Certifying sets of quantum observables with any full-rank state

Certifying Sets of Quantum Observables with Any Full-Rank State Zhen-Peng Xu,1, 2, ∗Debashis Saha,3, †Kishor Bharti,4, ‡and Ad´an Cabello5, 6, § 1School of Physics and Optoelectronics Engineering, Anhui University, 230601 Hefei, People’s Republic of China 2Naturwissenschaftlich-Technische Fakult¨at, Universit¨at Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany 3School of Physics, Indian Institute of Science Education and Research Thiruvananthapuram, Kerala 695551, India 4Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Republic of Singapore 5Departamento de F´ısica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain 6Instituto Carlos I de F´ısica Te´orica y Computacional, Universidad de Sevilla, E-41012 Sevilla, Spain We show that some sets of quantum observables are unique up to an isometry and have a contextuality witness that attains the same value for any initial state. We prove that these two properties make it possible to certify any of these sets by looking at the statistics of experiments with sequential measurements and using any initial state of full rank, including thermal and maximally mixed states. We prove that this “certification with any full-rank state” (CFR) is possible for any quantum system of finite dimension d≥3 and is robust and experimentally useful in dimensions 3 and 4. In addition, we prove that complete Kochen-Specker sets can be Bell self-tested if and only if they enable CFR. This establishes a fundamental connection between these two methods of certification, shows that both methods can be combined in the same experiment, and opens new possibilities for certifying quantum devices. Introduction.—Nonlocality [1] and contextuality [2] are two fundamental predictions of quantum theory. Quantum theory also predicts that, in certain cases, there is an essentially unique way to achieve some specific nonlocal [3–5] or contextual [6–8] correlation. Consequently, the observation of this specific correlation allows us to infer which quantum state has been prepared and which quantum observables have been measured, without making assumptions about the functioning of the devices used in the experiment [3–8]. However, none of the existing “device-independent” (DI) certification methods work if the fidelity of the prepared state with respect to a specific pure state is below a certain threshold. It is this specific pure state that guarantees the uniqueness of the quantum realization in the noiseless (ideal) case. In particular, none of the methods works if the prepared state is maximally mixed. This leads to the question of whether it would be possible to certify quantum observables using correlations produced by measurements on unspecified mixed states, including the maximally mixed state. That, in quantum theory, this question may have an affirmative answer is suggested by the observation that, for any quantum system of finite dimension d≥3, there exist finite sets of observables that produce contextual correlations for any quantum state [9–12]. These sets of observables are called state-independent contextuality (SI-C) sets [2,13,14]. SI-C sets have fundamental applications in quantum information [15–28] and have been experimentally tested [29–34]. ∗[email protected] †[email protected] ‡[email protected] §[email protected] But the existence of SI-C sets itself leads to another question: are there SI-C sets that are unique up to an isometry? This question is particularly relevant for understanding the mathematical structure of the set of quantum observables. Interestingly, if the answer to this question is positive, then there may be a connection to the question of whether there are quantum observables that can be certified with arbitrary mixed states. In this Letter, we first show that there are SI-C sets that (i) are unique up to an isometry, and (ii) have a SI-C witness Wthat achieves the same value for every initial quantum state. These SI-C sets have therefore a characteristic signature that can be experimentally tested: the relations of compatibility between the observables (which are encoded in the expression of W) and the stateindependent value of W. Next, we will show that SI-C sets with properties (i) and (ii) can be certified from the correlations of experiments with sequential measurements performed on any full-rank mixed state, including thermal and maximally mixed states. As soon as a mixed state of full rank gives the characteristic value for W, any other state will do so. This leads to a method for certifying quantum observables from correlations that is fundamentally different than self-testing based on Bell inequalities [3–5], state-dependent contextuality [6–8], prepare-and- measure [35,36], and steering [37–39]. There are two fundamental differences: (a) The initial state required for the certification is not determined by the set of observables to be certified; any state of full rank can be used. (b) The certification guarantees the state-independent uniqueness (up to an isometry) of the set of observables. In addition, we show that this new method, named “certification with any full-rank state” (CFR), is possible in every finite dimension d≥3, and provide a way to arXiv:2309.05735v2 [quant-ph] 8 Apr 2024 2 v1v2v3v4v5v6v7v8v9v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 vi10 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 vi2111000¯ 1 ¯q¯g0101q g 1q g 1q g vi3¯ 1 ¯q¯g¯ 1 ¯q¯g0 0 0 0 0 1 1 g q q 1g g q 1 wi4 4 4 4 4 4 4 4 4 7 7 7 7 7 7 7 7 7 7 7 7 TABLE I. BBC-21. Each column vicorresponds to one observable represented by the projector |vi⟩⟨vi|. The column vij gives the components of |vi⟩(unnormalized). ¯x=−x,q=e2π i /3, and g=q2. Compatible observables correspond to orthogonal vectors. The last row contains optimal weights wifor a SI-C witness Wof the form (1). The weights wij in (1) can be chosen in any way that satisfies wij ≥max{wi, wj}. obtain sets of observables that enable CFR in any d≥3. We also prove that CFR is robust against experimental imperfections using examples in d= 3 and 4, and show how to test the robustness in any other case. Finally, we show that, for a fundamental class of SI-C sets, CFR is a necessary condition for Bell self-testing. This points out a connection between two different forms of certification and shows that these two forms can be applied simultaneously in Bell experiments with sequential measurements. This opens up some interesting possibilities which are discussed. Certification with any full-rank state.—Unless otherwise indicated, hereafter we will focus on SI-C sets of projectors (rather than general self-adjoint operators) and on a special type of contextuality witness that can be defined from them using the following result, which is a generalization of a result in [40], whose proof is in [13]. Lemma 1.—Given a finite set of observables {Πi}, with possible results 0 or 1, and graph of compatibility G(in which each Πiis represented by a vertex i∈Vand there is an edge (i, j)∈Eif Πiand Πjare compatible), the following inequality holds for any noncontextual hiddenvariable (NCHV) theory: W:= X i∈V wiPi−X (i,j)∈E wijPij NCHV ≤α(G, w),(1) where w ={wi}i∈Vis a set of positive weights for the vertices of G,wij ≥max (wi, wj), Pi=P(Πi= 1) is the probability of obtaining outcome 1 when measuring observable Πi,Pij =P(Πi= 1,Πj= 1) is the probability of obtaining outcomes 1 and 1 when measuring Πiand Πj, and α(G, w) is the weighted independence number of Gwith vertex weight vector w (see [13] for the definition). Our first result is the following. Result 1.—For any quantum system of any finite dimension d≥3, there is a finite set of observables S={Πi}n i=1 and a functional Wsuch that, for any quantum state ρ,W(S, ρ) = Q, and, if W(S′, ρ′) = Qfor a set of observables S′={Π′ i}n i=1 and a state ρ′of full rank in dimension D, then S′and Sare equivalent in the sense that there is a unitary transformation Uthat, for all i, Πi⊗1d1⊕Π∗ i⊗1d2=UΠ′ iU†,(2) where 1d1is the identity in dimension d1, with d1+d2= D/d, Π∗ iis the conjugate of Πi,⊗denotes tensor product, ⊕denotes direct sum, and U†is the conjugate transpose of U. Moreover, Wis a SI-C witness since Q > C and W ≤ C(3) is a state-independent noncontextuality inequality. For the witnesses Wof the form (1), C=α(G, w). If those d-dimensional Πiare real (rather than complex), then Eq. (2) becomes Πi⊗1(D/d)=UΠ′ iU†.(4) The practical consequence of Result 1 is that if, in an ideal experiment with sequential measurements, a set of n measurement devices (one for each observable), combined in sequences as dictated by the form of W, yields W=Q for a state of full rank, then we can be sure that these devices implement S[or an equivalent set in the sense of Eqs. (2) or (4)]. Then, we will say that Senables CFR. The case of nonideal experiments will be discussed later. Proof.—The proof is based on identifying sets enabling CFR in any dimension d≥3. We will name the SI-C sets using the initials of the authors and the number of projectors in the set. For example, BBC-21 [41], CEG- 18 [42], and YO-13 [11]. In other cases, we use the full name rather than the initial, as in Peres-24 [43]. In other cases, we use the standard name, as in the Peres-Mermin square [44,45]. While the details of the proof are specific for each SI-C set, a common step in all proofs is showing that the violation of a full-rank state ρ′implies the same violation for any state. The proof starts by showing that, in d= 3, the set of 21 rank-one projectors in Table Ienables CFR. This set, hereafter called BBC-21, was introduced in [41] and is the smallest SI-C set of rank-one projectors requiring complex numbers known. The proof that BBC-21 is unique up to unitary transformations, which guarantees that condition (i) for CFR holds, is in [13]. Us- 3 ing the weights in the last row of Table I, the noncontextual bound of the witness Wdefined in Eq. (1) is α(G, w) = 36, while, for any initial quantum state, the value of Wis ϑ(G, w) = 40. This proves that BBC-21 also satisfies condition (ii) for CFR. In d= 4, we show that three related fundamental SI-C sets enable CFR: (I) CEG-18 [42], which is the smallest KS set [13] of rank-one projectors in any dimension (as proven in [46]), (II) Peres-24 [43], which is the smallest complete KS set (see Definition 4) of rank-one projectors known, and (III) the Peres-Mermin square [44,45], which is the smallest SI-C set of arbitrary self-adjoint operators (rather than projectors) known. The proofs that these sets are unique up to unitary transformations and the corresponding optimal state-independent contextuality witnesses yielding the same value for any state are in [13]. Finally, for any finite dimension d≥5, we prove (see [13]) that each of the members of a family of SI-C sets of rank-one projectors generated from Peres-24 using a method introduced in [47] is unique up to unitary transformations and has a SI-C witness producing the same value for any initial state. While existing correlation-based certification methods require preparing a state with a high overlap with a target pure state, the SI-C sets that enable CFR can be certified using any unspecified full-rank state, something that is easier to prepare. A simple strategy is to let an arbitrary state go through randomly chosen measurements [34], resulting in a maximally mixed state. Another strategy is to let the system interact with the environment, resulting in a thermal state. Both types of states are of full rank. Not all SI-C sets enable CFR. For example, Peres-33 [43], which is the KS set of rank-one projectors in d= 3 with the smallest number of bases known, is not unique up to unitary transformations. Interestingly, YO-13 [11], which is the SI-C set with smallest number of rank-one projectors in any dimension (as proven in [48]) and is a subset of Peres-33, enables CFR if two additional conditions are satisfied: (I’) The relations of orthogonality between the elements S′are the same as the relations of orthogonality between the elements S, and (II’) for ρ′, the probabilities are normalized for every set of mutually orthogonal projectors summing up to the identity. This is shown in [13]. Both (I’) and (II’) can be experimentally tested (as in [34]). Robustness.—The possibility of CFR of SI-C sets is a prediction of quantum theory. Now the question is whether this prediction can be tested in actual experiments or it requires idealizations that cannot be achieved in realistic experiments such as the requirement of perfectly sharp and compatible measurements for all pairs of compatible observables in the SI-C set. In other words, the question is whether CFR is robust against experimental imperfections. Answering this question requires an additional analysis based on semidefinite programming whose size is related to the size of the SI-C sets. Here, we have performed this analysis for three of the SI-C sets, in dimensions 3 and 4, that we have proven that enable CFR. In all cases, the analysis was performed on a laptop computer and the computational execution time was less than 1 h. The analysis of the robustness of the CFR of the other SI-C sets can be carried out using higher computational power. Our result here is that the CFRs based on BBC-21, CEG-18, and Peres-24 are robust. We will also show that the CFR of YO-13 is robust under an extra assumption. Our result requires introducing some definitions. Definition 1.— A set of projectors {Πi}is said to be a (θ, ϵ) realization of a SI-C set with respect to a contextuality witness Wof type (1) if, for all states |ψ⟩, n X i=1 wi⟨ψ|Πi|ψ⟩⩾θ > α(G, w),(5a) ⟨ψ|ΠiΠjΠi|ψ⟩⩽ϵ, (5b) whenever iand jare adjacent in G(i.e., whenever the corresponding projectors are orthogonal). The conditions in Eqs. (5a) and (5b) are related to the sum of probabilities PiwiPiand joint probability Pij in Eq. (1). In the ideal case, θ=Q(defined in Result 1), and ϵ= 0, which implies that the quantum value of W is Q. As θis close enough to Qand ϵis close enough to 0, the projectors {Πi}have the same rank. See [13] for details. Definition 2.—A noncontextuality inequality of the form (1) provides an (ϵ, r)-robust CFR of a (Q, 0) realization {Πi}of a SI-C set, if, for any (Q−ϵ, ϵ) realization {Π′ i}of the SI-C set, there is an isometry Φ such that |Φ(Πi)−Π′ i| ≤ O(ϵr).(6) Result 2.—The contextuality witnesses Wof the form (1) for BBC-21, CEG-18, Peres-24, and YO-13 used in Result 1 provide (ϵ, 1/2) robustness when ϵis smaller than 0.132, 0.134, 0.177, and 0.208, respectively. For YO-13, the proof requires the extra assumption that the probabilities of every three mutually orthogonal projectors sum 1. For more details on the proof, see [13]. Any witness Wof the form (1) can be expressed with the joint probabilities of the outcomes of two sequential measurements from {Aj}. From the observed values satisfying conditions related to the ideality and the orthogonality relations of the projectors, one can certify the projectors and the measurements Ai. Moreover, when the experimental value of Wis close enough to the quantum value, the robustness of the CFR is also ensured. See [13] for more details. Bell self-testing and CFR.—Bell self-testing [3] is the task of certifying quantum states and measurements using only the statistics of Bell experiments. One advantage of Bell self-testing with respect to CFR is that the former does not require projective measurements. One disadvantage, however, is that Bell self-testing requires 4 spacelike separation between the tests. Therefore, an interesting question is whether SI-C sets that allow for CFR can be Bell self-tested and, if so, what is the relation between Bell self-testing and CFR. To address these questions, the following definitions will be useful. Definition 3.—(Generalized KS set) A generalized Kochen-Specker (KS) set is a set of projectors of arbitrary rank (not necessarily of rank-one as it is the case in a KS set [49]) which does not admit an assignment of 0 or 1 satisfying that: (I) two orthogonal projectors cannot both be assigned 1, (II) for every set of mutually orthogonal projectors summing up to the identity, one and only one of them must be assigned 1. Definition 4.—(Complete KS set) The complete KS set associated to a generalized KS set Sis the set obtained by adding to Sthe projectors 1−Πi−Πjfor every pair of orthogonal projectors (Πi,Πj) in Sthat does not belong to a complete basis. For example, Peres-24 is a complete KS set, but CEG- 18 and Peres-33 are not (BBC-21 and YO-13 are not KS sets). A complete KS set enables CFR if it satisfies properties (i) and (ii). Now we need a way to produce Bell nonlocality using a complete KS set. For that aim, we will define the following nonlocal game. Definition 5.—(Context-projector KS game [16,17, 21]) In each round of the game, a referee gives to one of the players, Alice, one of the contexts (i.e., a set of commuting projectors summing up the identity) of a complete KS set Sand asks her to output one of the projectors of this context. In the same round, the referee gives to one spatially separated player, Bob, one of the projectors of the same context and asks him to output 1 or 0. Alice and Bob win the round either if Alice outputs the projector given to Bob and Bob outputs 1, or if Alice outputs a projector different than the one given to Bob and Bob outputs 0. This is a game that cannot be won with probability 1 with classical resources and no communication, but that can be won with probability 1 if the parties share copies of a qudit-qudit maximally entangled state with d≥3 and measure a complete KS set in dimension d. Now, we can address the question of whether the SI-C sets that allow for CFR can be Bell self-tested. Result 3.—The projectors of a complete KS set can be Bell self-tested if and only if the KS set enables CFR. The proof is in [13]. Here, we will focus on some implications of this result. One is that Bell self-testing and CFR can be accomplished simultaneously in an experiment that combines Bell and sequential tests [50– 52]. Consider two spatially separated parties, Alice and Bob 1, sharing copies of a qudit-qudit maximally entangled state and performing local measurements of the projectors of a complete KS set S. In addition, consider a third party, Bob 2, that receives the system that Bob 1 has measured (we assume that Bob 1’s measurements are nondemolition measurements [34,53]). Suppose that Bob 2 measures elements of Son this system. Then: (a) The Alice-Bob 1 statistics can Bell self-test Sin Alice’s and Bob 1’s sides. (b) The Bob 1-Bob 2 statistics enable CFR of Sin Bob 1’s and Bob 2’s sides (and the Alice-Bob 1 Bell self-test can guarantee that Bob 1’s input state is of full rank). (c) The Alice-Bob 2 statistics conditioned to that Bob 2’s measurement is compatible to Bob 1’s can Bell self-test Sin Alice’s and Bob 2’s sides. This allows for the simultaneous certification of Bob 1’s Sby two different methods and opens new possibilities. Conclusions and future directions.—In this Letter, we have presented three results that push the field of certification of quantum processes based only on correlations beyond its established limits. Results 1 and 2 allow us to circumvent a conceptual limitation of existing methods, namely, the need of targeting specific pure states. We have proven that this is not necessary: for any quantum system of any finite dimension d≥3, there are sets of quantum observables that can be certified using any full-rank quantum state. This “certification with any full rank state” offers interesting possibilities. For example, suppose that the same preparation is used to certify via CFR two sets of observables: one of them in dimension d1 and the other in dimension d2. This automatically certifies via CFR that the dimension of the system is lower bounded by the lowest common denominator of d1and d2. This provides a method to certify quantum systems of high dimensions, something that is difficult in a DI way [54,55]. Moreover, in principle, CFR becomes more useful as the dimension grows, since preparing a full-rank mixed state is easier than preparing a state with a high overlap with a pure target state. Result 3 pushes the field in a different sense. It shows that, for a general class of sets of observables, CFR is possible if and only if Bell self-testing is possible. This indicates that there may be a general unified framework for certification based solely on correlations, so that all existing methods can be viewed as particular cases. The precise characterization of this framework constitutes an interesting challenge. On the other hand, Result 3 shows that there are sets of observables that can be simultaneously Bell self tested (using Alice-Bob 1 correlations) and certified via CFR (using Bob 1-Bob 2 correlations). This is interesting as it may lead to a robust method for self-testing L¨uders processes [56,57] in any finite dimension d≥3 (which is where observables represented by rank-one projectors have one outcome whose quantum post-measurement state depends on the input state). In the framework of general probabilistic theories, L¨uders processes correspond to “ideal (or sharp) measurements” [58]: processes that yield the same outcome when repeated and are minimally disturbing (only disturb incompatible observables). The existence of ideal measurements is “one of the fundamental predictions of quantum mechanics” [57]. The DI certification of ideal measurements in arbitrary (finite) dimension would require the DI certification of the corresponding quantum instruments (which capture both the classical outputs and the corresponding quantum post-measurement 5 states [59–61]). Previous works have explored the DI [62] and semi-DI [63] certification of instruments corresponding to nonideal qubit measurements. The DI certification of ideal measurements would operationally “bridge the gap between general probabilistic theories and the DI framework” [64], blurring the boundaries between three different approaches for understanding quantum theory: DI, general probabilistic theories, and general Bayesian theories [65], where ideal measurements are central. Future research should go in these directions. This work was supported by the EU-funded project FoQaCiA and the MCINN/AEI (Project No. PID2020- 113738GB-I00). Z.-P. X. acknowledges support from the National Natural Science Foundation of China (Grant No. 12305007), Anhui Provincial Natural Science Foundation (Grant No. 2308085QA29), and the Alexander von Humboldt Foundation. SUPPLEMENTAL MATERIAL Appendix A: Concepts Definition 1 (Ideal measurement [64]).An ideal measurement of an observable Ais a measurement of Athat yields the same outcome when it is repeated on the same system and does not disturb any observable compatible with A. Definition 2 (Compatible observables).Two observables Aand Bare compatible or jointly measurable if there exists an observable Csuch that, for every initial state, for every outcome aof A, the probability of obtaining outcome afor Ais P(A=a) = X o∈ca P(C=o) (A1) and, for every outcome bof B, P(B=b) = X o∈cb P(C=o),(A2) where the disjoint union of {ca}aand the disjoint union of {cb}bare both equal to the complete set of outcomes of C.Cis called a refinement of A(and B). A(and B) is called a coarse-graining of C. Therefore, two observables are compatible when they have a common refinement or are both coarse-grains of the same observable. Definition 3 (Ideal observable).An ideal or sharp observable is one that can be measured with ideal measurements, and that all its possible coarse-grained versions can also be measured with ideal measurements. In quantum theory, ideal observables are represented by self-adjoint operators. Definition 4 (SI-C set).A state-independent contextuality (SI-C) set in dimension dis a set of ideal observables that produces contextuality for any initial state in dimension d. In particular, a set of nideal observables represented in quantum theory by nprojectors {Πi}n i=1 is a SI-C set if there is a set of weights w ={wi}n i=1 for the vertices of the graph Gof compatibility of {Πi}n i=1 (in which vertices represent observables and edges connect pairwise compatible observables) such that a noncontextuality inequality of the form (1) is violated by any quantum state in dimension d. Definition 5 (KS set).A Kochen-Specker (KS) set is a SI-C set of rank-one projectors which does not admit an assignment of 0or 1satisfying that: (I) two orthogonal projectors cannot both be assigned 1, (II) for every set of mutually orthogonal projectors summing the identity, one of them must be assigned 1. There are SI-C sets of rank-one projectors that are not KS sets. Examples are YO-13 [11] and BBC-21 [41]. Definition 6 (Egalitarian SI-C set).A SI-C set is egalitarian if it produces, for any state, the same violation of a given noncontextuality inequality. In particular, a SI-C set {Πi}is egalitarian if there is a set of weights w ={wi}for the vertices of the graph Gof compatibility of {Πi}such that, for any quantum state, the left-hand side of (1) yields Q > α(G, w). Definition 7 (Independence number).The independence number α(G, w)of a vertex-weighted graph (G, w) is the maximum w(S) = Pv∈Sw(v)taken over all independent sets Sof G. A set of vertices of Gis independent if all the vertices in it are pairwise nonadjacent. Definition 8 (Lov´asz number).The Lov´asz number ϑ(G, w)of a vertex-weighted graph (G, w)is the maximum of Piwi|⟨vi|ψ⟩|2over all unit vectors |ψ⟩and |vi⟩ such that ⟨vj|vi⟩= 0 whenever iand jare adjacent vertices of G. Definition 9 (Egalitarian Lov´asz-optimum SI-C set). An egalitarian SI-C set {Πi}is Lov´asz-optimum if, for any quantum state, the left-hand side of (1)equals the Lov´asz number of the weighted graph (G, w), denoted ϑ(G, w), where w is a set of weights for which the SI- C set is egalitarian. Appendix B: Tools used in the proof of Result 1 1. Proof of Lemma 1 Let us denote by athe maximum of the left-hand side of (1) that is achievable by a noncontextual hiddenvariable (NCHV) theory. Since set of correlations for 6 NCHV theories forms a convex polytope, acan be obtained with a deterministic assignment of outcomes to the observables. For a given deterministic assignment achieving a, if Πi= 1 and Πj= 1 for one (i, j)∈E, then let us consider the part in Eq. (1) which contains Πj= 1, wjP(Πj= 1) −X k:(k,j)∈E wkjP(Πk= 1,Πj= 1) ≤wjP(Πj= 1) −wijP(Πi= 1,Πj= 1) =wj−wij ≤0.(B1) This implies that, by setting Πj= 1 to be 0 in the deterministic probability assignment, the value does not decrease. Hence, the maximal value acan always be achieved by one deterministic probability assignment where Πiand Πjare not both assigned 1 if (i, j)∈E. Therefore, acan only be α(G, w). In the case that wij >max{wi, wj}, by setting Πj= 1 to be 0 in the deterministic probability assignment, the value increases. Hence, the maximal value acan never be achieved by the deterministic assignment where Πi= 1 and Πj= 1 for one (i, j)∈E. 2. Common procedure in all the proofs of uniqueness up to unitary transformations Let us first consider the case in which the graph of compatibility Gis equal to the graph of orthogonality of the observables/projectors {Πi}n i=1. That this is the case, can be experimentally tested as described in Lemma 2 in the main text. In some special cases, this can even be tested using the maximal quantum violation of the witness Wof the form (1). Theorem 1. For any noncontextuality inequality of the form (1), if there exists a set of observables {Πi}such that, for any quantum state, the left-hand side of (1) equals the maximum value attainable in quantum theory, denoted by Q, then Q=ϑ(G, w),(B2) where ϑ(G, w)is the Lov´asz number of the graph Gwith weights w, and the observables {Πi}must be of the type represented by projectors {Πi}(here, we use the same symbol for the observable and the projector that represents it). Moreover, if wij >max{wi, wj}, then, for any (i, j)∈E, ΠiΠj= 0.(B3) Proof. Let us first consider the case in which wij > max{wi, wj}. Consider the state ρi= Πi/tr(Πi) associated to Πi, with i∈V. For this state, Pρi(Πi= 1) = 1,(B4) and, for any jsuch that (i, j)∈E, Pρi(Πi= 1,Πj= 1) = Pρi(Πj= 1).(B5) If there is Πjsuch that (i, j)∈Eand ΠiΠj= 0, then the terms in the left-hand side of (1) that contain Πjsatisfy wjPρi(Πj= 1) −X k:(k,j)∈E wkjPρi(Πk= 1,Πj= 1) ≤wjPρi(Πj= 1) −wijPρi(Πi= 1,Πj= 1) =(wj−wij)Pρi(Πj= 1) <0.(B6) Hence, by setting Πj= 0, the quantum value of the lefthand side of (1) for state ρiincreases, which contradicts the assumption that Qis the maximum quantum value. Therefore, we can conclude that, for all (i, j)∈E, ΠiΠj= 0.(B7) Under this condition, for a given state ρ, the left-hand side of (1) equals to Piwip(Πi= 1)ρ, which is upper bounded by ϑ(G, w). Therefore, q≤ϑ(G, w) (see Sec. A). On the other hand, by the definition of ϑ(G, w), there is always {Πi}i∈Vsuch that the quantum value is ϑ(G, w), which implies q≥ϑ(G, w). Therefore, we can conclude that q=ϑ(G, w). Similarly, Q=ϑ(G, w) also holds in the case that wij = max{wi, wj}. However, in this case, ΠiΠj= 0 does not need to hold. The requirement of a full-rank state in our main result’s proof stems from the fact that if a full-rank state attains the maximal value Q, then any state achieves Q. Consequently, as per the above theorem, the projectors will satisfy the orthogonality relation according to G. To be precise, Eq. (B7) implies that the compatibility graph Grepresents also the orthogonality relations between the observables {Πi}. This step is crucial in the proof of Result 1. In general, Qmay be difficult to determine. Then, one cannot decide whether or not Qis achievable for all the states. However [66], for wij >max{wi, wj}, Q≤ϑ(G′, w′)−X (i,j)∈E wij,(B8) where (G′, w′) is the (weighted) graph of exclusivity of the weighted events {wi(Πi= 1)}i∈Vand {wij(Πi= 0,Πj= 0), wij(Πi= 0,Πj= 1), wij(Πi= 1,Πj= 0)}(i,j)∈E. Therefore, if for any quantum state, the lefthand side of (1) equals the right-hand side of (B8), we can conclude that Q=ϑ(G, w) and that ΠiΠj= 0 for any (i, j)∈E. In fact, this is the case for the optimal vertex-weighted graphs of compatibility of BBC-21, CEG-18, and Peres-24. For the one of YO-13, we need wij ≥2 max{wi, wj}in the form (1) and the extra normalization conditions in Eq. (B106a). The computations needed for checking Peres-39 and beyond it cannot be carried out with a laptop computer. The proof of uniqueness up to unitary transformations of a given SI-C set {Πi}n i=1 is then based on two facts: 1. All the projectors in {Πi}have the same rank, that we call κ. 7 2. If (i, j)∈ Ein the graph of orthogonality G, then rank(ΠjΠiΠj) = rank(Πi) = rank(Πj). The first fact is ensured by the second one if the complement graph of Gis connected, which is indeed true for all the cases considered here. The second fact holds also for all the cases considered here. This can be verified by semi-definite programming (SDP). To be more explicit, if there is iand jsuch that (i, j)∈ Eand rank(ΠjΠiΠj)<rank(Πj), then there should exist a state |s⟩such that ⟨s|Πj|s⟩= 1 and ⟨s|Πi|s⟩= 0. Notice that, {Πi}is a SI-C set with some weights w and quantum violation Q. Denote Tij =⟨s|Πi−1Πj−1|s⟩, where Π0=1. Then, the matrix Tsatisfies the following conditions: T⪰0, T11 = 1, T1k=Tkk,∀k, Tkl = 0,if (k, l)∈E, T1j= 1, T1i= 0,X k≥2 wk−1T1k=Q. (B9) For YO-13, we have to add some extra linear conditions [see Eq. (B106a)]. To check whether or not condition 2 holds, we can check whether or not the SDP in Eq. (B9) is feasible, which is a relaxation of the original problem. In all the cases considered here, the relaxation in Eq. (B9) is not feasible. This implies that rank(ΠjΠiΠj)<rank(Πj) cannot be true for (i, j)∈ E. Therefore, the second fact is also ensured. Then, we can choose a complete basis as the computational basis such that {Πi}c i=1 are the projectors into the subspace occupying dimensions from (c−1)κ+ 1 to cκ. Notice that it is always possible to choose a complete basis for the SI-C sets considered here. For any other projector Πi, we have Πi=L′† iL′ i, L′ i= [B1i, B2i, . . . , Bci].(B10) The reason is as follows. Since Πiis a projector of rank κ, it can be written as Πi=Pκ t=1 |vi⟩⟨vi|, where the |vi⟩’s are orthogonal normalized vectors. Denote L′ i= [|v1⟩,...,|vκ⟩]†, we have Πi=L′† iL′i. Here, the dimension of L′ iis κ×d, where dis the dimension and d=cκ. Hence, we can always write L′ ias [B1i, B2i, . . . , Bci], where Bti is a κ×κmatrix for any t. Then, for t= 1, . . . , c, we have ΠtΠiΠt=B† tiBti. Since ΠtΠiΠtis of rank κwhen (t, i)∈ E, we have Bti to be invertible in this case. In the case that (t, i)∈E, we have ΠtΠiΠt= 0. Consequently, in this case, Bti = 0. By definition of L′ i,L′ iL′† i=1κ. If B1iis invertible, then we introduce Li:= B−1 1iL′ i= [1, B−1 1iB2i, . . . , B−1 1iBci].Further, we can verify that L† i(LiL† i)−1Li=L′† i(L′iL′† i)−1L′i= Πi.(B11) In addition, LiL† j= 0 ⇔L′ iL′ j†= 0 ⇔ΠiΠj= 0.(B12) Hence, in the proofs of uniqueness, we will adopt {Li} for convenience. From the uniqueness of {Li}, we can recover the uniqueness of {Πi}. For completeness, a similar result is proven when point (III) of Lemma 1is proven. 3. Proof that BBC-21 is unique up to unitary transformations The uncharacterized projectors in this case are {Πi} that are defined by the {Li}operators in Eq. (B11). For convenience, we relabel {Li}by following the same order of the vectors {vi}21 i=1 in Table Ias follows: y0 1, y1 1, y2 1, y0 2, y1 2, y2 2, y0 3, y1 3, y2 3, z1, z2, z3, h0 1, h1 1, h2 1, h0 2, h1 2, h2 2, h0 3, h1 3, h2 3.(B13) For example, Π10 =z† 1(z1z† 1)−1z1, where z1=L10. These operators satisfy the additional conditions in Eq. (B12) due to the orthogonality relations of the projectors {Πi}. Without loss of generality, we assume that {z1, z2, z3} forms a complete basis. Moreover, the fact that z1⊥y0 1, z2⊥y0 2,z1⊥y1 1, we can take z1= [1,0,0], z2= [0,1,0], z3= [0,0,1], y0 1= [0,1, A], y0 2= [1,0, B], y1 1= [0,1, C],(B14) where A,B, and Care matrix variables to be determined. The reason is the following. The vertices related to z1, z2, z3form a clique of size 3, and the corresponding three projectors sum up to identity. Hence, as argued around Eq. (B10), we could have the first line of Eq. (B14). As argued around Eq. (B11) and Eq. (B12), we could have the second line of Eq. (B14). Notice that the zero matrix in y0 1follows from the fact that the projector related to y0 1is orthogonal to the one related to z1. Similarly for the others. The same reasoning is used in the proofs for the other SI-C sets. Since h0 1⊥y0 1and h0 1⊥y0 2, h0 1= [−B†,−A†,1].(B15) The reason is the following. Denote h0 1= [M1, M2, M3], where Mi’s are invertible since the corresponding projector is not orthogonal to any projectors corresponding to the zi’s. From the discussion around Eqs. (B11) and (B12), we can set M3=1. From the orthogonality relation between projectors, we know that h0 1⊥y0 1. Thus, 0 = h0 1y0 1 †=M2+A†, which implies that M2=−A†. Similarly, the orthogonality relation h0 1⊥y0 2leads to M1=−B†. Similarly, y0 3⊥z3and y0 3⊥h0 1implies y0 3= [−AB−1,1,0],(B16) h2 3⊥y1 1and h2 3⊥y0 2implies h2 3= [−B†,−C†,1],(B17) 8 y2 3⊥h2 3and y2 3⊥z3implies y2 3= [−CB−1,1,0],(B18) h0 2⊥y1 1and h0 2⊥y0 3implies h0 2= [(BA−1)†,1,−(C−1)†],(B19) y1 2⊥h0 2and y1 2⊥z2implies y1 2= [AB−1,0, C],(B20) h1 3⊥y0 1and h1 3⊥y1 2implies h1 3= [−(BA−1C)†,−A†,1],(B21) y1 3⊥h1 3and y1 3⊥y1 2implies y1 3= [−C−1AB−1, A−1,1],(B22) h1 1⊥y1 1and h1 1⊥y1 3implies h1 1= [(BA−1CA−1)†,1,−(C−1)†],(B23) y2 2⊥h1 1and y2 2⊥z2implies y2 2= [C−1AC−1AB−1,0,1],(B24) h1 2⊥y0 2and h1 2⊥y1 3implies h1 2= [1,(AC−1AB−1)†,−(B−1)†],(B25) y2 1⊥h1 2and y2 1⊥z1implies y2 1= [0,1, AC−1A],(B26) h0 3⊥y0 3and h0 3⊥y2 2implies h0 3= [1,(AB−1)†,−(C−1AC−1AB−1)†],(B27) h2 1⊥y1 2and h2 1⊥y2 1implies h2 1= [−(BA−1C)†,−(AC−1A)†,1],(B28) h2 2⊥y0 1and h2 2⊥y2 3implies h2 2= [(BC−1)†,1,−(A−1)†].(B29) Then, h2 1⊥y2 3implies that (AC−1)3=1.(B30) In addition, h0 1⊥h1 1implies B†BA−1CA−1+A†+C−1= 0,(B31) h0 1⊥h2 1implies B†BA−1C+A†AC−1A+1= 0,(B32) h1 1⊥h2 1implies (BA−1CA−1)†(BA−1C)+AC−1A+(C−1)†= 0.(B33) By making use (AC−1)3=1, we obtain B†B+A†C+C−1A= 0 (B34) B†B+A†AC−1A+A−1C= 0,(B35) C†+A−1=C†CA−1+C−1.(B36) Similarly, h0 2⊥h1 2implies B†B+A†AC−1A+ (C−1A)†= 0,(B37) h1 2⊥h2 2implies B†B+ (C†AC−1A)†+A−1C= 0.(B38) Hence, we have A†A=C†C, AA†=CC†, C†+A−1=A†+C−1.(B39) Since we still have the freedom to choose the basis for the subspaces related to z2and z3, we can assume that Ais diagonal and non-negative. Since Ais invertible, all the diagonal items are positive. We claim that A=1, otherwise, without loss of generality, denote A=X0 01,(B40) where Xis a diagonal matrix whose diagonal terms are positive and different than 1. By solving Eq. (B39), we obtain that C=X0 0C22,(B41) where C22 is invertible. Hence, there is a state |s⟩in the subspace corresponding to the block Xsuch that A|s⟩=C|s⟩=C†|s⟩=x|s⟩,(B42) where x > 0. Therefore, ⟨s|(B†B+A†C+C−1A)|s⟩=⟨s|B†B|s⟩+(x2+1) >0, (B43) which contradicts Eq. (B34). Hence, A=1, CC†=1, C3=1, B†B+C+C−1= 0.(B44) Note that, if we rotate the basis of the subspaces span by z2and z3with the same unitary U, this does not affect A=1and Cis changed to UCU†. By choosing a suitable U,UCU†is a diagonal matrix according to the spectral theorem of norm matrix. Therefore, we can assume that Cis diagonal. Notice that 1−C3= (1− C)(1+C+C2) = 0, then we have 1+C+C2= 0, otherwise 1=C, which leads to the contradiction that BB†=−21. Consequently, all the diagonal terms in C are e±2π i /3and B†B=1. That is, Bis unitary. Since we 9 v1v2v3v4v5v6v7v8v9vAvBvCvDvEvFvGvHvI vi1100011¯ 1 1 t0t¯ 1 1 1 t0 0 0 vi201001¯ 1 1 1 0 t0 1 1 ¯ 1 0 t t 0 vi300100t t 0 1 ¯ 1¯ 1 0 t0¯ 1¯ 1 1 1 vi40001000t¯ 1¯ 1 1 t0t¯ 1 1 ¯ 1 1 wi1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 TABLE II. CEG-18. Each column vicorresponds to one observable represented by the projector |vi⟩⟨vi|. The labels correspond to those in with Fig. 1. The rows vij give the components of |vi⟩(unnormalized). ¯ 1 = −1 and t=√2. The last row contains the weights of the optimal SI-C witness of the form (1). The weights wij in (1) can be chosen in any way that satisfies wij ≥max{wi, wj}. With these weights, α(G, w) = 4 and Q(G, w) = ϑ(G, w) = 9 2. still have the freedom to choose the basis of the subspace spanned by z1, we can assume that B=1. Although we cannot change all the diagonal terms e±2π i /3in Cto be e2π i /3with unitary, it can be done with the time-reversal operator in some dimensions which changes i into − i . The time-reversal operator is also isometric. Therefore, up to isometry, Πi=|vi⟩⟨vi| ⊗ 1, where |vi⟩is the normalized vector of the i-th column, ¯x=−x,q=e2π i /3, and g=q2. 4. CEG-18 and its SI-C witness CEG-18 is the set of 18 rank-one projectors in d= 4 shown in Table II. Its graph of compatibility of CEG-18 is depicted in Fig. 1. CEG-18 was introduced in [42] and is an egalitarian Lov´asz-optimum SI-C set and a critical KS set. It can be proven that CEG-18 is the KS set of rank-one projectors with the smallest possible cardinality (in any dimension!) [46]. 1 4 7 A D G 2 3 5 6 8 9B C E F H I FIG. 1. Graph of compatibility of CEG-18. Nodes represent observables and edges connect compatible observables. The labels refer to the observables in Table II. The last row of Table II provides the weights corresponding to the optimal SI-C witness of the form (1). 5. Proof that CEG-18 is unique up to unitary transformations Suppose that you have the L1, . . . , L18 with the same relations of orthogonality as the vectors v1, . . . , v9, vA, . . . , vIin Table II and Fig. 1. Without loss of generality, we can assume that L1= [1,0,0,0], L2= [0,1,0,0], L3= [0,0,1,0], L4= [0,0,0,1].(B45) Without loss of generality, we can assume that L5= [1, A, 0,0].(B46) Then, L6⊥L4and L6⊥L5imply L6= [1,−(A−1)†, B, 0].(B47) Similarly, L7⊥L4and L7⊥L5imply L7= [1,−(A−1)†, C, 0].(B48) Since L7⊥L6, C=−[1+ (AA†)−1](B−1)†.(B49) From L18 ⊥L1and L18 ⊥L2, we can assume L18 = [0,0,1, D].(B50) Since L17 ⊥L1and L17 ⊥L18, L17 = [0, E, 1,−(D−1)†].(B51) In addition, L16 ⊥L1and L16 ⊥L18 imply L16 = [0, F, 1,−(D−1)†].(B52) Since L17 ⊥L16 and L16 ⊥L7, F= [1+(DD†)−1](E−1)†=−B−1(A+(A−1)†).(B53) 16 v1v2v3v4v5v6v7v8v9v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 v24 v25 v26 v27 v28 v29 v30 v31 v32 v33 vi1100001111 0 0 ¯ 1 1 ¯ 1100tttt¯ 1¯ 111¯ 1¯ 111tttt vi201011¯ 1 1 0 0 ¯ 1100tttt0 0 ¯ 1 1 ¯ 1 1 ¯ 1 1 tttt¯ 1¯ 111 vi30 0 1 ¯ 1 1 0 0 ¯ 1 1 tttt0 0 ¯ 1 1 ¯ 1100tttt¯ 1 1 ¯ 1 1 ¯ 1 1 ¯ 1 1 wi333111111111111111111111111111111 TABLE VI. Peres-33. Each column vicorresponds to one observable represented by the projector |vi⟩⟨vi|. The rows vij give the components of |vi⟩(unnormalized). ¯ 1 = −1 and t=√2. The last row contains the weights wiof the optimal SI-C witness of the form (1), where the weights wij in (1) can be chosen in any way that satisfies wij ≥max{wi, wj}. With these weights, α(G, w) = 12 and Q(G, w) = ϑ(G, w) = 13 for all qutrit states. to the subspace spanned by the image of ΠjΠiΠj. This leads to the linear conditions ⟨v|Πj|v⟩= 1,⟨v|ΠjΠiΠj|v⟩=⟨v|Πi|v⟩= 0.(C6) That is, Tjj = 1, Tii = 0,(C7) where Tlk =⟨v|ΠlΠk|v⟩. By definition, Tlk is positive semidefinite. In addition, the orthogonal relations imply that Tkl = 0,∀(k, l)∈E. (C8) The quantum violation of state |v⟩is a linear function of Tkk whose upper bound can be calculated through the SDP under the conditions ΠiΠj= 0 for any (i, j)∈E and (C6). It cannot always be smaller than Q−ϵfor all (i, j)∈ E. Hence, if, for any quantum state, the violation is not smaller than Q−ϵ, then rank(ΠjΠiΠj) = rank(Πj) for any (i, j)∈ E. Proof of (II). From (I), ∀(i, j)∈ E, rank(Πi)≥rank(ΠjΠiΠj) = rank(Πj),(C9) which implies that rank(Πi) = rank(Πj) for any (i, j)∈ E. Since the complement of Gis connected, we conclude that the rank of all the projectors are same. Proof of (III). Due to (I) and the existence of complete basis with cprojectors, we have the relation d=cκ. Then, each rank-κprojector Πican be decomposed as Πi= (L′ i)†L′ i, L′ i= [B1i, B2i, . . . , Bci],(C10) where L′ iare some d×κmatrices and Bji’s are some κ×κ matrices. In the scenarios we considered, we can always choose {Πj}c j=1 to be the complete basis and take it to be the standard basis, that is, L′ 1= [1κ,0,...,0],(C11a) L′ 2= [0,1κ,...,0],(C11b) . . . L′ c= [0,0,...,1κ].(C11c) Using the above decomposition, we have ΠjΠiΠj=B† jiBji.(C12) Due to (I) and Eq. (C12), for (i, j)∈ E,j∈ {1, . . . , c}, the rank of Bji should be κand hence, this matrix Bji is invertible. While for (i, t)∈Eand t∈ {1, . . . , c}the left-hand-side of Eq. (C12) is zero and therefore Bti = 0. Using that Bji’s are either invertible or zero-matrices, a straight-forward calculation shows that ∀i, Π2 i= Πi=⇒L′ i(L′ i)†=1κ(C13) and ΠiΠt= 0 =⇒L′ i(L′ t)†= 0.(C14) By taking Li=B−1 ji L′ ifor some invertible Bji and by using Eq. (C13), we can express Πias Πi= (L′ i)†(L′ i(L′ i)†)−1L′ i =L† i(LiL† i)−1Li.(C15) Finally, notice that Eq. (C14) implies LiL† t= 0. 1. Robustness with imperfect orthogonality relations To analyze the robustness when the orthogonality relations have not been exactly ensured, we introduce the following lemmas and one assumption. Lemma 2. For a given matrix T⪯λ11in H, if ⟨s|T|s⟩=λ1,∀|s⟩ ∈ S, where Sis a linear basis of H, we have T=λ11in H. Proof. Obviously, λ1should be the maximal eigenvalue of T. Consequently, ⟨s|T|s⟩=λ1implies that |s⟩ ∈ Eλ1, where Eλ1is the eigenspace of Tfor the eigenvalue λ1. Hence, by assumption, S⊆Eλ1. By definition, H= span(S), which implies H ⊆ Eλ1.(C16) Thus, Eλ1=H. Equivalently, T=λ11. Lemma 3. For any two projectors Π1,Π2from a given setting, if rank(Π1)>rank(Π2), then ∃|s⟩ ∈ S s.t. ⟨s|Π1|s⟩= 1,⟨s|Π2|s⟩= 0.(C17) 17 Proof. The condition rank(Π1)>rank(Π2) implies that the intersection S0of the subspace s represented by Π1 and 1−Π2is not empty, denote Π0the projector onto subspace S0. Von Neumann has proven that [70] Π0= lim n→∞[Π1(1−Π2)]n,(C18) where Πi, i = 0,1,2 represent the actions of projection instead of matrices. By definition, there should be at least one state |s0⟩ ∈ Ssuch that Π0|s0⟩ = 0, since Π0= 0. Thus, by repeating the projection Π1,Π2with the initial state |s0⟩, we can finally obtain a state |s⟩such that ⟨s|Π0|s⟩= 1. By definition of Π0, we know that Π1⪰Π0,Π2⪰Π0. Hence, we have ⟨s|Π1|s⟩=⟨s|1−Π2|s⟩= 1.(C19) Lemma 4. For any two projectors Π1and Π2from a given setting, ∀ϵ > 0,∃|s⟩∈Ssuch that ⟨s|Π1Π2Π1|s⟩ ≥ λ1(Π1Π2Π1)−ϵ, (C20) where λ1(·)is the maximal eigenvalue. Proof. If Π1⊥Π2, any choice of |s⟩gives the conclusion. Otherwise, λ1(Π1Π2Π1)>0. In addition, there is a state |s⟩such that Π1Π2Π1|s⟩ = 0. Denote by |sn⟩the post-measurement state after the repetition, ntimes, of measurements Π2and Π1, that is, |sn⟩ ∝ (Π1Π2Π1)n|s⟩.(C21) Since limn→∞(λ/λ1)n= 0, for any eigenvalue λof Π1Π2Π1which is less than λ1, we that Π1Π2Π1 λ1(Π1Π2Π1)n →Π0, n → ∞,(C22) where Π0is the projector of the eigenspace of Π1Π2Π1 with the maximal eigenvalue λ1. Therefore, ⟨sn|Π1Π2Π1|sn⟩ → λ1(Π1Π2Π1), n → ∞.(C23) For a given c-dimensional SI-C set which is considered here, and another realization of its orthogonality relations and quantum violation of a given witness with errors in experiment, we make the assumption of complete context for this realization. Assumption 1 (Complete context).There is a complete context, i.e., a context with cprojectors, in which the relations of mutual exclusivity are perfect. This can be guaranteed if we have a device whose different outcomes correspond to different projectors in this complete context. With out loss of generality, we label this complete context with {1,2, . . . , c}. As we can see, the proof of self-test only relies on two conditions: 1. The relations of exclusivity hold, i.e., ΠiΠj= 0, ∀(i, j)∈E. 2. Each projector Πican be decomposed into block form on a complete basis (a complete context in the scenario). That is, Πi=L† iLi, Li= [B1i, B2i, . . . , Bci],(C24) where Bti is square invertible if (i, t)∈ E, otherwise, Bti = 0. In actual experiments, the first condition may not strictly hold due, i.e., to noise, so two mutually exclusive events of the ideal scenario may no be mutually exclusive. Furthermore, the second condition is linked to the first and to the violation of the SI-C inequality. If the first condition is valid, and the violation of the SI-C inequality is not too far from the violation in an ideal experiment, then the second condition is likewise true. As a result, we have CFR as we have shown before. For a given (Q−ϵ, ϵ)-SI-C realization, the relation of exclusivity ΠiΠj= 0 is (ϵ, 1/2)-robust ∀(i, j)∈E. Since ΠiΠjΠi= (ΠjΠi)†(ΠjΠi), the maximal singular value of ΠjΠiis upper bounded by √ϵ. That is, the first condition holds up to O(√ϵ). Now we show that in a (Q−ϵ, ϵ)-realization, the second condition holds if ϵ < min{ϵτ, ϵν}, where ϵτand ϵνindicate the invertibility of the blocks Bki in the decomposition and the completeness of the basis for the decomposition. Definition 10. For a given graph Gand a set of weights {wi}, the tolerance function τij(θ, ϵ)is defined as follows: τij(θ, ϵ) := min Xij subject to Xn k=1 wkXkk =θ, Xkk =X0k,1≤k≤n, Xij ≥0, X00 =X0i= 1, |Xkt| ≤ ϵ, (k, t)∈E, X∈ S1+n +.(C25) For CEG-18 and Peres-24, τij(Q, 0) is a strictly positive constant ∀(i, j)∈ Ein both cases. For YO-13, we reach the same conclusion with the extra assumption in Eq. (C3). When ϵis not so large in comparison with τij(Q, 0), then the critical values of ϵτsuch that min(i,j)∈Eτij(Q−ϵ, ϵ)>0 are given in Table VII. For a given (Q−ϵ, ϵ)-realization, τij(Q−ϵ, ϵ)>0 implies that, for any state |s⟩such that ⟨s|Πi|s⟩= 1, we have ⟨s|ΠiΠjΠi|s⟩>0. Therefore, ΠiΠjΠiis positive definite in the subspace corresponding to Πi. Hence, rank(ΠiΠjΠi) is no less than the dimension of this subspace, which is rank(Πi). On the other hand, rank(ΠiΠjΠi)≤rank(Πi). Therefore, rank(ΠiΠjΠi) = rank(Πi).(C26) 18 Hence, rank(Πj)≥rank(Πi). If τji(Q−ϵ, ϵ)>0 is also true, then we have rank(Πj) = rank(Πi). If the complement graph of the exclusivity Gis connected and τij(Q−ϵ, ϵ)>0, ∀(i, j)∈ E, we know that all the projectors should be of the same rank. If ϵ < ϵτ/2, there is τ0>0 such that τij(Q−ϵ, ϵ)> τ0. This implies that the projection of Πjinto the subspace spanned by Πishould be invertible and the inverse is bounded by 1/τ0. In the case that i= 1, . . . , c, min τij(Q−ϵ, ϵ) = min λmin(B† tiBti). Hence, λmin(B† tiBti)> τ0implies that λmax[(B−1 ti )†B−1 ti ]≤1/τ0. Therefore, the maximal singular value σmax(B−1 ti )≤1/√τ0. Definition 11. For a given graph Gand set of weights {wi}, the completeness function ν(θ, ϵ)is defined as follows: ν(θ, ϵ) := min c X k=1 X0k subject to Xn k=1 wkXkk =θ, Xkk =X0k,1≤k≤n, Xij ≥0, X00 = 1, |Xkt| ≤ ϵ, (k, t)∈E, X∈ S1+n +.(C27) For a (Q−ϵ, ϵ)-realization, ν > 0 for any small enough ϵ. This means that any state is not orthogonal to all the projectors in the context {1,2, . . . , c}. Under the complete context assumption that all the projectors in this context are orthogonal to each other, we have, Pc i=1 Πi=1. Consequentially, we have the block decomposition on this complete basis {Π1,Π2,...,Πc}. Since B† tiBti is the non-trivial part of ΠtΠiΠtwhose maximal (diagonal) element is upper bounded by ϵ,|Bti|max ≤ q|B† tiBti|max ≤√ϵif (i, t)∈E. Here |M|max is the max norm of the matrix M. In fact, rows of Liare a orthonormal basis of the subspace represented by Πi. Then ΠiΠjΠi≤ϵ1implies LiΠjL† i≤ϵ1, which leads to σmax(LiL† j)≤√ϵ, (i, j)∈E, (C28) Here we introduce the main idea of robustness analysis. As an example, we give the detailed analysis for YO-13 in the next section. The proof of self-testing in the ideal case implies that, the solution of Eq. (C28) is unique if ϵ= 0. Since each Bti in Lithere is invertible and the inverse is bounded, those facts lead to equations similar as in Eq. (B84) but with error O(√ϵ). Consequently, this results in the conclusion that all Πiis close to the ideal one up to O(√ϵ). Note that, since a lot of substitution has been done in the proof of the self-testing in the ideal case, the constant in O(√ϵ) depends on the number of equation in the proof of self-testing, i.e., the number of edges in the corresponding exclusivity graph. Apart from that, it depends on the upper bound of the inverse of Bti SI-C set BBC-21 CEG-18 Peres-24 YO-13 ϵτ0.00359 0.00557 0.00562 0.00296 ϵν0.00832 0.01527 0.01954 0.02325 TABLE VII. Critical values of ϵτand ϵνsuch that min τij(Q− ϵτ, ϵτ)>0 and ν(Q−ϵν, ϵν)>0. also for (i, t)∈ E, which in turn depends on the error ϵas we discussed before. All in all, our conclusion is that the self-testing is √ϵ-robust when the error ϵis small enough as suggested in Table VII. 2. Robustness analysis for YO-13 As we discussed before, when the error ϵ≤ϵτ/2 as shown in Table VII, we can still have the decomposition Πi=LiL† i, Li= [B1i, B2i, B3i],(C29) where σmax(Bti)≤√ϵif (t, i)∈E, otherwise Bti is invertible and the inverse is bounded, i.e., σmax(B−1 ti )≤ 1/√τ0. Therefore, we can still, for example, change Li to B−1 ki Li, where (k, i)∈ E, to simplify the the procedure of proof. Consequently, σmax(Bti)≤pϵ/τ0if (t, i)∈E. (C30) Without loss of generality, we assume that L1= [1d,0,0], L2= [0,1d,0], L3= [0,0,1d].(C31) For simplicity, in the following derivation, we will omit the subindex d. Since L1⊥L4and L1⊥L5up to some error (for simplicity, hereafter we will omit ‘up to some error’), L4= [O(√ϵ),1, A], L5= [O(√ϵ),1, A′],(C32) where O(√ϵ) represents either a number or a matrix whose maximal singular value is not larger than O(√ϵ). For brevity, hereafter we will denote o=O(√ϵ). L4⊥L5 implies σmax(AA′+1+o) = o. (C33) Therefore, σmax(A′+A−1)≤O(pϵ/τ0) = o. (C34) In this sense, we denote A′=−A−1+o. Similarly, L6= [1, o, B], L7= [1, o, −(B−1)†+o], L8= [1, C, o], L9= [1,−(C−1)†+o, o].(C35) Let us assume that L10 = [1, D, E], L11 = [1, F, G], L12 = [1, H, I], L13 = [1, J, K].(C36) 19 Then, L10 ⊥L6and L10 ⊥L8imply D=−(C−1)†+o, E =−(B−1)†+o, (C37) L11 ⊥L7and L11 ⊥L9imply F=C+o, G =B+o, (C38) L12 ⊥L6and L12 ⊥L9imply H=C+o, I =−(B−1)†+o, (C39) L13 ⊥L7and L13 ⊥L8imply J=−(C−1)†+o, K =B+o. (C40) In addition, L4⊥L10 and L4⊥L11 imply D=−EA†+o, F =−GA†+o. (C41) Therefore, (C−1)†=−(B−1)†A†+o, C =−BA†+o, (C42) which implies C=−BA−1+o=−BA†+o. (C43) Then, L4⊥L12 and L5⊥L13 imply H=IA−1+o, J =KA−1+o. (C44) This implies C=−(B−1)†A−1+o, −(C−1)†=BA−1+o, (C45) which implies C=−(B−1)†A−1+o=−(B−1)†A†+o. (C46) Hence, AA†=1+o, BB†=1+o. (C47) Since we still have the freedom to rotate the subspaces corresponding to L2and L3, we can set Aand Bto be Hermitian. Therefore, the square of any eigenvalue of A and Bis 1 + o, which means that the eigenvalues of A, B are either 1 + oor −1 + o. Without loss of generality, we assume they are all −1 + o. Hence, A=−1+oand B=−1+o. Consequently, C, F, G, H, K =−1+o, D, E, I, J =1+o. (C48) Then, by definition of Li, we know that its difference between the ideal one is also o, i.e., O(√ϵ). So do LiL† i and its inverse since each element in Liis bounded. The fact that Πi=L† i(LiL† i)−1Li(C49) implies that Πiis also O(√ϵ) close to the ideal one in the sense of maximal singular value. Appendix D: Tools used in the proof of Result 3 Initially, it is important to note that a witness can be naturally constructed for any given complete KS set. Furthermore, the maximal violation of this witness is attained by any set of projectors in an arbitrary dimension that satisfies the orthogonality and completeness relations according to the orthogonality graph Gassociated with the given KS set. In Result 3, we specifically refer to the CFR of a complete KS set with respect to the following particular witness. Given a complete KS set, consider an SI-C witness of the form WKS =X i wiP(Πi= 1) ⩽α(G, w),(D1) where wiis the number of bases in which projector Πi appears, and α(G, w) is the independence number of the orthogonality graph Gof the projectors {Πi}. Let us denote the maximal cliques, each of which corresponds to a complete basis, by the subsets of the set of vertices Tx⊂V, wherein x= 1, . . . , m, and there are mnumber of complete bases. Now the quantity X i wiΠi=X x X i∈Tx Πi!=m1,(D2) where we have used the fact that the sum of projectors in each of these maximal cliques is the identity. Hence, the quantum value of the witness (D1) is mfor any state. Note that mis strictly greater than α(G, w), which follows from the definition of the KS set. Furthermore, since the maximal value of Pi∈TxP(Πi= 1) cannot be more than 1, mserves as the quantum upper bound. On the reverse direction, the witness (D1) is maximally violated by a set of projectors {Πi}only when X i∈Tx Πi=1,∀x. (D3) Additionally, these projectors adhere to the orthogonality graph. Thus, any set of projectors providing the maximum violation of witness (D1) must be a KS set according to the orthogonality graph G. 1. Concepts and previous results Definition 12 (Bipartite game).A bipartite game G= (X×Y, A ×B, W)is a game involving two players, Alice and Bob. In each round of the game, Alice receives an input x∈Xand provides an output a∈A, and Bob receives an input y∈Yand provides an output b∈B. Alice and Bob win the round if the inputs and outputs satisfy a winning condition W⊆(X×Y)×(A×B). Definition 13 (Wining strategy).A winning strategy for a bipartite game G= (X×Y, A ×B, W )is a strategy 20 according to which for every (x, y)∈X×Y, Alice and Bob output aand b, respectively, such that (x, y, a, b)∈ W. It is known (see, e.g., [21]) that, for any KS set, there exists a context-projector KS game with a quantum winning strategy and no classical winning strategy. Consider the graph of orthogonality Gof a complete KS set of n projectors {Πi}n i=1 in C d. Each projector is represented by a vertex in the graph, and orthogonal projectors are represented by adjacent vertices. A clique in Grepresents a set of mutually orthogonal projectors. A maximal clique of Gis a clique that cannot be extended by including one more adjacent vertex. Let Txdenote maximal cliques of G, having |Tx|distinct elements. The elements of Txare {Tx,a}, where xis the label of the clique and a= 1,...,|Tx|indicates the elements in that clique. In other words, Tx:= {Tx,1, . . . , Tx,|Tx|}. Suppose there are mdifferent maximal cliques, i.e., x= 1, . . . , m. According to the graph G, Alice is given a maximal clique from the set of maximal cliques and has to output one of the vertices from that clique, while Bob receives a vertex from that clique and outputs either 0 or 1. They win the game if Alice outputs the vertex that is given to Bob and Bob outputs 1, or if Alice outputs a vertex that is not given to Bob and Bob outputs 0. Using the formal notation, Alice receives x∈ {1, . . . , m}and outputs a∈ {1,...,|Tx|}, while Bob receives y∈ {1, . . . , n}and outputs b∈ {0,1}. The payoff function that they aim to maximize is given by B=X a,b,x,y ca,b,x,y p(a, b|x, y) (D4) where ca,b,x,y =     1,if y∈Tx, y =Tx,a and b= 0, 1,if y∈Tx, y =Tx,a and b= 1, 0,otherwise. (D5) The following quantum strategy achieves perfect winning. Alice and Bob share the maximally entangled state |ϕ+ d⟩=1 √d d−1 X i=0 |ii⟩.(D6) Alice measures the following observable Axfor input x that corresponds to the basis Tx, Ax={Π∗ Tx,1,...,Π∗ Tx,|Tx|}.(D7) And Bob measures By={1−Πy,Πy}(D8) for input y. Using the fact that (A⊗B)|ϕ+⟩= (1⊗ BAT)|ϕ+⟩for any operators A, B, we obtain the winning conditions for every pair of inputs. To show that no winning classical strategy (without communication) exists, note that the best classical (local) strategy can be assumed to be deterministic, in which Bob assigns 0 or 1 values to the nvertices. To win every round of the game, no two orthogonal projectors can be assigned both 1, since every pair of projectors belongs to at least one basis in a complete KS set. Moreover, only one projector is assigned 1 in every context. Such an assignment is impossible for a generalized KS set. 2. Proof of Result 3 The implication — if a complete KS set cannot be certified with CFR then the corresponding context-projector KS game does not admit Bell self-testing — is straightforward. The uncharacterized KS set acting on C Dis denoted by {Πi}, and the reference KS set acting on C d is denoted by {Πi}. Let us assume that these two KS sets are not connected by unitary transformations, that is, ∄U, UΠiU†= Πi⊗1⊕Π∗ i⊗1,∀i. (D9) Now, consider the following two realizations of a contextprojector KS game where the local observables are constructed from {Πi}and {Πi⊕Π∗ i}according to (D7)- (D8), and the shared states are |ϕ+ D⟩and (1/2) |ϕ+ d⟩ ⊕ |ϕ+ d⟩, respectively. Due to (D9), there is no local unitary transformation on each side that can map the local measurements in one quantum strategy to the other. Therefore, the context-projector KS game does not admit Bell self-testing. To show the reverse implication, it suffices to establish that in any quantum winning strategy, Bob’s measurements (D8){1−By, By}and Alice’s measurements {Aa x}should be projective and constitute a KS set. Consequently, it follows that if there is no unitary transformation between two quantum strategies, then the KS set does not satisfy CFR. Let ρbe the shared state in the quantum strategy, and dAand dBbe the local dimension of the reduced states. Without loss of generality, we can assume that the POVMs of Alice and Bob are in their respective dA- dimensional and dB-dimensional Hilbert spaces. If not, we can consider the projection of these POVMs into the corresponding subspaces. Bob’s measurements. Let us denote ρa x= trA(ρAa x), the unnormalized reduced state on Bob’s side when outcome ais observed for the measurement setting x. The fact that PaAa x=1implies X a trB(ρa x) = trB X a ρa x!= tr ρX a Aa x!= 1. (D10) From the winning conditions of the context-projector game, we know that p(¯a, 1|x, y) + X a=¯a p(a, 0|x, y)=1,(D11) 21 where y=Tx,¯a, which translates to tr(ρ¯a xBy) + X a=¯a tr (ρa x(1−By)) = 1.(D12) We represent Sx,a as the subspace spanned by ρa x, where 1x,a stands for the identity operator in Sx,a. Additionally, B|Sdenotes the restriction of the operator Bto the subspace S. By combining Eqs. (D10) and (D12), we obtain tr(ρ¯a xBy) = tr(ρ¯a x) + tr  X a=¯a ρa xBy .(D13) Since By⪯1, it follows from Eq. (D13) that By|Sx,¯a=1x,¯a, By|˜ Sx,¯a= 0,(D14) where ˜ Sx,¯a=⊕a=¯aSx,a.(D15) Therefore, Sx,a ⊥Sx,¯aif a= ¯a. Moreover, since ˜ Sx,¯a⊕ Sx,¯ais the full space of Bob’s local system, By=1x,¯a.(D16) This leads to the orthogonality conditions, ByBy′= 0 whenever y=y′and y, y′∈Tx. In addition, due to the fact that Paρa xis of full rank, the normalization condition also holds. That is, X y∈Tx By=X a 1x,a =1.(D17) Alice’s measurements. Similarly, let us denote the reduced states on Alice’s side by σy= trB(ρBy),¯σy= trB(ρ(1−By)), for the measurement setting y, such that tr(σy) + tr(¯σy)=1.(D18) The winning condition Eq. (D12) implies tr(σyA¯a x) + X a=¯a tr(¯σyAa x) = 1,(D19) which leads to tr(σyA¯a x) = tr(σy),tr(¯σyA¯a x) = 0.(D20) Hence, A¯a x|Sy=1y, A¯a x|˜ Sy= 0,(D21) where Sy,˜ Syare the space spanned by σy,¯σy, respectively. Moreover, we can infer that Sy⊥Sy′∀y, y′∈ Tx, y =y′. Since σy+ ¯σy= trB(ρ) is of full rank, we know that A¯a x=1y.(D22) Subsequently, we have the orthogonality condition 1y11y2= 0 whenever y1=Tx,a1, y2=Tx,a2and a1=a2, and the normalization condition X a Aa x=X y∈Tx 1y=1.(D23) In total, Eqs. (D16), (D17), (D22), and (D23) imply that {By}and {A¯a x}are two realizations of the KS set, and thus, if the local measurements in a quantum winning strategy cannot be Bell self-tested, then there exist inequivalent realizations of the KS set. 3. Bell self-testing of the maximally entangled state If specific conditions are met by the complete KS set, it becomes possible to self-test the maximally entangled state. To simplify matters, here, we will not delve into the necessary and sufficient conditions. Instead, we will demonstrate that the maximally entangled state can be reliably self-tested if a complete KS set admits CFR and satisfies the following criteria: In one realization of the KS set, there exist two bases made of rank-one real projectors, {|i⟩⟨i|}d i=1 and {|vi⟩⟨vi|}d i=1, such that ⟨i|vj⟩ = 0 for all i, j. Note that, without loss of generality, we can consider one basis to be the computational basis. To prove it, we suppose that the uncharacterized projectors {A¯a x}and {By}act on C dAand C dB, respectively, and the unknown shared state is ρ. Since the set of projectors admits Bell self-testing, there exist local unitaries UAand UBsuch that UA(A¯a x)U† A= Πy⊗1dA d , UB(By)U† B= Πy⊗1dB d ,(D24) wherein {Πy}are the projectors in those two bases in the reference d-dimensional KS set. Let us denote the reduced state of (UA⊗UB)(ρ)(UA⊗UB)†onto the subspace C d⊗ C dby ρAB. The winning conditions given by Eq. (D11) implies p(a, 1|x, y) = 0,if y=Tx,a.(D25) The relation must hold for any state |ϕ⟩that belongs to the support of ρAB, that is, ∀(y, y′)∈E, (Πy⊗Πy′)|ϕ⟩= 0.(D26) After substituting |ϕ⟩=Pi,j cij|ij⟩and the computational basis {|i⟩⟨i|} in Eq. (D26), we obtain that cij = 0,for i=j. (D27) Therefore, we can express |ϕ⟩as |ϕ⟩= (1⊗S)|ϕ+ d⟩,(D28) where Sis a diagonal matrix whose elements can be taken to be positive by exploiting the freedom of local unitary. 22 Taking yand y′from the other basis {|vi⟩⟨vi|}d i=1, it follows from (D25) that (|vi⟩⟨vi|⊗|vj⟩⟨vj|)(1⊗S)|ϕ+ d⟩= 0.(D29) Using the fact that (A⊗B)|ϕ+⟩= (1⊗BAT)|ϕ+⟩, we obtain (1⊗|vj⟩⟨vj|S|vi⟩⟨vi|)|ϕ+ d⟩= 0,(D30) which implies ⟨vj|S|vi⟩= 0,∀i=j. (D31) This relation can be rephrased as ⟨j|U†SU|i⟩= 0,∀i=j, (D32) where Uis the unitary such that |vi⟩=U|i⟩. Therefore, U†SU is also diagonal. Furthermore, since unitary does not change the eigenvalues, there exists a permutation matrix Tsuch that V=UT and V†SV =S⇔[S, V ]=0.(D33) Finally, due to the fact that ⟨i|vj⟩ = 0 for all i, j, all the elements in Uare non-zero, and thus, all the elements in Vare also non-zero. Consequently, the commutation relation [S, V ] = 0 holds only if all the eigenvalues of S should be the same, i.e., S=1. 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