Device-independent quantiﬁcation of measurement incompatibility

,

In this sense, a violation of a Bell inequality is also a DI witness of incompatibility, as incompatible measurements are necessary to observe it [14,37,38].In this work, we address the quantitative question: How incompatible the underlying measurements have to be in order to observe a certain quantum violation of a Bell inequality?A central tool in our investigation is the notion of moment matrix, that has wide applications in the characterization of quantum correlations and DI approaches [30,[39][40][41].Here, we introduce the measurement moment matrix (MMM) that allows us to quantify several quantities that are formulated via semidefinite programming (SDP) [42] in terms of measurement effects, such as incompatibility robustness [43][44][45], genuinemultipartite incompatibility [46], and similar quantities [33,47,48].
Our results allows for investigations beyond the DI scenario.In fact, due to its generality the idea of MMMs can be straightforwardly extended to the semi-DI approach, i.e., where the dimension of quantum system is assumed to be known [49,50], to investigate the role of dimension constraints or even non-projectivness in measurement incompatibility, and it can be extended even to the prepare-and-measure scenario.

II. INCOMPATIBLE MEASUREMENTS
Let us start by briefly reviewing the concept of measurement incompatibility.Consider a quantum measurement described by a positive-operator-valued measure (POVM) {E A a|x } a for a given x, where the indices x ∈ X and a ∈ A, label the measurement settings and outcomes of the measurement, respectively.The operators E A a|x , called effect operators are positive semidefinite, i.e., E A a|x 0 ∀a, x, and satisfy the normalization condition, x is called a measurement assemblage [51].A measurement assemblage is said to be compatible or jointly measurable if it can be written as [52,53] where {G λ } λ is a valid POVM and P(a|x, λ) are nonnegative numbers such that ∑ a P(a|x, λ) = 1 for all x, λ.Physically, joint measurability means that the statistic of each POVM in the assemblage can be obtained by classically post-processing the statistic of a parent POVM {G λ } λ , irrespective of the state.Several incompatibility measures have been proposed in the literature (see Ref. [54] for an overview).Here, we choose the incompatibility robustness [43][44][45] defined as where the minimum is take w.r.t.any arbitrary assemblage {N a|x } a,x .Here, IR is related to the minimum noise necessary for {E A a|x } a,x to become jointly measurable.From a quantum information perspective, IR quantifies the advantage that {E A a|x } a,x provides w.r.t.jointly measurable assemblages for a certain state-discrimination task [19][20][21][22][23][24][25].Moreover, it can be efficiently computed via SDP [42,44]: where λ := (λ 1 , λ 2 , ..., λ |X | ), λ i ∈ A, encodes the deterministic strategies.

III. THE MEASUREMENT MOMENT MATRICES
As first noted by Moroder et.al. [30], moment matrices can be interpreted as the application of a completely positive map to a (set of) positive operator(s), such as a quantum state [30,39,40,55] or steering state ensembles [34,35].Here, we define the measurement moment matrices (MMMs) by applying a completely positive map on POVMs where the map is obtained by first embedding the system A in the tensor product with a second identical system B, i.e., E A a|x → E A a|x ⊗ 1 1 B , which is a completely positive map, and then applying the Kraus operators K n : AB → AB defined as K n := ∑ i |i ABAB n|( AB ) 1 2 S i , with {|i } i and {|n } n the orthonormal bases for the output space AB and the input space AB, respectively, and {S i } is a sequence of operators to be specified later.In this way, one obtains a moment matrix for each a, x.In what follows, we simply use the symbol χ[E A a|x ], or even χ, when there is no risk of confusion.The MMM χ is a type of localizing matrix, proposed in the context of noncommutative polynomial optimization [41], but here we define them from the perspective of measurement effects.In particular, their formulation is independent of the standard Navascués-Pironio-Acín (NPA) moment matrix [40,55].
We choose the operators {S i } as products of POVM elements, e.g., ), etc.}, and following the convention of Ref. [30]: a level is denoted by {S a|x 's and E B b|y 's.Even though the operators AB , {E A a|x } a,x and {E B b|y } b,y are uncharacterized, one is still able to obtain specific entries in χ, such as those corresponding to accessible statistics in a DI setting, i.e., P(a, b|x, y) = tr(E A a|x ⊗ E B b|y AB ).
Moreover, by the Neumark dilation [56], any POVM can be realized by a projective measurement in a higher dimensional space, implying conditions such as 0 = tr Moreover, since the MMMs are obtained by applying a completely positive map on valid POVMs (see Eq. ( 4)), each χ is positive semidefinite by construction.It is convenient to decompose χ into the charac-terized parts and unknown parts [30]: where all of F a,b,x,y and F v are symmetric matrices.The complex numbers u v represent all the uncharacterized variables.

IV. DEVICE-INDEPENDENT QUANTIFICATION OF MEASUREMENT INCOMPATIBILITY
Via the MMM, we are able to define, for any SDP involving effect operators, its DI relaxation, i.e., another version of the problem involving only DI assumptions.As an example, we will show below how to define the incompatibility robustness.Several other examples, such as incompatibility jointly measurable robustness, incompatibility probabilistic robustness, incompatibility random robustness, and the incompatibility weight, are described in App. A. The problem in Eq. ( 3) is mapped to where χ[G λ ] 1 1 := tr(G λ ⊗ 1 1 B AB ).The objective function is the same as that of Eq. (3) due to the fact that tr The first three constrains are directly obtained from the three constraints in Eq. ( 3).The rest are associated with, respectively, normalization of POVMs, positivity of POVMs, and the observed nonlocal correlation.The above problem is not an SDP yet, since the third constraint in Eq. ( 7) is quadratic.To tackle this problem, we relax the third constraint by keeping only the characterized terms in χ[ 1 1].Namely, the relaxed constraint becomes: where, with some abuse of notation (since no elements in χ[G λ ] are actually fixed), we mean to retain only the constraints associated with entries in χ[1 1] fixed as in Eq. ( 6), i.e., with the observed probabilities P obs (a, b|x, y).

Given
P obs (a, b|x, y) The solution obtained above, denoted by IR DI , is a lower bound on IR of the underlying measurement assemblage.In other words, it tells us the minimum degree of measurement incompatibility present when observing a certain nonlocal correlation.
An analogous SDP can be used for bounding from below the measurements incompatibility necessary for a given violation of Bell inequality.In this case, only the Bell value, i.e., I(P), is given and not P obs (a, b|x, y).As a consequence one simply removes entirely the third constraint in Eq. ( 7), as χ[1 1] fixed is not characterized.Alternatively, by changing the objective function one may ask what is the maximal violation of a Bell inequality for a given value IR 0 of the robustness.It can be easily shown that for each pair (I(P), IR 0 ) a feasible solution of one SDP is also a feasible solution of the other, hence, they characterize the same set.See App.B for more details.
The formulation with the fixed IR 0 , however, turns out to more more convenient, as it removes the nonlinearity in the previous SDP.In fact, the substitution ∑ λ χ[G λ ] 1 1 − 1 = IR 0 , allows us to write the third constraint of Eq. (7) as We apply this method to the tilted-CHSH inequality [57], see the next section for a detailed explanation and Fig. 2 for a summary of the results.What we want to highlight now, is that for the simple case analyzed in Fig. 2, the SDP in Eq. ( 7) already provides an exact solution, despite the relaxation of the nonlinear constraint.In contrast, for the case of genuine multipartite incompatibility robustness, discussed in Sec.VI below, we see that different bounds arise when the same constraint is taken into account or not, see also Apps.C and D.
For each value of α one can obtain the optimal state and the optimal pair of measurements (unique up to local isometries) providing the maximal quantum violation.The value of Bob's robustness for a given θ coincides with its DI bound computed via the MMM assuming the corresponding distribution P(a, b|x, y) (see Fig. 2).In the same figure, we also plot the DI bound of IR obtained via the nonlocality robustness (NLR) [33] method.The NLR method, as well as another method proposed for the DI lower bound of incompatibility, i.e., the assemblage moment matrix (AMM) [34,35] method, are based on the connection between steering and incompatibility [15,16,44].In contrast, the MMM relies on the construction of a moment matrix directly from the measurement operators.In App.E, we show that the AMM can be identified with a special case of a MMM.Hence, it can never provide a better bound for incompatibility.In addition, we explicitly show via the I 3322 inequality [60], that the MMM provides strictly better bounds.

VI. QUANTIFICATION OF GENUINE MULTIPARTITE INCOMPATIBILITY ROBUSTNESS
Here, we show how the MMM can be used to quantify the genuine multipartite incompatibility robustness (GMIR) recently introduced by Quintino et al. [46].An example is provided in Fig. 3 for different Bell inequalities.All the results presented, use the maximization of the Bell violation for a given robustness, see Eq. ( 13) below.As we discuss in App.C, the results obtained with this method are provable better than those obtained minimizing the robustness for a given Bell violation.Finally, in addition to being able to quantify the GMIR, our method can also improve the thresholds for its detection.We compare ours with those computed in Ref. [46].
A measurement assemblage of three measurements {{E a|x } a } x=1,2,3 is said to be genuinely triplewise incompatible [46] if it is impossible to write it as a convex mixture of three measurement assemblages, each containing a different pair of compatible measurements [46].More concretely, if there exists three assemblages (1,3), (2,3) such that {J st a|s } a and {J st a|t } a are jointly measurable for any pair s, t and each E a|x can be written as for some probabilities p 12 , p 23 , and p 13 that respect p 12 + p 23 + p 13 = 1, we will say that {{E a|x } a } x=1,2,3 are not genuine triplewise incompatible.This condition can be written in a SDP form (see Ref. [46] and App.C for a brief self-contained summary), which leads to a SDP formulation of the robustness as Given {E a|x } a,x , and variables Applying the same argument as the one for the standard incompatibility robustness above SDP can have a DI relaxation via moment matrices Given P obs (a, b|x, y), and for (s, t, x) = (1, 2, 3), (1, 3, 2), (2, 3, 1); for all x, P(a, b|x, y) = P obs (a, b|x, y).
Again, one can compute the maximum of a Bell in-  [46]) and better or equal than the NPA hierarchy with additional commutativity constraints (the set Q 2conv J M defined in [46], see App.D for details).We recall that the bound for I E is tight, as proven in [46].equality I(P) for a given robustness IR 0 as Given IR 0 , and with sum over (s, t) = (1, 2), (1, 3), (2,3); for (s, t, x) = (1, 2, 3), (1, 3, 2), (2, 3, 1); As we mention above, in this case one can show that the problem in Eq. ( 13), namely, the maximization of the Bell violation for a given robustness, provides better bounds than the inverse problem, namely, the minimization of the robustness for a given Bell violation.This is due to the possibility of removing the nonlinear constraint present in the intermediate formulation.More details can be found in App. C.
In addition to the quantitative results plotted in Fig. 3, our method is also able to improve the numerical thresholds for the detection of GMI previously found in Ref. [46], see Tab.I and App.D for more details.

VII. SEMI-DEVICE-INDEPENDENT APPROACH AND PROJECTIVE MEASUREMENTS
Another advantage of our method is that it admits a direct extension to semi-device-independent (SDI) characterization of incompatibility.This can be achieved by employing ideas from the Navascués-Vertesi hierarchy [61], which generalizes the NPA hierarchy and aims to bound the set of finite dimensional quantum correlations.The key idea of this generalization comes from the fact that moment matrices generated by states and measurements of a given Hilbert space dimension d span only a subspace S d of the whole space of moment matrices.One can then try to add the corresponding constraint to the problem in Eq. (7).In practice, this is achieved by generating a basis of random moment matrices (e.g., by means of the Gram-Schmidt process) by sampling states and measurements of a given dimension.
In contrast to the DI approach, in which all POVMs can be dilated to projective measurements by increasing the system's dimension, in the SDI approach one can additionally impose the constraint that the measurements E A a|x are projective.
We tried several approaches to the SDI quantification of measurement incompatibility, with and without the assumption of projective measurements.A few of them, which work in the case of Bell inequalities [61], do not generalize to the case of incompatibility quantification, either for fundamental reasons or because they fail to provide an improvement in the numerical results for the cases analyzed.A summary of these approaches is given in App.F.
The most successful approach is the one in which dimension constraints are imposed by requiring that the observed probabilities are generated by a system of bounded dimension.In this case, since we are restricting ourselves to dichotomic measurements, we can use the fact that correlations generated by projections are extremal.Let us denote by Γ ∈ S d the moment matrix generated via the NV method, assuming that the measurements are projective, and Γ P(a,b|x,y) the matrix entry corresponding to the observed probability P(a, b|x, y).The SDP for the computation of the minimal robustness associated to a violation K of a Bell inequality I(P), can be written as where P(a, b|x, y) denotes the entries in the MMM {χ[E A a|x ]} a,x , in the usual DI approach, corresponding to the probability P(a, b|x, y), and Γ, as discussed above, is generated by sampling moment matrices generated with dichotomic projective measurements in dimension d.
Equivalently, one can fix the robustness IR 0 and maximize the Bell inequality violation, as follows

Given
IR 0 max with the same use of notation as above.
In order to compare the different methods, we computed different lower bounds on the incompatibility robustness for a given violation of the I 3322 inequality.First, we tried the dilation method presented in Eq. (F4) (in App.F) for d = 2, which gave no improvement over the standard DI approach.In contrast, the SDP in Eq. ( 14), for d = 2, provided a substantially improved lower bound on the robustness, with respect to the DI case.In addition, we also compare the SDI approach with the one where the additional condition of projective measurements is assumed.With the assumption of projective measurements, we were able to obtain a substantially improved bound for the case of Finally, an analogous procedure allows us to extend the MMM to another typical SDI scenario, namely the prepare-and-measure scenario.More details can be found in App.G.

VIII. CONCLUSIONS AND OUTLOOK
We proposed a framework, the MMM, to quantify the degree of (several notions of) measurement incompatibility in a DI manner.The main idea behind our method is to construct moment matrices by applying a completely positive map on POVMs.Due to the operational characterization of the incompatibility robustness [19][20][21][22][23][24][25], our result also bounds, in a DI scheme, the usefulness of a set of POVMs in the problem of quantum state discrimination.In contrast to previous DI bounds of incompatibility in Refs.[33,34], our method does not rely on any concept of steering, but provides a direct interpretation of the moment matrix as a completely positive mapping of the measurement operators.Our MMM method is shown to outperform both methods in the quantification of incompatibility in simple examples, and we rigorously proven that it always performs better or equal to the method in Ref. [34].Moreover, the MMM method provides a DI bound of the genuine multipartite incompatibility, a recently introduced notion, for which no DI quantifier was known so far, and it improves the known thresholds for the its detection.Finally, given its generality our method is straightforwardly adaptable to include additional constraints such as the system dimension (semi-DI approach), the assumption of projective measurements, and it is applicable to the prepare-and-measure scenario (see the discussion in App.G).
We leave as an open problem to determine the convergence of the proposed hierarchy.Since we could not give either a positive or negative answer to this question, we used the term "relaxation" for the optimization problems throughout the text.However, we would like to point out that at least in the case of tilted CHSH, for which an analytical solution is known, our method recovers the exact relation between the incompatibility robustness and nonlocality (see Fig. 2).
As a future research direction, we would like to investigate the connection between DI and SDI quantifier of incompatibility and self-testing of measurements (see Ref. [62] for a related approach).In fact, in Ref. [54], the authors showed that for the incompatibility robustness, pairs of measurements associated with mutually unbiased bases (MUBs) are the most incompatible in any dimension, even if it is not proven that they are the only ones.In the CHSH scenario, our calculation showed that IR DI saturates IR of a pair of qubit-measurements corresponding to the MUB for the maximal quantum violation of the CHSH inequality (Ref.[34] also saurates this bound).For high dimensional cases, one can use the family of Bell inequalities in Ref. [63] to compute IR DI .Due to the limitation of our computational capacity, we leave this issue for the potential future research.This intuition is further strengthened by the work of Ref. [64], which showed that the assemblage moment matrices proposed in Ref. [34] can be used to self-test state assemblages.Therefore it is natural to ask if the MMMs can be analogously used to self-test quantum measurements.Finally, a possible further extension of this work is in the direction of the SDI characterization of incompatibility in the prepare-and-measure scenario.In fact, it is believed that incompatible measurements are necessary for quantum advantage in the so-called random access codes [65].

Appendix A: Different measures of global incompatibility
In this section, we consider other measures of incompatibility and explicitly write down their DI quantifications in the SDP form.There are robustness-based measures: is jointly measurable, (A1) where the noisy models {N a|x } a,x satisfy different constraints (see, e.g., [54]) and each type of models is denoted by superindices i.The last measure we consider is the incompatibility weight [47].For the simplicity of formulation of the following SDPs we will not write explicitly the variables of optimization.Instead, we specify the input to each SDP next to "Given".

The incompatibility jointly measurable robustness
The noisy assemblage {N a|x } a,x for the incompatibility jointly measurable robustness IR J [33] admits a jointly measurable model.As such, IR J can be com-puted via the following SDP: By applying the MMM and removing the constrains containing quadratic free variables, the solution of the following SDP gives a lower bound on IR J :1 Given P obs (a, b|x, y) a|x ] 0, ∀a, x, P(a, b|x, y) = P obs (a, b|x, y), ∀a, b, x, y, (A3) where χ[G λ ] fixed and χ[H λ ] fixed in the fourth and fifth constraints respectively denote, as in the main text, χ[G λ ] and χ[H λ ] retaining entries whose indices correspond to non-vanishing terms in χ[1 1] fixed .

The incompatibility probabilistic robustness
The noisy model for the incompatibility probabilistic robustness IR P [66] is defined as N a|x = p(a|x) • 1 1 for all a, x, with real numbers p(a|x) satisfying p(a|x) ≥ 0 for all a, x, and ∑ a p(a|x) = 1 for all x.The associated SDP can then written as: By applying the MMM, a DI lower bound can be computed via the following SDP: x, P(a, b|x, y) = P obs (a, b|x, y) ∀a, b, x, y. (A5)

The incompatibility random robustness
The final robustness-based measure is the incompatibility random robustness IR R [44,48], where the noisy assemblage is composed of the white noise: (A6) With the same technique, a DI lower bound on IR R can be computed via the following SDP: a|x ] 0, ∀a, x, P(a, b|x, y) = P obs (a, b|x, y), ∀a, b, x, y. (A7)

The incompatibility weight
The last measure of incompatibility we consider is the incompatibility weight IW [47].Consider that one decomposes E A a|x into E A a|x = tO a|x + (1 − t)N a|x , where {O a|x } a,x is any valid quantum measurement assemblage and {N a|x } a,x is a jointly measurable measurement assemblage.IW is defined as the minimum ratio of O a|x , i.e., the minimum value of t, required to decompose E A a|x .Consequently, IW can be computed via the following SDP: Following the same procedure as in the previous sections, we obtain the following SDP, which can be used to compute a DI lower bound on IW: Given P obs (a, b|x, y) Note that all of above SDPs that compute DI lower bounds on the degree of incompatibility require the detailed information about the observed correlation {P obs (a, b|x, y)} a,b,x,y .If one is merely concerned with a Bell inequality violation without the specific characterization of {P obs (a, b|x, y)} a,b,x,y , the constrains containing χ[1 1] fixed have to be fully removed.

Appendix B: Different constraints on incompatibility robustness
As we discussed in the main text, different relaxations of the following problem exist min a|x ] 0 ∀a, x, P(a, b|x, y) = P obs (a, b|x, y) ∀a, b, x, y, (B1) which are necessary to remove the nonlinear constraint: Moreover, the problem in Eq. (B1) assume the knowledge of the full distribution of probabilities {P obs (a, b|x, y)} a,b,x,y , whereas in some cases, we may want to estimate the robustness simply from the violation of a Bell inequality.
In this case, we want to characterize the set of all possible pairs (IR, I(P)), where IR represents the incompatibility robustness and I(P) the value of some Bell expression.Notice that, even if I(P) is evaluated on a probability distribution P, we are not assuming that such P is directly accessible, the parameter in our problem is only the value of the Bell expression.
The set of valid (IR, I(P)) can be defined by the following SDP (feasibility problem): where the matrices W a,x and the coefficients α a,x are properly chosen to extract the Bell expression from the terms corresponding to probabilities appearing in {χ[E A a|x ]} a,x .It is then clear, then, the (nontrivial) extreme points of this set are equivalently characterized by the following two problems: minimize IR given I(P), and maximize I(P) given IR. (B3) In fact, one may have highly incompatible observables and fail to obtain a highly violation of a Bell inequality due to the low entanglement in the shared state.The problems in Eq. (B3) can be directly solved by transforming the feasibility problem in Eq. (B2).By construction, a feasible solution of one problem is also a feasible solution for the other one, so they characterize the same set of pairs (IR, I(P)).It is important to remark that here we are not using the full duality properties of the SDP, but simply the relation between IR and I(P) encoded in Eq. (B2) and the fact that the problems in Eq. (B3) are sufficient to characterize the nontrivial part of this set.
The formulation with the fixed IR 0 , however, provides an advantage since an extra condition can be imposed.In fact, the substitution ∑ λ χ[G λ ] 1 1 − 1 = IR 0 , allows us to write the third constraint of Eq. (B1) as effectively removing the nonlinearity appearing in the SDP in Eq. (B1).We then have (B4) The fact that the SDP in Eq. (B4) provides a better characterization of the set (IR, I(P)) is confirmed by numerical calculations.First, the incompatibility robustness has been analyzed in Fig. 2, where this distinction is not relevant.However, a characterization analogous to that in Eq. (B3) appears also for the genuine multipartite incompatibility robustness.For that case, we can see directly that the use of the two different formulations provides different results and that the computation for a fixed robustness IR 0 provides a better bound.More details can be found in App. C.

Appendix C: SDP formulation for genuine-multipartite incompatibility
In the following, we recall several results from Ref. [46], in particular the SDPs (C2) and (C4), and discuss their DI relaxation via the MMM.
Following [46], we recall that genuine triplewise in-compatibility , namely, the impossibility of writing a|x 0, ∀a, x; ∑ a J 13 a|x = p 13 1 1, ∀x, where δ a,λ x is the deterministic strategy that assign probability 1 if the x-th component of λ is equal to a.
Clearly, the same argument can be extended to define genuine multipartite incompatibility beyond the triplewise case.
In simple terms, this set is obtained by the NPAhierarchy constraints plus linear constraints corresponding to Bell inequalities involving only x = 1, 2 and all possible dichotomized measurements on Bob's side.
The above definition can be extended to the convex hull of L 12 , L 13 , L 23 , i.e., L 2conv as follows P(ab|xy) belongs to L Q 2conv if: P(ab|xy) ∈ Q, P(ab|xy) = µ 12 P 12 (ab|xy) + µ 13 P 13 (ab|xy) + µ 23 P 23 (ab|xy), As noticed in Ref. [46], imposing locality constraints at the level of the observed distribution is not the same as imposing constraints on the joint measurability of observables in the NPA hierarchy approximating the set Q.For instance, consider the set Q 12 J M defined as follow P(ab|xy) belongs to Q 12 J M if: In other words, the two measurements {E a|1 } a and {E a|2 } a are substituted by a single joint measurement M 12  aa .In terms of the NPA hierarchy, this can be simply obtained by taking the moments involving M 12 aa instead of {E a|1 } a and {E a|2 } a .Similarly, the convex hull Q 2conv J M can be defined as P(ab|xy) belongs to Q 2conv J M if: P(ab|xy) ∈ Q, P(ab|xy) = µ 12 P 12 (ab|xy) + µ 13 P 13 (ab|xy) + µ 23 P 23 (ab|xy), The SDP approximation of this set involves computing three different NPA moment matrices, one for each distribution P ij (ab|xy).
It is important to remark that the NPA hierarchy can be computed by assuming the dilation of the POVMs to projective measurements.It is also important to remark that, even if some structure of measurement incompatibility require POVMs (e.g., the hollow triangle), in Eq. (D6) only pairwise JM conditions arise, one for each P ij .A total JM measurability condition among a measurement assemblage {E a|x } a,x is equivalent to the existence of a common dilation in which the measurements are represented by commuting projective measurements.In this sense, due to the convex nature of the genuine multipartite incompatibility problem, there is no contradiction between the use of the dilation and the fact that non-trivial compatibility structures necessarily require POVMs.Comparison between lower bounds on IR in the I 3322 scenario [60].The blue-solid and black-dashed curves represent, respectively, lower bounds obtained from our method and from the method of the assemblage moment matrices [34].
The local and quantum bounds for the I 3322 inequality are, respectively, 0 and around 0.250875561 [68].The levels of the hierarchy of the semidefinite relaxation used to carry out the computation in both methods are the 3rd level.
construction of moment matrices.In the DI scenario, all measurements can be assumed to be projective due to the Neumark dilation, as discussed in the main text.Such a dilation, however, requires to increase the Hilbert space dimension and is, thus, not always possible if the dimension the system is constrained as in the SDI scenario.In some cases, however, projective measurements can be recovered by a convexity argument.For instance, for dichotomic measurements, it is known that they are all convex mixtures of projective measurements (intuitively, it is sufficient to decompose the 0-outcome element), so we can restrict ourselves to projective measurements if the objective function we wish to minimize is linear in the POVMs operator.This is the case for, e.g., Bell inequalities as noted in [61], but it is also the case for the incompatibility robustness.In order to show that, it is useful to introduce first some slack variables ({S a,x } a,x ), namely, G λ 0, S a,x 0, ∀ λ, a, x, It is clear that if, for a given x, E a|x = ∑ i µ i P i a|x , the minimal robustness will be obtained for a given projective measurement {P i a|x } a,x .It is not obvious, however, what happens if one tries to minimize the robustness for a fixed Bell inequality violation.In fact, by choosing one element of the decomposition as above, we may decrease both the robustness and the Bell inequality violation.
A possible approach to the problem by dilation of the measurements of both Alice and Bob, has been already proposed in Ref. [61], while generating a basis for S d , the space of moment matrices corresponding to dimension d, one should sample Alice's and Bob's measurements of the form with random unitaries U x and U y .Random states should then be taken of the form ρ = |0 0| A ⊗ |0 0| B ⊗ |ψ ψ| AB .Since we are interested in dichotomic measurements, dimension of the auxiliary spaces A and B is 2 in both cases.In practice, however, this method was not able to provide a better bound of the SDI bound in d = 2 for the I 3322 inequality.
Appendix G: Extension of MMM method to the prepare-and-measure scenario The prepare-and-measure (P-M) scenario, e.g., the one given by random access codes [49,65], is a paradigm often considered in quantum information processing as an alternative to the Bell scenario.The P-M scenario is a one-way communication scenario in which one party, let us say Bob, prepares a physical system in a state ρ y chosen from a finite set indexed by y and sends it to the other party, Alice.Alice measures this system with a choice of measurement specified by x.The conditional distribution P(a|x, y), where a is the outcome of Alice's measurement, is then used to semi-device-independently characterize the states and measurements in this scenario.The classical distribution P(a|x, y) is the one produced by states and measurements which can be simultaneously diagonalized in some basis of the Hilbert space in which they are defined.
One important distinction between the P-M and Bell scenario is that parties' measurements do not need to be space-like separated.However, in order to observe a gap between classical and quantum strategies some form of restriction on the communication needs to be imposed [69,70].Here, we consider the most common type of restriction, an upper-bound on the Hilbert's space dimension in which the states and measurements are defined.This enables us to use the hierarchy of Ref. [61] to approximate the set of quantum correlations P(a|x, y) and subsequently map the incompatibility robustness SDP to MMM SDP.
The map is a direct extension, merely a simplification of Eq. ( 4) of the main text, and can be written as follows: where K n := ∑ i |i AA n|S i , and {S i } is the following sequence of operators: {S i } = {1 1 A , E A a|x , ρ y , E A a|x ρ y , etc.}.The MMM can then be defined as which is a direct analogy of Eq. ( 5) of the main text.
Using this map, one can formulate an SDI relaxation of incompatibility robustness SDP, which reads min χ[E A a|x ] ∈ S d , ∀a, x, χ[1 1] ∈ S d , P(a|x, y) = P obs (a|x, y) ∀a, x, y, (G3) where S d is a subspace of moment matrices spanned by those corresponding to states and measurements defined on Hilbert space of dimension d.

FIG. 3 .
FIG.3.MMMs can also be used to compute lower bounds on genuine triplewise IR in a DI setting.(a) DI lower bounds on genuine triplewise IR in the elegant Bell scenario.(b)The black-dashed, red-dash-dotted, and blue-solid curves represent, respectively, DI lower bounds on genuine triplewise IR in the I 3322 , I 3 3422 , and I 2 3422 scenarios.The SDP carrying out the computation can be found in Eq. (13).

2 FIG. 4 . 4 .
FIG.4.Quantification of incompatibility robustness for given violation of I 3322 .Blue dash-dotted line: SDI approach for d = 2. Black solid line: DI approach.A similar curve (with a difference of the order of 10 −5 ) corresponds to the SDI calculation with the additional assumption of projective measurements for d = 3, 4 (identical up to numerical precision).Red dashed line: SDI approach with additional assumption of projective measurements for d = 2.
FIG. 5.Comparison between two SDPs computing the bounds.The solid curves are bounds obtained by fixing a Bell violation and minimizing the robustness (cf.Eq. (12)) while the dashed curves are bounds obtained by fixing a robustness and maximizing the Bell violation (cf.Eq. (C5)).

2 FIG. 6 .
FIG.6.Comparison between lower bounds on IR in the I 3322 scenario[60].The blue-solid and black-dashed curves represent, respectively, lower bounds obtained from our method and from the method of the assemblage moment matrices[34].The local and quantum bounds for the I 3322 inequality are, respectively, 0 and around 0.250875561[68].The levels of the hierarchy of the semidefinite relaxation used to carry out the computation in both methods are the 3rd level.
s.t.A k , X = b k , ∀ k X 0. (F2)It is then clear that the entries of the POVM elements E a|x will appear in the vector b, and consequently in the objective of the dual problem max E a|x = p 12 J 12 a|x + p 23 J 23 a|x + p 13 J 13 a|x (C1) for some probabilities p 12 , p 23 , and p 13 with p 12 + p 23 + p 13 = 1, is equivalent to the infeasibility of the following SDP: Given {E a|1 } a , {E a|2 } a , {E a|3 } a