C k be a unit vector with Schmidt decomposition

y = k i=1 λ i e i ⊗ f i ,
where { e i } k i=1 and { f j } k j=1 are orthonormal families in C d and C k , respectively. Further, λ i ≥ 0 and such that k i=1 λ i = 1. We then have

y, Y y = k i,j=1 λ i λ j (e i ⊗ f i ), Y (e j ⊗ f j ) .
Let Ω = k i=1 g i ⊗ g i be an unnormalized maximally entangled state with an orthonormal basis

{ g i } k i=1 of C k . Moreover, let V : C k → C d and Q : C k → C k be defined as V = k i=1 e i g * i , Q = k j=1 λ j f j g * j .
Then, V :

C k → C d is an isometry. Therefore, (V * ⊗ I k )Y (V ⊗ I k ) ≥ 0 by assumption, as X ∈ C. Hence, y, Y y = Ω, (I k ⊗ Q * )(V * ⊗ I k )Y (V ⊗ I k )(I k ⊗ Q)Ω ≥ 0.
Thus, Y is positive semidefinite, because y was arbitrary. where { e i } g i=1 is the standard basis in R g . We have

Theorem 5.3. Let E ∈ (M sa d ) g and let 2E -I := (2E 1 -I d , . . . , 2E g -I d ). We have (1) D ,g (1) ⊆ D 2E-I (1) if and only if E 1 , . . . , E g are quantum effects. (2) D ,g ⊆ D 2E-I if and only if E 1 , . . . , E g are jointly measurable quantum effects. (3) D ,g (k) ⊆ D 2E-I (k) for k ∈ [d] if an only if for any isometry V : C k → C d , the induced compressions V * E 1 V, . . . V * E g V
e i ∈ D E (1) ⇐⇒ 2E i -I ≤ I ⇐⇒ E i ≤ I -e i ∈ D E (1) ⇐⇒ -(2E i -I) ≤ I ⇐⇒ E i ≥ 0,
proving the first claim.

We now characterize the free spectrahedral inclusion from the second point. In the following, we will identify diagonal matrices with vectors and the subalgebra of diagonal 2 g × 2 g -matrices with C 2 g . The operator system associated with D ,g is C 2 g ⊇ OS ,g := span{v 0 , v 1 , . . . , v g }, where, indexing the 2 g coordinates by sign vectors ε ∈ {±1} g ,

v 0 (ε) = 1 v i (ε) = ε(i) ∀i ∈ [g].
Here, ε(i) is the i-th entry of the vector ε. The dimension of this operator system is g + 1. We define a map Φ : OS ,g → M d by

v 0 → I v i → 2E i -I ∀i ∈ [g].
The spectrahedral inclusion D ,g ⊆ D 2E-I holds if and only if the map Φ is completely positive (Lemma 4.4). If this is the case, Arveson's extension theorem (see [Pau03, Theorem 6.2] for a finite-dimensional version) guarantees the existence of a completely positive extension Φ of this map to the whole algebra C 2 g , because OS ,g is an operator system. As C 2 g is a commutative matrix algebra, it is enough to show that the extension is positive [Pau03, Theorem 3.11]. To find such an extension Φ : C 2 g → M d , we consider the basis (g η ) η∈{±1} g of the vector space C 2 g which is defined as follows:

g η (ε) = 1 ε=η ≥ 0.
Here, 1 ε=η = 1 if ε = η and zero otherwise. Let us write G η := Φ(g η ); since the g η are positive, the (complete) positivity of Φ is equivalent to G η ≥ 0, for all η.

We have, for all ε ∈ {±1} g ,

1 = v 0 (ε) = η 1 ε=η = η g η ( ),
and thus we can rewrite

Φ(v 0 ) = I ⇐⇒ η G η = I.
We also have

v i (ε) = ε(i) = 21 ε(i)=+1 -1 = 2 η : η(i)=+1 1 ε=η -1 = 2 η : η(i)=+1 g η ( ) - η g η ( ),
and thus we have, for all i ∈ [g],

Φ(v i ) = 2E i -I ⇐⇒ η : η(i)=+1 G η = E i .
Collecting all these facts, we have shown that the map Φ extends to a (completely) positive map on the whole C 2 g if and only if there exist operators (G η ) η∈{±1} g such that

∀η ∈ {±1} g , G η ≥ 0 η G η = I ∀i ∈ [g], η : η(i)=+1 G η = E i ;
but these are precisely the conditions for the joint measurability of the effects E 1 , . . . , E g and we are done with the second point. For the third assertion, it follows from Lemma 5.2 that

D ,g (k) ⊆ D 2E-I (k) holds if and only if D ,g (k) ⊆ D 2V * EV -I (k) for any isometry V : C k → C d .
Further, Corollary 4.6 asserts that this is equivalent to D ,g ⊆ D 2V * EV -I for all isometries V as above. The claim then follows from the second assertion of this theorem.

Remark 5.4. The fact that the second point of the theorem above implies the third point, read on the quantum effects side of the equivalence, is a well known fact1 : compressions of jointly measurable effects are jointly measurable.

Remark 5.5. If E ∈ (M sa d ) g is a g-tuple of pairwise commuting matrices, then D ,g (1) ⊆ D 2E-I (1) implies D ,g ⊆ D 2E-I . This is true, because the effects on the right hand side generate a commutative matrix subalgebra. Inclusion at level one then implies that the corresponding map Φ is positive. As its range is contained in a commutative matrix algebra, this also implies that Φ is completely positive [Pau03, Theorem 3.9], which then yields the inclusion at the level of free spectrahedra. This recovers the well-known result from quantum information theory that pairwise commuting effects are jointly measurable.

Remark 5.6. We can recover another result from quantum information theory, namely that effects of the form aI, a ∈ [0, 1], are trivially compatible with any effect. This corresponds to the fact that

(4) D ,g ⊆ D 2E-I ⇐⇒ D ,g+1 ⊆ D (2E-I,αI)
for any E ∈ (M sa d ) g . Here, we write α = 2a -1, i.e. α ∈ [-1, 1]. This can best be seen at the level of maps. It is easy to see that the v i defining D ,g (c.f. proof of Theorem 5.3) can be written as

v i = i-1 j=1 I 2 ⊗ diag[+1, -1] ⊗ g j=i+1 I 2 .
Let Φ g be the map corresponding to the left hand side and Φ g+1 be the one corresponding to the right hand side of Equation (4). For the " only if "-direction, we can simply define Φ g (A) = Φ g+1 (A⊗I 2 ), where A ∈ C 2 g . For the "if "-direction, we define the linear map Ψ : C 2 → C as

Ψ : (1, 0) → α + 1 2 , Ψ : (0, 1) → 1 -α 2 .
This map is unital, positive and therefore also completely positive. We can then set Φ g+1 = Φ g ⊗ Ψ.

It can then be checked using the above expression for the v i that this map has indeed the desired properties.

Theorem 5.7. It holds that Γ(g, d) = ∆(g, d) and that Γ 0 (g, d) = ∆ 0 (g, d).

Proof. Let s ∈ R g . Then s ∈ Γ(g,
s • D ,g ⊆ D 2E-I ⇐⇒ D ,g ⊆ D s•(2E-I) ⇐⇒ D ,g ⊆ D 2E -I ,
where

E i = s i E i + (1 -s i )I/2 for all i ∈ [g]. Therefore, Theorem 5.3 implies that s ∈ Γ(g, d) is equivalent to the implication D ,g (1) ⊆ D 2E-I (1) ⇒ s • D ,g ⊆ D 2E-I for all E ∈ (M sa d ) g . This is equivalent to s ∈ ∆(g, d), because X → 2X -I is a bijection on M sa d . The second assertion follows since X → 2X-I is a bijection between { X ∈ M sa d : tr[X] = d/2 } and { A ∈ M sa d : tr[A] = 0 }.

Lower bounds from cloning

In this section we provide, using known facts about the set Γ clone , lower bounds for Γ lin and Γ. We start by recalling the main results from the theory of symmetric cloning.

F c,1 (T ) = inf j∈[g] inf σ pure tr σ (j) T (σ ⊗N )
where σ (j) = I ⊗(j-1) ⊗ σ ⊗ I ⊗(g-j) ∈ M ⊗g d and

F c,all = inf σ pure tr σ ⊗g T (σ ⊗N ) .
These quantities are both maximized by the optimal quantum cloner

T (ρ) = d[N ] d[g] S g (ρ ⊗ I)S g .
Here, d[g] = d+g-1 g is the dimension of the symmetric subspace ∨ g C d ⊆ (C d ) ⊗g and S g is the corresponding orthogonal projection. We can therefore identify E = γE + (1 -γ)I E , where I E is the trivial effect tr{E}/dI depending on E. Therefore, γ is a lower bound on the joint measurability of a family of effects,

T * E (i) 1 i ⊗ . . . ⊗ E (j) g j
being the joint observables.

Inserting the expression for γ from the symmetric cloning bounds and using Proposition 3.4, we obtain the following result; note that below, the second quantity is always larger than the third one. Proposition 6.2. For all g, d ≥ 2,

g + d g(1 + d) (1, 1, . . . , 1 g times ) ∈ Γ clone (g, d) ⊆ Γ lin (g, d) ⊆ Γ all (g, d) g + 2d g(1 + 2d) (1, 1, . . . , 1 g times ) ∈ Γ clone (g, 2d) ⊆ Γ 0 (g, 2d) ⊆ Γ(g, d) ⊆ Γ all (g, d) g + d g + d(2g -1) (1, 1, . . . , 1 g times ) ∈ F (Γ clone (g, d)) ⊆ F (Γ all (g, d)) ⊆ Γ(g, d).
In the general, non-symmetric case, the exact form of the set Γ clone (g, d) has been computed, by different methods, in [Kay16] and [S ĆHM14]; the following restatement of the optimal cloning probabilities is taken from the former reference.

Theorem 6.3. [Kay16, Theorem 1, Section 2.3] For any g, d ≥ 2, Γ clone (g, d) = s ∈ [0, 1] g : (g + d -1) g -d 2 + d + (d 2 -1) g i=1 s i (5) ≤ g i=1 s i (d 2 -1) + 1 2 .
Using the variables t i := s i (d 2 -1) + 1 ∈ [1, d 2 ], we have the simpler expression

Γ clone (g, d) = s ∈ [0, 1] g : t 1 - t 1/2 g + d -1 ≤ d(d -1) ,
where • p denotes the p -quantity on R g :

t p = ( g i=1 |t i | p ) 1/p .
Proof. The formula is exactly [Kay16, Equation (5)], after the change of variables

F i = s i +(1-s i )/d, for all i ∈ [g].
Remark 6.4. Note that the symmetric cloning optimal probability is recovered by setting s 1 = s 2 = • • • = s g in the result above, yielding the the maximal value

s max = g + d g(d + 1)
.

Remark 6.5. In the regime d → ∞, the left hand side of (5) behaves like d 3 ( s 1 -1), whereas the right hand side behaves like d 2 s 1/2 . Hence, asymptotically, the achievable cloning probabilities should satisfy i s i ≤ 1; the set such values is the probability simplex, i.e. the convex hull of the points {e i } g i=1 , where e i is the basis vector having a 1 in position i and zeros elsewhere.

We discuss next the special cases of pairs and triplets, i.e. g = 2, 3. The most studied for (asymmetric) cloning is the g = 2 case (see, e.g. [START_REF] Cerf | Asymmetric quantum cloning in any dimension[END_REF] or the more recent [Has17, Theorem 3]). We plot in the left panel of Figure 2 the sets Γ clone (2, d) for various values of d, as subsets of Γ(2, d) = QC 2 . Proposition 6.6. For all d ≥ 2, we have

Γ clone (2, d) = {(s, t) ∈ [0, 1] 2 : s + t - 2 d (1 -s)(1 -t) ≤ 1}.
Proof. To see that the condition above is equivalent to equation (5) for g = 2, one can solve both for t and show that the answer is the following:

t ≤ 1 -s - 2(1 -s) d 2 + 2 1 -s + (d 2 -1)s(1 -s) d 2 .
The case g = 3 is also worth mentioning, since one can obtain manageable expressions for the set Γ clone (3, d). In the right panel of Figure 2, we plot the slice Γ clone (3, d) ∩ {(s, t, t)} for various values of d (this corresponds to asking that the "quality" of the second and third clones are identical), against the Euclidean ball (see Section 7.2 for the relevance of the quarter-circle). Proposition 6.7. For all d ≥ 2, we have, either in explicit or in parametric form 7. Lower bounds from free spectrahedra 7.1. Dimension dependent and symmetric lower bounds. This part basically reproduces the proof of Theorem 1.4 in [START_REF] Helton | Dilations, linear matrix inequalities, the matrix cube problem and beta distributions[END_REF] and shows that making minor changes, the proof also works in the case where the spectrahedra are given in terms of complex instead of real matrices. Note that in this case, we obtain an inclusion constant of 2d instead of merely d. Let us first recall a lemma from [START_REF] Helton | Dilations, linear matrix inequalities, the matrix cube problem and beta distributions[END_REF], which was proved there for real matrices but carries over without change.

[IAC + 05] Γ clone (3, d) = (s, t, u) ∈ [0, 1] 3 : (d + 2) 3 -d 2 + d + (d 2 -1)(s + t + u) ≤ (d 2 -1)s + 1 + (d 2 -1)t + 1 + (d 2 -1)u + 1 2 = 1 -b 2 -c 2 - 2bc d + 1 , 1 -c 2 -a 2 - 2ca d + 1 , 1 -a 2 -b 2 - 2ab d + 1 : a 2 + b 2 + c 2 + 2(ab + bc + ca)/d ≤ 1, a, b, c ≥ 0 . 0.
Lemma 7.1 (Lemma 8.2 from [START_REF] Helton | Dilations, linear matrix inequalities, the matrix cube problem and beta distributions[END_REF]). Suppose T = (T j,l ) is a k × k block matrix with blocks of equal size. If T j,l ∞ ≤ 1 for every j, l

∈ [k], then T ∞ ≤ k. Proposition 7.2. Let A ∈ (M sa D ) g and B ∈ (M sa d ) g . Suppose further that -D A ⊆ D A and that D A (1) is bounded. If D A (1) ⊆ D B (1), then D A ⊆ 2dD B .
Proof. Fix some level n and consider {e l } n l=1 , the standard orthonormal basis for

C n . Fix 1 ≤ s = t ≤ n and set p ± s,t = 1/ √ 2(e s ± e t ) ∈ C n , φ ± s,t = 1/ √ 2(e s ± ie t ) ∈ C n . Further, let P ± s,t = I D ⊗ p ± s,t , Φ ± s,t = I D ⊗ φ ± s,t . Then P ± s,t * P ± s,t = I D = Φ ± s,t * Φ ± s,t . Let M ∈ M sa n , C ∈ M D . Then P + s,t * (C ⊗ M )P + s,t -P - s,t * (C ⊗ M )P - s,t = 2C ⊗ Re(M ) s,t Φ + s,t * (C ⊗ M )Φ + s,t -Φ - s,t * (C ⊗ M )Φ - s,t = -2C ⊗ Im(M ) s,t . Let X ∈ D A (n)
and let Z = j A j ⊗ X j . By hypothesis, -X ∈ D A (n) as well, thus ±Z ≤ I Dn . By the above calculations, also ± P ± s,t * ZP ± s,t ≤ I D . Therefore, ± p ± s,t * Xp ± s,t ∈ D A (1) for all s = t.

Convexity of D A (1) together with the above implies ±Re(X) s,t = ±(Re(X 1 ) s,t , . . . , Re(X g ) s,t ) ∈ D A (1). The same holds true for s = t if one chooses p t = e t and makes the necessary adjustments in the above argument. Considering φ ± s,t , we find that ±Im(X) s,t ∈ D A (1) for all s, t ∈ [n] as well. Now set

T s,t = j B j ⊗ (X j ) s,t . It holds that T s,t ∞ ≤ j B j ⊗ Re(X j ) s,t ∞ + k B k ⊗ Im(X k ) s,t ∞ .
Moreover, we know that

-I n ≤ j B j ⊗ Re(X j ) s,t ≤ I n -I n ≤ k B k ⊗ Im(X k ) s,t ≤ I n
as the real and imaginary parts of the entries of X have been found to be in D A (1) and therefore also in D B (1) by hypothesis. Combining the two findings, it follows that T s,t ∞ ≤ 2. An application of Lemma 7.1 to T /2 allows us to conclude that T ∞ ≤ 2n. Thus,

-I dn ≤ 1 2n j B j ⊗ X j ≤ I dn which implies ±1/(2n)X ∈ D B (n). At level n = d, this implies D A (d) ⊆ 2dD B (d). Since B ∈ (M sa d )
g , an application of Corollary 4.6 proves the assertion.

Remark 7.3. The assumption that D A (1) is bounded is not necessary and does not appear in [START_REF] Helton | Dilations, linear matrix inequalities, the matrix cube problem and beta distributions[END_REF], see Remark 4.7.

Exploiting the link between inclusion of free spectrahedra and joint measurability, the previous proposition corresponds to: Corollary 7.4. Let s = (1/(2d), . . . 1/(2d)). Then s ∈ Γ(g, d).

Proof. The matrix diamond is symmetric, i.e. it holds that -D ,g = D ,g . Further, D ,g (1) is bounded. From Proposition 7.2, it follows that D ,g (1) ⊆ D A (1) implies D ,g ⊆ 2dD A for any A ∈ (M sa d ). Thus, s ∈ ∆(g, d). The claim then follows from Theorem 5.7.

Remark 7.5. In [HKMS14], Proposition 7.2 was proven for spectrahedra defined by real matrices and with d instead of 2d. We point out that this result cannot hold in the complex case. Consider

d = 2, g = 3, A i = σ i and B i = σ i for i ∈ [3].
Here, σ i are the usual Pauli matrices. In this case, the operator system spanned by the A i is the whole matrix algebra. Let s 0 = (s, s, s) with s ∈ [0, 1]. For s 0 D A ⊆ D B to hold, the map

Φ (A) = sA + (1 -s) tr[A]/2I
for A ∈ M 2 must be completely positive. This follows from Lemma 4.4 (see also [HKMS14, Section 1.4]). A short calculation shows that the minimal eigenvalue of the Choi matrix [Wat18, Section 2.2.2] of this map is -s/2 + (1 -s)/4. For the Choi matrix to be positive, we thus require s ≤ 1/3, which is fulfilled by s = 1/2d but not by s = 1/d. However, this calculation does not imply that 2d is optimal in Proposition 7.2, leaving this question open.

7.2. Dimension independent lower bounds. We restate in this section one implication of [PSS18, Theorem 6.6], which, interpreted in terms of inclusion constants, yields Theorem 7.7. For the convenience of the reader and for the sake of being self-contained, we reproduce the proof with several simplifications and written in a language more familiar to quantum information specialists. Let MatBall g be the matrix ball (see [Pis03, Chapter 7], [HKMS14, Section 14], [DDOSS17, Section 9] for the different operator structures one can put on the 2 -ball)

MatBall g := {(X 1 , . . . , X g ) ∈ (M sa d ) g : g i=1 X 2 i ≤ I}.
We recall the following result [PSS18, Lemma 6.5].

Lemma 7.6. For all g ≥ 1, D ,g ⊆ MatBall g .

Proof. Let (X 1 , . . . , X g ) ∈ D ,g . By definition, we have that, for all ε ∈ {±1} g ,

g i=1 ε i X i ≤ I.
Squaring the relation above, we get i

X 2 i + i =j ε i ε j X i X j ≤ I.
Averaging the above inequality for all values of ε, we are left with i X 2 i ≤ I, which is the claim we aimed for.

Define the "quarter-circle" (6) QC g := s ∈ R g + : s 2 1 + . . . + s 2 g ≤ 1 to be part of the unit disk contained in the positive orthant.

Theorem 7.7. [PSS18, Theorem 6.6] Let A ∈ (M sa d ) g . For any vector s ∈ R g + such that i s 2 i ≤ 1 and any spectrahedron D A , whenever D ,g (1) ⊆ D A (1), we have s • MatBall g ⊆ D A . In particular, s • D ,g ⊆ D A . In terms of inclusion constants, we have ∀g, d, QC g ⊆ ∆(g, d).

Proof. Using Lemma 7.6, under the hypotheses, we only need to show the inclusion sMatBall g ⊆ D A . To this end, consider a g-tuple of n×n self-adjoint matrices (X 1 , . . . , X g ) such that g i=1 X 2 i ≤ I. We claim that this inequality implies that, for all s as in the statement,

g i=1 s i |X i | ≤ I.
Indeed, this follows from the general matrix inequality

i B i C i ∞ ≤ i B i B * i 1/2 ∞ i C * i C i 1/2 ∞ .
The above inequality can be seen to hold by writing the B i in the first row of a larger matrix and the C i in the first column of another such matrix. Writing now

X i = Y i -Z i with positive semidefinite operators Y i , Z i in such a way that |X i | = Y i + Z i , we also have g i=1 s i Y i + g i=1 s i Z i ≤ I.

We interpret the last inequality as {s

i Y i } g i=1 {s i Z i } g
i=1 being a partial POVM, and we apply the Naimark dilation theorem (see [NC10, Section 2.2.8] or [Wat18, Theorem 2.42]). Hence, there exist an isometry V : C n → C n ⊗ C 2g+1 and 2g mutually orthogonal projections

P i , Q i ∈ M n(2g+1) such that s i Y i = V * P i V and s i Z i = V * Q i V . We thus have s i X i = V * (P i -Q i )V ,
and the operators R i := P i -Q i are commuting, normal and with joint spectrum in D ,g (1). Thus, with E the joint spectral measure of R,

g i=1 A i ⊗ s i X i = g i=1 A i ⊗ V * R i V = D A (1) g i=1 A i y i ⊗ V * dE(y)V ≤ I ⊗ I.
This shows that sMatBall g (k) ⊆ D A (k) and the assertion follows as k was arbitrary.

Remark 7.8. Corollary 7.17 of [START_REF] Davidson | Dilations, inclusions of matrix convex sets, and completely positive maps[END_REF] shows that (1/g, . . . 1/g) ∈ ∆ D A for all A ∈ (M sa d ) g such that D A (1) is invariant under projection onto some orthonormal basis. This result thus holds in particular for the matrix diamond, but also for more general spectrahedra. It corresponds to the observation of Remark 3.3 that (1/g, . . . 1/g) ∈ Γ(g, d). In the concrete situation that D A = D ,g , the statement of Theorem 7.7 is much stronger, as one might expect.

Upper bounds

We present in this section two upper bounds (i.e. containing sets) for the Γ and Γ 0 sets, one coming from quantum information theory [START_REF] Zhu | Information complementarity: A new paradigm for decoding quantum incompatibility[END_REF] and another one coming from matrix convex set theory [START_REF] Passer | Minimal and maximal matrix convex sets[END_REF]. These two upper bounds are interesting in two different regimes: the first one applies when the number of POVMs is larger than the dimension of the quantum system, while the second one applies in the complementary regime, where the dimension is large with respect to the number of POVMs. Another important difference between the two results below is that the first one (Theorem 8.2) deals with the set Γ lin , while the second one (Theorem 8.8) deals with the set Γ 0 . 8.1. Zhu's necessary condition for joint measurability. We start by recalling Zhu's incompatibility criterion from [START_REF] Zhu | Information complementarity: A new paradigm for decoding quantum incompatibility[END_REF]; see also [START_REF] Zhu | Universal steering criteria[END_REF] for the mathematical details. To do so, define for a non-zero operator

A ∈ M d , G(A) := |A A| tr[A] ∈ M sa d 2 ,
where |A ∈ C d 2 is the vectorization of the matrix A. In the same vein, if

A • := A -tr[A]/dI denotes the traceless version of A, let G(A) := |A • A • | tr[A] ∈ M sa d 2 .
We also extend additively the definitions above to POVMs

G # ({E i }) := i G # (E i ),
where G # denotes either G or G. Using the remarkable fact that the functions G # are subadditive, Zhu has showed the following result in [START_REF] Zhu | Information complementarity: A new paradigm for decoding quantum incompatibility[END_REF]equations (10,11)].

Proposition 8.1. If a set of g POVMs {E (1) }, . . . , {E (g) } on M d are compatible, then

min{tr[H] : H ≥ G({E (i) }), ∀i ∈ [g]} = 1 + min{tr[H] : H ≥ G({E (i) }), ∀i ∈ [g]} ≤ d.
It turns out that the semidefinite program appearing in the result above is particularly easy in the case where the d 2 × d 2 matrices G({E (1) }), . . . , G({E (g) }) have orthogonal supports; if that is the case, then the optimal H is the sum of the matrices G({E (i) }), and the condition above reads

g i=1 tr G({E (i) }) ≤ d -1.
In order to exploit this phenomenon, let G max (d) be the maximal integer g such that there exist E 1 , . . . , E g non-trivial orthogonal projections in M d with the property d tr[

E i E j ] = (tr[E i ])(tr[E j ]) for all 1 ≤ i < j ≤ g.
Theorem 8.2. For all dimensions d and all 1 ≤ g ≤ G max (d), we have

Γ lin (g, d) ⊆ √ d -1QC g .
Proof. Let g, d be as in the statement, and consider the 2-outcome POVMs

{E i , I d -E }, i ∈ [g],
where

E i are such that d tr[E i E j ] = (tr[E i ])(tr[E j ]) for all 1 ≤ i < j ≤ g. Since the effects E i
are non-trivial orthogonal projections, the previous condition is equivalent to tr E • i E • j = 0, for all i = j. Fix s ∈ Γ lin (g, d); from the definition of the set Γ lin (g, d), it follows that the effects

s i E i + (1 -s i )d -1 tr[E i ]I
g i=1 tr G({s i E i + (1 -s i )d -1 tr[E i ]I, s i (I d -E i ) + (1 -s i )d -1 (d -tr[E i ])I}) ≤ d -1.
Let us compute, for fixed i, the general term in the sum above. Start by computing

G({E i , I d -E i }) = |E • i E • i | tr[E i ] + |(I d -E i ) • (I d -E i ) • | d -tr[E i ] = d|E • i E • i | tr[E i ](d -tr[E i ])
.

Using that E i is a (non-trivial) projection, we get tr G({E i , I d -

E i }) = d tr (E • i ) 2 tr[E i ](d -tr[E i ]) = d[tr E 2 i -tr[E i ] 2 /d] tr[E i ](d -tr[E i ]) = 1.
For the noisy version, mixing with the identity does not change the trace, hence

G({s i E i + (1 -s i )d -1 tr[E i ]I,s i (I d -E i ) + (1 -s i )d -1 (d -tr[E i ])I}) = ds 2 i |E • i E • i | (tr[E i ])(d -tr[E i ]) = s 2 i G({E i , I d -E i }).
and thus, taking the trace tr G({s

i E i + (1 -s i )d -1 tr[E i ]I, s i (I d -E i ) + (1 -s i )d -1 (d -tr[E i ])I}) = s 2 i . Zhu's condition (7) implies thus g i=1 s 2 i ≤ d -1,
proving the claim.

Remark 8.3. An analysis of the proof above shows that the same result holds for the set Γ 0 instead of Γ lin , but with an extra restriction on the operators E i : we must ask that tr[E i ] = d/2 for all i ∈ [g]. We leave the existence of large tuples of such operators as an open problem.

Let us now discuss the function G max (d). First, note that in order for the upper bound in the result above to be non-trivial, we must have G max (d) ≥ d. Below, we give two lower bounds on the function G max , conditional on the existence of mutually unbiased bases (MUBs) and symmetric informationally complete POVMs (SIC-POVMs).

Recall that k orthonormal bases {x

(i) j } d j=1 , i ∈ [k]
are called mutually unbiased if and only if for all i 1 = i 2 and all j 1 , j 2 , | x

(i 1 ) j 1 , x (i 2 ) j 2 | = 1/ √ d.

Proof.

Let

y i = ( √ t, √ 1 -tx i ) for i ∈ [d 2 ]
, where t ∈ [0, 1] is a parameter. Using the fact that the x i are a SIC-POVM, we have that | y i , y j | 2 = t + (1 -t)/(d + 1), so for t = 1/d 2 , | y i , y j | 2 = 1/d. Setting E i = |y i y i | then proves the claim.

Pairwise anti-commuting unitary operators and spectrahedral inclusion constants.

We present now a different type of upper bound, this time on the set ∆ 0 = Γ 0 . What follows is based on [PSS18, Theorem 6.6]. We adapt the proof there to our setting by taking into account the system dimension d. The main ingredient of the construction in [START_REF] Passer | Minimal and maximal matrix convex sets[END_REF] is the following Hurwitz-Radon like result.

Lemma 8.6. [START_REF] Maxwell | Note on an algebraic theorem of Eddington[END_REF] or [Hru16, Theorem 1] For d = 2 k , k ∈ N 0 , there exist 2k+1 anti-commuting, self-adjoint, unitary matrices F 1 , . . . , F 2k+1 ∈ U d . Moreover, 2 k is the smallest dimension where such a (2k + 1)-tuple exists.

A (2k + 1)-tuple as above is sometimes called a spin system in operator theory, see, e.g. [Pis03]. One can easily construct such matrices recursively, as follows. For k = 0, simply take

F (0) 1 := [1]. For k ≥ 1, define F (k+1) i = σ X ⊗ F (k) i ∀i ∈ [2k + 1] and F (k+1) 2k+2 = σ Y ⊗ I 2 k , F (k+1) 2k+3 = σ Z ⊗ I 2 k ,
where σ X,Y,Z are the Pauli matrices

σ X = 0 1 1 0 , σ Y = 0 -i i 0 and σ Z = 1 0 0 -1 .
For example, we have F

(1)

1 = σ X , F (1)
2 = σ Y , F (1)
3 = σ Z , and

F (2) 1 = σ X ⊗ σ X , F (2) 2 = σ X ⊗ σ Y , F (2) 3 = σ X ⊗ σ Z , F (2)
4 = σ Y ⊗ I 2 , F (2)
5 = σ Z ⊗ I 2 . Remark 8.7. Note that our construction differs from the one in [START_REF] Passer | Minimal and maximal matrix convex sets[END_REF], because we aim for the smallest dimension which contains g anti-commuting, self-adjoint and unitary elements. This way, we obtain d ≥ 2 (g-1)/2 instead of d ≥ 2 g-1 in the next theorem.

Theorem 8.8. Let g ≥ 2, d ≥ 2 (g-1)/2 and consider s ∈ R g + such that for any spectrahedron D A defined by traceless matrices A i ∈ M d , D ,g (1) ⊆ D A (1) implies s • D ,g ⊆ D A . Then, i s 2 i ≤ 1. In terms of inclusion constants, we have ∀d ≥ 2 (g-1)/2 , ∆(g, d) ⊆ ∆ 0 (g, d) ⊆ QC g .

Proof. Let us consider g anti-commuting, self-adjoint, unitary matrices F 1 , . . . , F g ∈ U d as in the construction following Lemma 8.6; these matrices also enjoy the property of being traceless when g ≥ 2. Let D F be the spectrahedron defined by the matrices F i , where F i is the entry-wise complex conjugate of F i . Since the matrices F i are unitary, it is clear that D ,g (1) ⊆ D F (1). Assume now that s•D ,g ⊆ D F for some non-negative g-tuple s. Put ŝ := s/ s 2 . We claim that (ŝ i F i ) g i=1 ∈ D ,g . Indeed, for any choice of signs ε i , we have

g i=1 ε i ŝi F i ∞ = g i=1 ε i ŝi F i 2 1/2 ∞ = g i=1 ŝ2 i I + i =j ε i ε j ŝi ŝj F i F j 1/2 ∞ = g i=1 ŝ2 i I 1/2 ∞ = 1.
In the equality above, we have used the fact that the cross terms in the sum obtained by expanding the square vanish; it is this behavior of the matrices F i that renders them useful in operator theory.

From the hypothesis, it follows that (s i ŝi F i ) g i=1 ∈ D F ; in particular, we have

g i=1 s 2 i s 2 F i ⊗ F i ∞ = g i=1 s 2 i s 2 = s 2 ≤ 1,
which is the conclusion we aimed for. In the equation above, we have used the following fact (see [PSS18, equation (5.4)] for the corresponding statement): for non-negative scalars a 1 , . . . , a g ,

g i=1 a i F i ⊗ F i ∞ = g i=1 a i .
The fact that the left hand side in the equality above is smaller than the right hand side follows from the triangle inequality. The reverse inequality follows from taking the scalar product against the maximally entangled state

ω d := 1 d d i,j=1
(e i ⊗ e i )(e j ⊗ e j ) * , for some orthonormal basis {e

i } d i=1 of C d .
Putting together the result above with Theorem 7.7, we derive the following equality, one of the main results of this paper.

Corollary 8.9. For any g ≥ 2 and any d ≥ 2 (g-1)/2 , we have

∆(g, d) = Γ(g, d) = ∆ 0 (g, d) = Γ 0 (g, d) = QC g .
Remark 8.10. If the dimension bound d ≥ 2 (g-1)/2 holds, the matrices (F 1 + I d )/2 ⊕ 0, . . . , (F g -I d )/2 ⊕ 0 considered in this section are the most incompatible g-tuple of d × d quantum effects. Indeed, for any direction ŝ ∈ QC g , ŝ = 1, it follows from Corollary 8.9 that the g-tuple (t 1 ŝ1 (F 1 + I d )/2, . . . , t g ŝg (F g + I d )/2) is compatible if and only if t i ≤ 1 for all i ∈ [g]. We would also like to point out that, for d = 2 and g = 3, g = 2, the claim above corresponds to the maximal incompatibility of the measurements corresponding to the Pauli observables.

Discussion

In this final section, we would like to put the results obtained in this work in perspective, and compare them with previously known bounds. We also list and discuss some questions left open in this work. 9.1. The shape of the different compatibility regions. We start by listing some previously known results on the different sets Γ # considered in this work. Let us remind the reader that our primary focus was on the sets Γ(g, d), because of their connection to the inclusion problem for free spectrahedra. In the quantum information community, the sets Γ all and Γ lin play a very important role, because the most general type of trivial noise is allowed in the former case, and because of the linear structure in the latter case. Previously, mainly the cases of small g, d have been considered in the literature. General lower bounds have been shown mostly using tools from symmetric approximate cloning, while upper bounds were rarely considered in the general case (we are considering here only the case of 2-outcome POVMs).

Let us first discuss the results in the literature for small g, d. Using an argument which connects joint measurability with a violation of the CHSH inequality [WPGF09], it was shown in [BHSS13] that QC 2 ⊆ Γ(2, d) for all d ∈ N. Further, it was shown that also Γ all (2, 2)

⊆ QC 2 [BH08, Proposition 3] and Γ 0 (2, 2) ⊆ QC 2 [BH08, Proposition 4]. Therefore, for d ≥ 2, an application of Proposition 3.6 yields Γ(2, d) = Γ all (2, d) = Γ 0 (2, d) = QC 2 .
Less was known in the g ≥ 3 case, since the connection to the CHSH inequality no longer holds [BV18, HQB18]. From [START_REF] Busch | Unsharp reality and joint measurements for spin observables[END_REF] (see also [BLPY16, Section 14.4]), it follows that QC 3 ⊆ Γ 0 (3, 2). Moreover, [BA07] show that Γ 0 (3, 2) ⊆ QC 3 , hence Γ 0 (3, 2) = QC 3 . This was improved in [PG11, Section XI] to show also Γ all (3, 2) ⊆ QC 3 . Using the results of this paper and combining them with the findings above, we can prove a stronger statement. An application of Theorem 7.7 together with Proposition 3.6 yields

Γ(3, d) = Γ all (3, d) = Γ 0 (3, d) = QC 3 .
In the general case, the lower bounds came mainly from symmetric cloning [HSTZ14], see Proposition 6.2.

Let us now discuss the contributions of this paper to both the theory of joint measurability and free spectrahedra. As discussed in the introduction, our main insight, the relation between the joint measurability of 2-outcome POVMs and the inclusion problem for the matrix diamond, allows us to translate results from one field to the other. Arguably one of the main results in this work is the lower bound obtained in Theorem 7.7. Our theorem is based on results about inclusion of free spectrahedra derived in [START_REF] Passer | Minimal and maximal matrix convex sets[END_REF], which can be transferred to the quantum setting. Together with the upper bound from [START_REF] Passer | Minimal and maximal matrix convex sets[END_REF] and the lower bound from [START_REF] Helton | Dilations, linear matrix inequalities, the matrix cube problem and beta distributions[END_REF], we obtain a much better understanding of the sets Γ(g, d). We present in Figure 4 our current picture of the sets Γ(g, d), or, equivalently, of the sets of inclusion constants ∆(g, d). The curves d = 2 (g-1)/2 and d = √ g/2 delimit three regions: above the first curve, we know that the set Γ(g, d) is equal to QC g , the positive part of the unit Euclidean ball, while below the second curve, we know the inclusion QC g ⊆ Γ(g, d) to be strict. Below the curve d = 2 (g-1)/2 , the upper bound from Theorem 8.8 does not apply, while below the second curve, d = √ g/2, the lower bound 1/(2d) in the symmetric case is larger than the lower bound 1/ √ g coming from the quarter-circle QC g . It is worthwhile to mention that the best lower bound for the sets Γ(g, d) coming from symmetric cloning (second line in Proposition 6.2) is worse than the best of the two bounds coming from spectrahedron theory:

g + 2d g(1 + 2d) ≤ max 1 √ g , 1 2d .
However, in the asymmetric regime, cloning gives non-trivial lower bounds, since the 1/(2d) bound from Proposition 7.2 is not applicable for asymmetric tuples. We expect to obtain non-trivial results as soon as g+2d g(1+2d) > 1/ √ g, that is as soon as g > 4d 2 . As an example, we plot in Figure 3, for even g = 2g 0 , the set of points of the form (s, . . . , s g 0 times , t, . . . , t g 0 times

) belonging to Γ(g, 4) and to QC 4 . . The range of parameters (s, t) for which the point (s, . . . , s, t, . . . , t) ∈ [0, 1] g , having the same number of s and t, belongs to Γ clone (g, 4) ⊆ Γ(g, 2) (red curve) and QG g (blue, dashed curve), for g = 2, 16, 20, 40. Note that the bound from asymmetric cloning becomes better for g > 4d 2 = 16.

The main question left open in this work is to compute the sets Γ(g, d) = ∆(g, d) for the range of parameters g, d where the upper and lower bounds from [START_REF] Passer | Minimal and maximal matrix convex sets[END_REF] do not agree. Question 9.1. Compute, for d < 2 (g-1)/2 , the sets Γ(g, d) = ∆(g, d).

Regarding the sets Γ lin (g, d), the lower bounds coming from cloning (see Section 6) were already known in the literature; in particular, we recall that, in the symmetric case, ∀d ≥ 2 (g-1)/2 , Γ lin (g, d) ⊆ Γ 0 (g, d) ⊆ QC g .

Finally, regarding the sets Γ all , allowing the most general type of noise, the new lower bounds obtained in this work are precisely the same as the ones for the sets Γ. Importantly, for all g, d, we have QC g ⊆ Γ all (g, d). Note that the bound above was previously known only in the case g = 2. Moreover, in the symmetric case, we have max 1 √ g , 1 2d

(1, 1, . . . , 1 g times

) ⊆ Γ all (g, d).

Upper bounds can be obtained via the map F from Proposition 3.4 from upper bounds for Γ. For example, in the symmetric case, using Theorem 8.8 we get ∀d ≥ 2 (g-1)/2 , s(1, 1, . . . , 1 g times

) ⊆ Γ all (g, d) =⇒ s ≤ 2 1 + √ g , which is roughly two times the lower bounds above. 9.2. The shape of the inclusion sets. Hitherto, we have discussed the implications of this work for quantum information theory. However, our results also shed new light on ∆(g, d) and ∆ 0 (g, d), the sets of inclusion constants for the matrix diamond. As ∆ 0 (g, d) = Γ 0 (g, d) by Theorem 5.7, we have the lower bounds ∀g, d ≥ 1 QC g ⊆ Γ(g, d) ⊆ ∆ 0 (g, d)

∀g, d ≥ 1 Γ clone (g, d) ⊆ ∆ 0 (g, d).

Looking at the symmetric case, for which s(1, . . . , 1) ∈ Γ clone if and only if s ≤ (g + d)/(g(1 + d)), we see that this is larger than 1/ √ g if and only if d ≤ √ g. Therefore, both lower bounds are non-trivial. We remark that for all d, g ≥ 1,

1 2d ≤ g + d g(1 + d) .
Therefore, the result from symmetric cloning is always stronger than the one from [START_REF] Helton | Dilations, linear matrix inequalities, the matrix cube problem and beta distributions[END_REF] (see Corollary 7.4). In terms of upper bounds, we only have that ∀d ≥ 2 (g-1)/2 ∆ 0 (g, d) ⊆ QC g from the work of Passer et al. (Theorem 8.8).

Regarding the sets ∆(g, d) one obtains new lower bounds in the asymmetric setting, using cloning and the inclusion Γ clone (g, 2d) ⊆ ∆(g, d), see Figure 3. 9.3. Outlook: POVMs with more outcomes. In this work, we have focused on binary measurements. However, our methods also work for measurements with more outcomes. For example, consider the case of a binary POVM { E, I -E } and a POVM with three outcomes { F 1 , F 2 , I -F 1 -F 2 }. Then, it can be shown that joint measurability is equivalent to the inclusion problem of the free spectrahedra defined by

  Lemma 4.4. Let A ∈ (M sa D ) g and B ∈ (M sa d ) g . Further, let D A (1) be bounded. Then D A (n) ⊆ D B (n) holds if and only if Φ as given above is n-positive. In particular, D A ⊆ D B if and only if Φ is completely positive.

  Corollary 4.6. Let A ∈ (M sa D ) g and B ∈ (M sa d ) g . Moreover, let D A (1) be bounded. Then D A (d) ⊆ D B (d) if and only if D A ⊆ D B . Proof. From Lemma 4.4, D A (d) ⊆ D B (d) is equivalent to Φ being d-positive. Since Φ maps to M d , this is equivalent to complete positivity of the map [Pau03, Theorem 6.1]. The claim then follows by another application of Lemma 4.4.

  Theorem 6.1 ([Key02, Theorem 7.2.1]). For a quantum channel T : M ⊗N d → M ⊗g d , consider the quantities

From [ Wer98 ,

 Wer98 Equation 3.7], we know further that ρ := tr i c [T (ρ)] = γρ + (1 -γ)I/d, where γ = (g + d)/(g(1 + d)) (for N = 1). Here, tr i c means tracing out all systems but the i-th one. Going to the dual picture, we can compute that for E ∈ M d tr[ρE] = tr ρ γE + (1 -γ) tr E d I .

Figure 2 .

 2 Figure 2. Left: the sets Γ clone (2, d) for d = 2, 3, 5, 10, 100 as subsets of Γ(2, d), which is a quarter-circle for all d ≥ 2 (see Corollary 8.9). Right: the cuts Γ clone (3, d)∩ {(s, t, t)} for d = 2, 3, 5, 10, 100.

  Figure3. The range of parameters (s, t) for which the point (s, . . . , s, t, . . . , t) ∈ [0, 1] g , having the same number of s and t, belongs to Γ clone (g, 4) ⊆ Γ(g, 2) (red curve) and QG g (blue, dashed curve), for g = 2, 16, 20, 40. Note that the bound from asymmetric cloning becomes better for g > 4d 2 = 16.

Figure 4 .

 4 Figure 4. The sets Γ(g, d).

  -1, -1, 2, -1, -1], A 2 = 2 3 diag[-1, 2, -1, -1, 2, -1], A 3 = diag[1, 1, 1, -1, -1, -1]andB 1 = 2E -I, B 2 = 2F 1 -That is, D A (1) ⊆ D B (1) if and only if E, I -E, F 1 , F 2 and I -F 1 -F 2 are quantum effects and D A ⊆ D B if and only if { E, I -E } and { F 1 , F 2 , I -F 1 -F 2 } are jointly measurable POVMs.Inclusion constants then correspond again to mixing with I/2 (for the binary POVM) and I/3 (for the three-outcome POVM), respectively. This idea is explored in detail in [BN18].

  Let further E 1 , . . . , E g ∈ Eff d . By the choice of s and t, we know that the s i E i + (1 -s i )I/2 and the t i E i + (1 -t i )I/2 are each jointly measurable and give rise to joint POVMs R i 1 ,...,ig and R i 1 ,...,ig , respectively. Then, λR i 1 ,...,ig + (1 -λ)R i 1 ,...,ig

	Proposition 3.2. Γ # (g, d) is convex for d, g ∈ N and # ∈ { all, ∅, lin, 0, clone }.
	Proof. We only prove the proposition for Γ(g, d) here, because the proofs for the other sets are very
	similar. Let s, t ∈ Γ(g, d) and λ ∈ [0, 1].

  are jointly measurable quantum effects.Proof. Let us start with the first point. Since D ,g (1) is a convex polytope, we need to check inclusion only at extreme points. That means that the first assertion holds if and only if ±e i ∈ D 2E-I for all i ∈ [g],

  are compatible, and thus, by Proposition 8.1,

	(7)

  The maximal number of MUBs in C d is d + 1, and this bound is attained if d is a prime power; very few other existence results are known, see [DEB Ż10] for a review. If there exist k MUBs in dimension d, then G max (d) ≥ k. This follows by setting E i = j∈J i |x Recall that a unit rank POVM {d -1 |x i x i |} d 2 i=1 is called symmetric and informationally complete if and only if for all i = j, | x i , x j | 2 = 1/(d + 1). Whether a SIC-POVM exists in dimension d is a challenging question and an ongoing research subject. Analytic examples of SIC-POVMs have been constructed for d = 1, . . . , 21, 24, 28, 30, 31, 35, 37, 39, 43, 48, 124, 323, and numerical constructions exist for much larger values of d, see [SG10] for a review and [ACFW18] for some recent progress. The use of SIC-POVMs yields the following corollary of Theorem 8.2. Corollary 8.5. If there exists a SIC-POVM {x i } d 2 i=1 in dimension d, then G max (d + 1) ≥ d 2 and thus, for all g ≤ d 2 , we have Γ lin (g, d + 1) ⊆

	(i) j	x	(i) j |, where {x (i) j } d j=1 is the i-th MUB and J i is some non-trivial subset
	of [d]. We record next the following consequence of Theorem 8.2.
	Corollary 8.4. If d = p n is a prime power, then, for all g ≤ d + 1,
			Γ lin (g, d) ⊆	√	d -1QC g .
				√
				dQC g .

We thank Teiko Heinosaari for pointing this out to us.

Acknowledgments. A.B. would like to thank Michael M. Wolf for pointing out the connection between joint measurability and the CHSH inequality. A.B. acknowledges support from the ISAM Graduate Center at the Technische Universität München. I.N. would like to thank Teiko Heinosaari for many inspiring discussions about joint measurability; in particular, he pointed out to us the fact in Remark 5.4. I.N.'s research has been supported by the ANR projects StoQ (grant number ANR-14-CE25-0003-01) and NEXT (grant number ANR-10-LABX-0037-NEXT), and by the PHC Sakura program (grant number 38615VA). I.N. also acknowledges the hospitality of the Technische Universität München. A.B. and I.N. would like to thank the Institut Henri Poincaré in Paris for its hospitality and for hosting the trimester on "Analysis in Quantum Information Theory", during which part of this work was undertaken.