Converting contextuality into nonlocality

Converting Contextuality into Nonlocality Adán Cabello * Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain and Instituto Carlos I de Física Teórica y Computacional, Universidad de Sevilla, E-41012 Sevilla, Spain (Received 30 November 2020; revised 20 April 2021; accepted 7 July 2021; published 11 August 2021) We introduce a general method which converts, in a unified way, any form of quantum contextuality, including any form of state-dependent contextuality, into a quantum violation of a bipartite Bell inequality. As an example, we apply the method to a quantum violation of the Klyachko-Can-Binicioğlu-Shumovsky inequality. DOI: 10.1103/PhysRevLett.127.070401 Introduction.—Nonlocal games [1,2] provide an intuitive understanding of where the advantage of quantum resources lies and a framework, used in computer science [3],to analyze quantum protocols. Contextuality is known to be a crucial resource for some forms of computation with quantum speed up [4–6]. However, although some forms of contextuality can be converted into nonlocal games, there is no universal method for converting any form of contextuality into a nonlocal game. The aim of this Letter is to introduce a unified method that achieves this task. The research on how contextuality can be converted into nonlocality started with the works of Stairs [7] and Heywood and Redhead [8], extending the proofs of the Kochen-Specker (KS) theorem [9] to bipartite scenarios with entanglement and has evolved in many ways in connection to extensions of the KS theorem [10,11], Bell inequalities [12–14], and nonlocal games [15–20]. So far, the forms of contextuality that can be converted into nonlocality are (i) Those forms of state-dependent contextuality (SD-C) corresponding to scenarios whose measurements can be distributed between two or more parties in such a way that each party has at least two incompatible measurements. (ii) Those that are produced by KS sets [9] (i.e., sets of rank-one projectors which do not admit a “KS assignment,”i.e., an assignment of 0 or 1 satisfying that two orthogonal projectors cannot both have assigned 1, and for every set of mutually orthogonal projectors summing the identity, one of them must be assigned 1) or by proofs of the KS theorem (e.g., [21,22]) that can be reduced to KS sets [23,24]. For methods of conversion, see, e.g., [1,14]. (iii) Those produced by some particular state-independent contextuality (SI-C) sets [25] (i.e., sets of projectors which produce noncontextual behaviors for any initial state) that are not KS sets. For methods of conversion, see [26]. (iv) In addition, some constraint satisfaction problems and local no-hiddenvariables proofs can be converted into nonlocal games [17–20]. In each case, “convert”may mean a different thing. Forms of contextuality that we do not know how to convert into nonlocality are those produced by sequentially measuring noncomposite systems initially prepared in specific states in contextuality scenarios which cannot be embedded in Bell scenarios. A particularly relevant example is the quantum violation of the KlyachkoCan-Binicioğlu-Shumovsky (KCBS) inequality [27] with single qutrits. This is arguably the most fundamental form of quantum SD-C produced by noncomposite systems, as the KCBS inequality is the only nontrivial tight noncontextuality inequality [28] in the scenario with the smallest number of measurements in which qutrits produce contextuality (qutrits are the quantum systems of smallest dimension that produce contextuality [9]), and because it plays a crucial role for understanding quantum contextuality [29–31]. The aim of this Letter is to provide a general unified method capable of converting any form of SD-C or SI-C into bipartite nonlocality. The philosophy behind the method is guided by the recognition of the singular role of SI-C, as pointed out in, e.g., [32] (“we argue that a primitive entity of contextuality should embrace stateindependence”). The method takes any set of measurements that provides SD-C and identifies the minimal extension of it that provides SI-C and then converts the SI-C into bipartite nonlocality preserving the gap between quantum and noncontextual theories in the SI-C (which becomes the gap between quantum and local theories). First, we describe the method, which has three steps. Then, we apply the method to a quantum violation of the KCBS inequality. Finally, we provide an intuitive explanation of how it works and discuss its virtues and limitations. Method.—An ideal measurement of an observable Ais a measurement of Athat yields the same outcome when repeated and does not disturb any compatible (i.e., jointly measurable) observable. A context is a set of ideal measurements of compatible observables. A scenario is characterized by a number of measurements, their PHYSICAL REVIEW LETTERS 127, 070401 (2021) 0031-9007=21=127(7)=070401(7) 070401-1 © 2021 American Physical Society outcomes, and relations of compatibility [31]. In quantum theory, every ideal measurement is represented by the spectral projectors of a self-adjoint operator, and compatible observables correspond to commuting operators. A behavior for a scenario (i.e., a set of probability distributions for each of its contexts) is contextual if the probability distributions for each context cannot be obtained as the marginals of a global probability distribution on all observables. Otherwise, the behavior is noncontextual. Contextuality is detected by the violation of noncontextuality inequalities whose bounds are derived solely from the assumption of outcome noncontextuality [25,27,33–35]. Any quantum contextual behavior can be produced by a set of rank-one projectors S¼fΠ1;…;Πngacting on a quantum state jψiin a Hilbert space of dimension d≥3[29].GivenS, contexts are subsets of Scontaining mutually commuting projectors. Step 1: A SI-C set is critical if by removing any of its elements the resulting set is not a SI-C set. A KS set is critical [36] if by removing any of its elements the resulting set is not a KS set. Here we show that every set Sproducing SD-C can be extended into a critical SI-C set S00 ¼S∪S0. To prove this, we need the following result [37,38].In d≥3, given any two nonorthogonal rank-one projectors ΠAand ΠB, there is a set of projectors Esuch that, for any KS assignment f,fðΠAÞþfðΠBÞ≤1. The set ΠA∪E∪ ΠBis called a true-implies-false set (TIFS) [39], definite prediction set [37], 01-gadget [38], or Hardy-like proof [40]. The construction of a critical SI-C set containing Sis as follows. Let Gbe the graph of orthogonality of S.LetNbe the minimum number of disjoint bases that cover all the vertices of G.IfSallows for SD-C, then N≥3[29].If N<dþ1, then we add disjoint bases until the total of number of disjoint bases is Nþ1. Then, we use the construction shown, for d¼3,inFig.1(a) and, for d¼4, in Fig. 1(d), and which works similarly for any d≥5, based on creating TIFSs between some specific nodes. If N>dþ1, then we use the construction shown, for different combinations of dand N,inFigs.1(b),1(c), or 1(e). In all cases, the resulting set is a critical KS set in dimension dfor the reasons explained in Fig. 1. If one removes any of the nodes in each of the constructions in Fig. 1, then the resulting set admits a KS noncontextual assignment. Some SI-C sets are not KS sets (e.g., [25,41,42]). Hence, the resulting critical KS set could, in principle, not be a critical SI-C set. However, this problem can be solved by suitably choosing the extra nodes used for the TIFSs in Fig. 1 [43]. A minimal critical SI-C set is a critical SI-C set of minimum cardinality. The previous proof guarantees that critical SI-C sets containing Sexist. However, the method used in the proof does not guarantee that the resulting critical SI-C set is minimal. To obtain a minimal critical SIC set S00 from S, we can use the following results. Let us call Gthe graph of orthogonality of S00, and let dbe the dimension of the Hilbert space. Necessary conditions for S00 to be a SI-C set are that the chromatic number of Gsatisfies χðGÞ>d [55] and that the fractional chromatic number satisfies χfðGÞ>d[56,57]. These conditions allow us to identify candidates to be minimal critical SI-C sets containing any given SD-C set. Then, we can use the necessary and sufficient condition for being a SI-C set [57] to check whether or not they are SI-C sets. This condition states that a set of rank-one projectors S00 ¼fΠi;…;Πngis a SI-C set if and only if there are nonnegative numbers w¼ ðw1;…;w nÞand a number 0≤y<1such that Pj∈Iwj≤ yfor all I, where Iis any independent set of G, and PiwiΠi≥1. In practice, finding a critical SI-C set containing Sis not a problem. However, proving that it has minimal cardinality may be difficult [43]. See [40,68] for examples of such proofs [43]. Nevertheless, minimality is only required for elegance; to connect SD-C to nonlocality, what matters is the criticality of the SI-C set. Step 2: As pointed out in [57], the weights wneeded to guarantee that S00 is a SI-C set generate a noncontextuality inequality violated by any quantum state. The results in [29,54] allow us to express this inequality as (a) (b) (c) (d) (e) FIG. 1. Every node represents a rank-one projector. A continuous vertical line between d≥3nodes indicates that they are mutually orthogonal. Hence, in dimension d, in any KS assignment, one of them has to be assigned 1. A dashed line between two nodes indicates that there is a TIFS between (and including) them. Hence, in any KS assignment, both of them cannot be assigned 1. Construction to obtain a critical KS set in dimension d≥3from N≥dþ1disjoint bases: (a) For d¼3and N¼dþ1. (b) For d¼3and N¼dþ2. (c) For d¼3and N¼dþ3. (d) For d¼4and N¼dþ1. (e) For d¼4 and N¼dþ2. The construction works similarly for any d≥ 3and N≥dþ1. In all cases, it is impossible to assign to the depicted nodes the values 0 or 1 satisfying that one of the dnodes in each continuous vertical line must be 1, while nodes connected by a dashed line cannot both be 1. However, such an assignment is possible whenever we remove any of the depicted nodes. PHYSICAL REVIEW LETTERS 127, 070401 (2021) 070401-2 X i∈VðGÞ wiPðΠi¼1Þ−X ði;jÞ∈EðGÞ maxðwi;w jÞPðΠi¼1;Πj¼1Þ≤ NCHVαðG;wÞ;ð1Þ where PðΠi¼1;Πj¼1Þis the probability of obtaining outcome 1 in the measurement associated to Πi(which has possible outcomes 1 and 0) and also in the measurement associated to Πj,VðGÞ,andEðGÞare the sets of vertices and edges of G, respectively, αðG;wÞis the independence number of ðG;wÞ[i.e., the graph in which weight wiisassignedtoeachi∈VðGÞ], and NCHV stands for noncontextual hidden-variable theories. The independence number of a (weighted) graph is the cardinality of its largest set of vertices (taking their weights into account) such that no two are adjacent. Step 3: This step has two ingredients. One is the following method, introduced in [7,8] and used extensively since then to embed a KS set in a bipartite Bell scenario. In each run of the experiment, we prepare a pair of particles in the two-qudit maximally entangled state jΨi¼ 1 ffiffiffi d pX d−1 k¼0jkki;ð2Þ distribute one particle to Alice and the other to Bob, and allow Alice (Bob) to freely and independently choose and perform one measurement from S00 (from the set obtained by taking the complex conjugate of the elements in S00). Here, we apply this embedding not only to KS sets but to any SI-C set. The second ingredient is the observation that the behavior produced by this state and these measurements violate the following Bell inequality: X i∈VðGÞ wiPðΠA i¼1;ΠB i¼1Þ−X ði;jÞ∈EðGÞ maxðwi;w jÞ 2½PðΠA i¼1;ΠB j¼1ÞþPðΠA j¼1;ΠB i¼1Þ ≤ LHVαðG;wÞ;ð3Þ where PðΠA i¼1;ΠB j¼1Þis the probability that Alice obtains outcome 1 for measurement Πion her particle and Bob obtains outcome 1 for measurement Πjon his particle. LHV stands for local hidden-variable theories. That (3) is a Bell inequality follows from the fact that, for LHV theories, the maximum of the lefthand side of (3) is always attained by a deterministic assignment for the outcomes of the elements of S00 in Alice’s particle and a deterministic assignment for the outcomes of the elements of the complex conjugate of S00 in Bob’s particle. To maximize the left-hand side of (3), we need to maximize (taking into account the weights) the number of projectors Πito which outcome 1 is assigned both in Alice’sandBob’s particles, while minimizing the number of adjacent Πjto which outcome 1 is assigned, which is exactly the definition of independence number of a (weighted) graph ðG;wÞ. We can translate this violation into a nonlocal game with quantum advantage following the method in [2] (Sec. II.B4). The interest of Bell inequality (3) comes from the following observations. Noncontextuality inequalities of the form (1) are in one-to-one correspondence with Bell inequalities of the form (3). The noncontextual bound in (1) is equal to the local bound in (3). The quantum violation of (1) for the maximally mixed state using S00 is equal to the quantum violation of the Bell inequality (3) for the maximally entangled state (2) and using S00 in Alice’s side and the complex conjugate of S00 in Bob’s side. Moreover, if S00 admits a weight wfor which the left-hand side of (1) is represented in quantum theory by λ1with λ>αðG;wÞ(as is the case in many critical SI-C sets, e.g., [25,58,78]), then, all quantum states violate inequality (1) by the same value, and this violation coincides with that of the Bell inequality (3) for state (2). Overall, step 3 is an interesting result by itself, as it applies to any SI-C set (and not only to sets that can be reduced to KS sets, as [1,14]) and preserves the gap between quantum and noncontextual theories (while previous methods [1,14,26] do not). Converting KCBS contextuality into nonlocality.—Here, we apply the method described above to a set of projectors [74,75] leading to a violation of the KCBS inequality [27]. The method works for any form of contextuality. The example has been chosen for its relevance and simplicity, as we can use a previous result [68] to identify S00. Consider S¼fΠ1;…;Π5g, where Πi¼jviihvij, with jv1i¼ð1;0;0ÞT;ð4aÞ jv2i¼ 1 ffiffiffi 2 pð0;1;1ÞT;ð4bÞ jv3i¼ 1 ffiffiffi 3 pð1;−1;1ÞT;ð4cÞ PHYSICAL REVIEW LETTERS 127, 070401 (2021) 070401-3 jv4i¼ 1 ffiffiffi 2 pð1;1;0ÞT;ð4dÞ jv5i¼ð0;0;1ÞT:ð4eÞ These measurements violate the KCBS inequality [27], which can be written [54] as X i∈VðGÞ PðΠi¼1Þ−X ði;jÞ∈EðGÞ PðΠi¼1;Πj¼1Þ≤αðGÞ; ð5Þ where Gis the graph in Fig. 2(a), for which αðGÞ¼2.For example [74,75], the state jψi¼ð1=ffiffiffi 3 pÞð1;1;1ÞTgives 2þ1 9, which violates inequality (5). Step 1: The smallest critical SI-C set S00 that contains Sis the Yu-Oh set [25]. This follows from the proof in [57] that the Yu-Oh set is the SI-C set of rank-1 projectors with minimum cardinality. Therefore, S0¼fΠ6;…;Π13g, where Πi¼jviihvij, with jv6i¼ 1 ffiffiffi 2 pð0;1;−1ÞT;ð6aÞ jv7i¼ 1 ffiffiffi 3 pð1;1;1ÞT;ð6bÞ jv8i¼ 1 ffiffiffi 2 pð1;−1;0ÞT;ð6cÞ jv9i¼ 1 ffiffiffi 2 pð1;0;−1ÞT;ð6dÞ jv10i¼ 1 ffiffiffi 2 pð1;0;1ÞT;ð6eÞ jv11i¼ð0;1;0ÞT;ð6fÞ jv12i¼ 1 ffiffiffi 3 pð−1;1;1ÞT;ð6gÞ jv13i¼ 1 ffiffiffi 3 pð1;1;−1ÞT:ð6hÞ The graph Gthat represents the relations of orthogonality between the projectors S00 ¼fΠ1;…;Π13gis shown in Fig. 2(b). Step 2: The set of weights fw1;…;w 13gleading to the largest gap between quantum and noncontextual theories for inequality (1) for S00 is wi¼2for i¼3, 7, 12, 13, and wi¼3, otherwise. See Fig. 2(c). This follows from the observation that, in this case, the noncontextuality inequality (1) has αðG;wÞ¼11, while it is violated by any quantum state of dimension d¼3, since, for any initial state (including the maximally mixed state), the left-hand side of (1) is 1 3ð2×4þ3×9Þ¼11 þ2 3. Step 3: Distributing pairs of particles in the maximally entangled state (2), with d¼3, between Alice and Bob and allowing each of them to perform a randomly chosen spacelike separated measurement from S00 (in this case, S00 and its complex conjugate are equal), we obtain a nonlocal behavior as the local bound of the Bell inequality (3) is αðG;wÞ¼11, while the value for the left-hand side of (3) is, again, 11 þ2 3. Explanation, virtues, and limitations.—Here, we give someintuitionofhowthe method works. The set ofstates(in dimension d≥3) that yield contextual behaviors grows as the set of measurements grows from Sto S00. For example, while the state jψ0i¼ð1=ffiffiffi 3 pÞð1;−1;1ÞTdoes not violate inequality (5), it violates a similar noncontextuality inequality replacing Sby fΠ1;…;Π9g[79]. When all the measurements in S00 are used, then even the maximally mixed state produces contextuality and weights can be adjusted [57] to produce equal state-independent violation of a noncontextuality inequality for all states [25,33–35]. The Bell inequality (3) follows from the SI-C inequality (1) by noticing that (1) can be tested in experiments consisting of two sequential measurements on a maximally mixed state. We can assume that each of these measurements is performed by a different party. Sometimes Alice is the first to measure and Bob the second, and sometimes vice versa. Sometimes both parties measure the same Πi, sometimes they measure different but compatible projectors. This view leads to the Bell inequality (3) which shares the classical bound and it is also violated by the same amount when preparing pairs in state (2) and giving one particle to Alice and the other to Bob, as, in this case, Alice’s and Bob’s outcomes are perfectly correlated, and Alice’s and Bob’s local states are maximally mixed states. FIG. 2. (a) Five-vertex graph Gthat represents the relations of orthogonality between the projectors S¼fΠ1;…;Π5gneeded to violate the KCBS inequality (5). Projector Πiis represented by vertex i, mutually orthogonal projectors are represented by adjacent vertices. (b) Extended 13-vertex graph Grepresenting the relations of orthogonality between elements of the smallest SI-C set S00 ¼fΠ1;…;Π13gthat contains S. (c) Vertex-weighted graph ðG;wÞwith the weights that produce the largest SI-C. Vertices in white have weight 2 and vertices in black have weight 3. These are the weights used in the SI-C inequality (1) and the Bell inequality (3). PHYSICAL REVIEW LETTERS 127, 070401 (2021) 070401-4 Virtues: (I) While inequalities (1) and (5) are noncontextuality inequalities that might only be testable by performing sequential nondemolition measurements on single systems [80–82], inequality (3) is a Bell inequality that can be tested by performing local measurements on spatially separated systems and can be converted into a nonlocal game. (II) The “compatibility”or “sharpness” loophole [83] in contextuality experiments with sequential measurements vanishes in the Bell test, as, there, measurements do not need to be ideal (or sharp) [31] and observables on different particles are automatically compatible. (III) The gap between quantum and local theories for the Bell inequality (3) is the same as the gap between quantum and noncontextual theories for the SI-C inequality (1), and both are produced using the same measurements. (IV) The violation of the Bell inequality (3) by the measurements in S00 and state (2) vanishes whenever we remove from S00 any element of S. This follows from the fact that, in that case, inequality (1) is not violated by the maximally mixed state. Therefore, the maximally entangled state (2) fails to violate the Bell inequality (3), as the local states of Alice and Bob are maximally mixed. This property follows from the fact that S00 is a critical SI-C set. (V) There is no “contextuality-nonlocality tradeoff”[84,85]. The quantum violations of the SI-C inequality (1) and the Bell inequality (3) can be tested simultaneously in the same experiment. According to quantum theory, the experiment would give (equal) violations of both inequalities. The violation of (1) can be observed by allowing one of the parties, e.g., Alice, to perform sequential measurements. The violation of (3) can be observed by considering the first (or second) measurements of Alice and the (only) measurements of Bob. It would be interesting to observe these simultaneous violations in an actual experiment. Limitations: Except for the case S¼S00, the nonlocal behavior resulting from the application of this method does not have the same gap between quantum and local theories than the gap between quantum and noncontextual theories of the original SD-C behavior. Arguably, no method exists that preserves this gap for all forms of state-dependent contextuality. We thank Costantino Budroni and Zhen-Peng Xu for comments on an earlier version. This work was supported by Project Qdisc (Project No. US-15097), with FEDER funds, MINECO Project No. FIS2017-89609-P, with FEDER funds, and QuantERA Grant SECRET, by MINECO (Project No. PCI2019-111885-2). *[email protected] [1] R. Cleve, P. Høyer, B. Toner, and J. Watrous, Consequences and limits of nonlocal strategies, in Proceedings. 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