Noncontextual wirings

Noncontextual Wirings Barbara Amaral,1,2,3 Adán Cabello,4Marcelo Terra Cunha,5and Leandro Aolita6,3,7 1Departamento de Matemática, Universidade Federal de Ouro Preto, 35400-000 Ouro Preto, Minas Gerais, Brazil 2Departamento de Física e Matemática, CAP—Universidade Federal de São João del-Rei, 36.420-000 Ouro Branco, Minas Gerais, Brazil 3International Institute of Physics, Federal University of Rio Grande do Norte, 59070-405 Natal, Brazil 4Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain 5Departamento de Matemática Aplicada, IMECC-Unicamp, 13084-970 Campinas, São Paulo, Brazil 6Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, Rio de Janeiro 21941-972, Brazil 7ICTP South American Institute for Fundamental Research, Instituto de Física Teórica, UNESP-Universidade Estadual Paulista R. Dr. Bento T. Ferraz 271, Bl. II, São Paulo 01140-070, São Paulo, Brazil (Received 6 June 2017; revised manuscript received 5 December 2017; published 29 March 2018) Contextuality is a fundamental feature of quantum theory necessary for certain models of quantum computation and communication. Serious steps have therefore been taken towards a formal framework for contextuality as an operational resource. However, the main ingredient of a resource theory—a concrete, explicit form of free operations of contextuality—was still missing. Here we provide such a component by introducing noncontextual wirings: a class of contextuality-free operations with a clear operational interpretation and a friendly parametrization. We characterize them completely for general black-box measurement devices with arbitrarily many inputs and outputs. As applications, we show that the relative entropy of contextuality is a contextuality monotone and that maximally contextual boxes that serve as contextuality bits exist for a broad class of scenarios. Our results complete a unified resource-theoretic framework for contextuality and Bell nonlocality. DOI: 10.1103/PhysRevLett.120.130403 Introduction.—Quantum contextuality refers to the impossibility of explaining the statistical predictions of quantum theory in terms of models where the measurement outcomes reveal preexistent system properties that are independent of the context, i.e., on which other compatible measurements are jointly performed [1,2]. Contextuality can be seen as a generalization of Bell nonlocality [3] to the case where the spacelike separation restriction is removed, so that single systems are included. It thus represents an exotic, intrinsically quantum phenomenon with both fundamental and practical implications. Contextuality has received lots of attention over the last decade. On one hand, it has been experimentally studied in a variety of physical setups [4–8]. On the other one, it is known to be a resource in magic-state [9–12] and measurement-based [13] quantum computing, for random number certification [14], and for several other information-processing tasks in the Bell scenario of spacelike separated measurements [15]. This has motivated considerable interest in resource theories of both contextuality [16–18] and Bell nonlocality [19–21]. Resource theories give powerful frameworks for the formal treatment of a physical property as an operational resource, adequate for its characterization, quantification, and manipulation [22,23]. Their central component is a special class of transformations, called the free operations, that fulfill the essential requirement of mapping every free (i.e., resourceless) object of the theory into a free object. Whereas resource-theoretic approaches for quantum nonlocality are highly developed [19–21,24–28], the operational framework of contextuality as a resource is still less developed. In Refs. [16,17], an abstract characterization of the axiomatic structure of a resource theory of contextuality was done. However, a concrete specification of the free operations of contextuality was not given. Without an explicit parametrization of a physically motivated class of free operations, a resource theory significantly loses applicability. For instance, in Refs. [16,17], an interesting measure of contextuality, called the relative entropy of contextuality, was proposed, but only partial monotonicity under a rather restricted subset of contextuality free operations was shown. Monotonicity (nonincrease under the corresponding free operations) is the fundamental requirement for a function to be a valid quantifier of a resource. Here, we fill this gap by introducing the class of noncontextual wirings. These are the natural noncontextuality preserving physical operations at hand in the deviceindependent scenario of black-box measurement devices, where one does not assume any a priori knowledge of the state or the observables in question. We derive a friendly analytical expression for generic noncontextual wirings applicable to all nondisturbing boxes, so that both quantum PHYSICAL REVIEW LETTERS 120, 130403 (2018) 0031-9007=18=120(13)=130403(6) 130403-1 © 2018 American Physical Society and postquantum boxes are covered. In addition, the framework is versatile in that it allows for transformations between systems with different numbers of inputs and outputs as well as different compatibility constraints. Furthermore, we show that, for the case of Bell tests, the wirings reduce to the canonical free operations of Bell nonlocality [19–21]. Hence, the framework constitutes a unified resource theory for both contextuality and Bell nonlocality in their most general forms. As applications, first we show that an important quantifier called relative entropy of contextuality is monotonic under all noncontextual wirings, a problem left open in Refs. [16,17]. Then, for the broad class of so-called cycle boxes, we show that contextality bits exists in the strongest possible sense: single boxes from which the entire nondisturbing set can be freely obtained with noncontextual wirings. Nondisturbing boxes.—We consider a measurement device with Nbuttons (inputs) and Mlights (outputs), with N,M∈N. Not all buttons are compatible, i.e., can be pressed jointly. Each subset of compatible buttons defines a context [29–31]. Let X¼f1;2;…;Ngrepresent the set of buttons. The contexts can be encoded in an input compatibility hypergraph IX≔fχj⊆Xgj¼1;…;jIXj, where each hyperedge χjcontains the buttons that can be jointly pressed in context j, with jIXjthe number of contexts [30,31]. We say that jis a maximal context if, for all 1≤j0≤jIXj,χj⊆χj0implies χj0¼χj. Similarly, not all lights can turn on jointly. Let A¼ f1;2;…;Mgbe the set of lights. Then, each kth button has a set AðkÞ⊆Aof lights associated, one—and only one—of which turns on upon pressing that button. The number of lights on is thus always equal to the number of buttons pressed. Hence, for the lights it is more convenient to work with mutual exclusivity constraints. These can be encoded in an output exclusivity hypergraph OA≔fAðkÞgk¼1;…;N, where AðkÞencodes the exclusivity hyperedge of button k∈X. We denote by AðχÞ≔⋃ k∈χAðkÞthe subset of lights associated with all the buttons in χ∈IX. In turn, note that different buttons may share associated lights. We refer to XðlÞ≔fk∈X∶l∈AðkÞgas the subset of buttons associated with light l∈A. We restrict throughout to the case where only incompatible buttons can have common associated lights. That is, for every l∈A,fk; k0g⊆XðlÞis allowed only if fk; k0g∩χ⊂fk; k0gfor all χ∈IX. For any input hypergraph IXand output hypergraph OA, we consider conditional probability distributions PAjX≔fpAjXða;χÞga∈f0;1gM;χ∈IX:ð1Þ The M-bit string a≔ða1;aMÞ∈f0;1gMrepresents the state of all Mlights: al¼0stands for “lth light off”and al¼1for “lth light on”. Hence, pAjXða;χÞis the probability of the lights being in state aupon pressing the buttons in the subset χ, which is nonzero only if aassigns the state “on”to one, and only one, of the lights associated with each button in χ. That is, for each χ∈Iχ, pAjXða;χÞ≠0only if kaðkÞkh¼1, with aðkÞ≔ðalÞl∈AðkÞ the substring of aof lights associated with button kand kaðkÞkhthe Hamming norm of (number of ones in) aðkÞ, for all k∈χ. We refer to any such PAjXas a box behavior relative to IXand OA. A specially relevant class is that of nondisturbing behaviors: PAjXis nondisturbing if, for all χ, χ0∈IXwith χ0⊂χ, X al∶l∉Aðχ0Þ pAjXða;χÞ¼pAðχ0ÞjX0ðaðχ0Þ;χ0Þ;ð2Þ with aðχ0Þ≔ðalÞl∈Aðχ0Þthe substring of aof lights associated with the buttons in χ0(instead of the entire context χ). The nondisturbance condition demands that whenever two contexts have buttons in common the marginal distribution over the common buttons is independent of the context. It is thus the analogue of the no-signaling condition in Bell scenarios [15]. With this, we can at last provide a precise formal definition of the general mathematical objects of the resource theory. Namely, we call every set of input and output hypergraphs IXand OA, respectively, together with a nondisturbing behavior PAjXrelative to them, a box, B≔fIX;OA;PAjXg:ð3Þ We call the set of all such nondisturbing boxes ND. In turn, the free objects of the theory, i.e., the resourceless ones, are given by the class NC ⊂ND of noncontextual (NC) boxes, defined by NC box behaviors. A behavior PAjXis NC if it admits a NC hidden-variable model, i.e., if, for all χ∈IXand a∈f0;1gMwe have pAjXða;χÞ¼X λ pΛðλÞY l∈A DlðaljχðlÞ;λÞ;ð4Þ where Λis the hidden variable, taking the value λwith probability pΛðλÞ,χðlÞ≔χ∩XðlÞis the single-element subset [32] of χassociated with light l, and DlðaljχðlÞ;λÞ≔δ(al;flðχðlÞ;λÞ), where δ(al;flðχðlÞ;λÞ), with δthe Kronecker delta, is the λth NC deterministic response function for the lth light given the input χðlÞ. The function flencodes the deterministic assignment of χðlÞ into alfor the λth global deterministic strategy incorporating the constraints of OA. That is, it is such that, for all λ, flð∅;λÞ¼0(lth light is off if no associated button is pressed, i.e., if χðlÞ¼∅) and flðχðlÞ;λÞ×fl0ðχðl0Þ;λÞ¼0, whenever fl; l0g⊆AðkÞfor any k∈X(no mutually exclusive lights simultaneously on). Note that, since fldepends only on χðlÞ(instead of the entire context χ), Dlcan only generate NC behaviors in Eq. (4). In fact, we show in PHYSICAL REVIEW LETTERS 120, 130403 (2018) 130403-2 Sec. IV of the Supplemental Material [33] that, when the contexts are defined by spacelike separated buttons, expression (4) reduces to the usual local hidden-variable models of Bell nonlocality [15]. Any box outside NC is called contextual. It is a well-known fact that measurements on quantum states can yield contextual boxes. Contextuality-free operations.—We consider compositions of the initial box Bwith a preprocessing box BPRE ≔fIY;OB;PBjYg∈NC;ð5Þ and a ðb;ψÞ-dependent postprocessing box BPOSTðb;ψÞ≔fIZ;OC;PCjZ;ψ;bg∈NC;ð6Þ for all b∈f0;1gjBjand ψ∈IY, as shown in Fig. 1.Yand Bare, respectively, the sets of buttons and lights of BPRE, and Zand Cthose of BPOSTðb;ψÞ. For the composition to be possible, we demand that the set of allowed outputs of BPRE is a subset of the allowed inputs of B, and the same for Bwith BPOST. To this end, we need to introduce the output compatibility hypergraph ¯ OAassociated to OA, given by all subsets α⊂Aof output lights with at most one light per exclusivity hyperedge in OA:¯ OA≔fα⊂A∶jα∩AðkÞj≤1; k¼1;…;Ng, and similarly for ¯ OB. That is, ¯ OAand ¯ OB give the compatible combinations of lights on, those not violating any of the constraints in OAand OB, respectively. Then, we demand that ¯ OB⊆IXand ¯ OA⊆IZ. Moreover, we allow PCjZ;b;ψto have only a restricted dependence on ðb;ψÞ, in such a way that each output light of the postprocessing box is causally influenced only by the inputs and outputs of the preprocessing box that are associated with it. That is, we demand that, for all b∈f0;1gjBj,c∈f0;1gjCj,ψ∈IY, and ζ∈IZ, pCjZ;b;ψðc;ζÞ¼X ϕ pΦðϕÞY n∈C DnðcnjζðnÞ;χðbÞ ½n;ψ½n;ϕÞ;ð7Þ with DnðcnjζðnÞ;χðbÞ ½n;ψ½n;ϕÞdefined analogously to DlðaljχðlÞ;λÞin Eq. (4). Similarly to χðlÞthere, ζðnÞis the single-element subset of ζassociated with light n∈C. In turn, we now introduce the short-hand notations χðbÞ ½nand ψ½n≔ψðχðbÞ ½nÞ[34]. The subset χðbÞ ½nis composed of the single button in χðζðnÞÞdirectly wired to some light on in b, whereas ψ½nis the single-button subset of ψassociated to the light directly wired to the button of χðbÞ ½n. These subsets are all well defined through the hypergraphs IXand OA, independently of the specific behavior PAjXin question, as shown in Sec. I of the Supplemental Material [33]. This is crucial for the composition not to create contextuality. With this, we are now in a good position to introduce the free operations of contextuality. Definition 1: Noncontextual wirings.—We define the noncontextual wiring with respect to the preand postprocessing boxes described above, as the linear map WNC that takes any initial box B∈ND, given by Eq. (3), into a final box Bf≔WNCðBÞwith Nf≔jYjbuttons and Mf≔jCjlights, with WNCðBÞ≔fIY;OC;PCjYg;ð8Þ where PCjYis the final behavior, given by pCjYðc;ψÞ ¼X a∈f0;1gjAj b∈f0;1gjBj pCjZ;b;ψðc;ζðaÞÞpAjXða;χðbÞÞpBjYðb;ψÞ;ð9Þ for all c∈f0;1gjCjand ψ∈IY. We denote the class of all such wirings by NCW. Self-consistency of the theory requires that NCW satisfies the following property, proven in Sec. II of the Supplemental Material [33]. Lemma 1: Nondisturbance preservation.—The class of boxes ND is closed under all wirings in NCW. In addition, to give valid free operations, NCW must fulfill the following requirement, proven in Sec. III of the Supplemental Material [33]. Theorem 1: Noncontextuality preservation.—The class of boxes NC is closed under all wirings in NCW. Intuitively, this is connected to the fact that the composition of any three independent noncontextual boxes yields a final box that is also noncontextual (with three independent noncontextual hidden variables). NCW is, however, FIG. 1. A noncontextual wiring WNC with respect to preand postprocessing boxes BPRE and BPOST, respectively, mapping an initial box Binto a final box WNCðBÞ. The buttons and lights of WNCðBÞare given by the buttons of BPRE and the lights of BPOST, respectively. Only the lights (buttons) of Bof the same color can be on (pressed) at the same time. The behavior of BPOST is causally influenced by BPRE, but in a restricted way such that the statistics of each output light of BPOST depends only on the buttons and lights of BPRE that are associated with it (see text). As a result, if Bis noncontextual so is WNCðBÞ. PHYSICAL REVIEW LETTERS 120, 130403 (2018) 130403-3 more powerful than such compositions because the preand postprocessing boxes here are not independent. Still, the restriction of Eq. (7) enables noncontextuality preservation (see Sec. III of the Supplemental Material). Finally, in Sec. IV of the Supplemental Material [33], we show that, for spacelike separated measurements, NCW reduces to local operations assisted by shared randomness, the canonical free operations of Bell nonlocality [19–21]. Contextuality monotones.—In Ref. [16], a measure of contextuality called the relative entropy of contextuality, RC, was introduced. For an arbitrary box B∈ND, RCðBÞ≔min B∈NCSðBkBÞ:ð10Þ SðBkBÞis the relative entropy of Bwith respect to B(see Sec. IVof the Supplemental Material [33]), which measures the distinguishability of Bfrom Bin a broad class of scenarios [21]. Hence, RCðBÞquantifies the distinguishability of Bfrom its closest (with respect to S) noncontextual box B, providing a direct generalization to contextuality of the statistical strength of Bell nonlocality proofs [35]. The essential requirement for a function to be a valid measure of a resource is that it is monotonic (i.e., nonincreasing) under the corresponding free operations. In Ref. [16], the authors show, for quantum boxes, monotonicity of RCunder probabilistic mixtures of independent channels on each quantum observable (each context). This corresponds to a restricted subset of NCW [36]. Here, we show monotonicity of RCunder the whole class NCW and for all boxes B∈ND. Lemma 2: Monotonicity of RC.—Let B∈ND. Then, RC½WNCðBÞ ≤RCðBÞfor all WNC ∈NCW. The proof (given in Sec. Vof the Supplemental Material [33]) relies explicitly on the parametrization of NCW in Eq. (9). Interestingly, also, another measure of contextuality, the contextual fraction CðBÞ[29,37], was recently shown to be monotonic under some specific classes of contextualityfree operations [18]. A straightforward calculation (see Sec. VI of the Supplemental Material [33]) shows that CðBÞ is also monotonic under the NCW class. Lemma 3: Monotonicity of C.—Let B∈ND. Then, C½WNCðBÞ ≤CðBÞfor all WNC ∈NCW. Contextuality bits.—The operational framework developed allows us to study contextuality interconversions. A natural question is whether there exists a box from which all boxes, for fixed input and output hypergraphs, can be obtained for free (i.e., through noncontextual wirings). This is intimately connected to quantification: such a superior box can be taken as a unit of contextuality, or contextuality bit, yielding a natural and unambiguous (measure-independent) definition of maximally contextual boxes. Here we answer that question affirmatively for a broad class given by the so-called N-cycle boxes (see Fig. 2). A N-cycle box has as many maximal contexts as buttons (N), each kth maximal context consists of two buttons (kand kþ1), each kth button belongs to two maximal contexts (χkand χk−1) and has two associated output lights, the (2k−1)th and the ð2kÞth lights, so that M¼2N. Modulo Nis implicitly assumed for the labels of buttons, contexts, and lights. These boxes admit 2N−1contextuality bits: Lemma 4: Existence of contextuality bits.—For any N≥3, all N-cycle boxes in ND can be freely obtained from an N-cycle box with behavior PðγÞ AjXof components pðγÞ AjXða;χÞ ≔1 2;if χ¼fk; k þ1gand a2k−s¼a2ðkþ1Þ−sþγk; 0;otherwise; ð11Þ for all s∈f0;1gand k∈X, with γ≔ðγ1;…;γNÞ, such that γk¼0or 1 and kγkhis an odd integer. Equation (11) describes any of the 2N−1contextual N-cycle behaviors extremal in ND, derived (in a different notation) and shown to be equivalent under noncontextual relabelings of outputs in Ref. [41]. The proof of the lemma, given in Sec. VII of the Supplemental Material [33], consists then of showing that any convex mixing of such relabelings is in NCW. For the particular case N¼4(the CHSH scenario), the behaviors in Eq. (11) become equivalent to the no-signaling extremal box of [42], known to generate all no-signaling boxes under local wirings assisted by shared randomness [19–21]. Lemma 5 thus generalizes this fact to arbitrary N≥3and noncontextual wirings. Finally, it is important to mention that, for even N, the buttons can be split into two disjoint subsets of N=2 incompatible buttons each, and the lights can be reduced from 2Nto only 4 (one mutually exclusive pair per subset of buttons), as in the chained inequalities [38]. This is an alternative representation of the same physical box. Our formalism is totally versatile in this sense, as it can directly deal with any chosen representation of a box. Final discussion.—Recent investigations suggest that contextuality may be a key resource for quantum FIG. 2. N-cycle graphs CNfor N¼3, 4, 5, and 6 buttons. A Ncycle box is such that the union of all hyperedges in IXequals CNand each input button has its own pair of output lights. For even N, the class is also intimately connected to the well-known chained inequalities of Bell nonlocality [38]. It includes the Clauser-Horne-Shimony-Holt (CHSH) scenario [39], where the inputs define the square C4, and the Klyachko-Can-Binicioˇ gluShumovsky one [40], where the inputs form the pentagon C5. For any N≥3, there exist contextuality bits, i.e., maximally contextual N-cycle boxes from which all other N-cycle boxes can be obtained for free (see text). PHYSICAL REVIEW LETTERS 120, 130403 (2018) 130403-4 advantages in various information-processing tasks [9–14]. Here we take a step forward towards contextuality as an operational resource by introducing and characterizing noncontextual wirings. In contrast to more abstract approaches [16,17], noncontextual wirings have a clear operational interpretation and admit a friendly analytical parametrization. This is useful to classify, quantify, and manipulate contextuality as a formal resource. For instance, the question of monotonicity of contextuality was until recently unclear. While in Refs. [16–18] monotonocity of the relative entropy of contextually and of the contextual fraction is proven under some specific operations, here we have settled the problem of monotonicity under all noncontextual wirings for both contextually measures. Furthermore, we have also shown that maximally contextual single boxes that serve as contextuality bits exist for all cycle boxes, which encompass important Bell scenarios [39,40] and play a crucial role in contextuality theory [43–48]. This result can also be extended to boxes with more outputs [24]. Interesting questions are, e.g., what the simplest box admitting inequivalent (not freely interconvertible) classes of contextuality is and what the simplest one allowing for contextuality distillation. Finally, we have shown that, for Bell scenarios, noncontextual wirings reduce to the usual free operations of Bell nonlocality [19–21], which is interesting in itself. Hence, our findings yield a main missing ingredient for a complete, unified resource theory of contextuality and Bell nonlocality. The present work was initiated during the workshop “Quantum Correlations, Contextuality, and All That… Again”at the International Institute of Physics (IIP), Natal, Brazil. The participants of the workshop as well as the hospitality of IIP are gratefully acknowledged. We thank D. Cavalcanti, C. Duarte, M. Pusey, R. Soares Barbosa, S. Mansfield and an anonymous Referee for fruitful discussions. B. A. thanks the Instituto de Matemática Pura e Aplicada (IMPA) for the hospitality at Rio de Janeiro, Brazil. B. A., M. T. C., and L. A. acknowledge financial support from the Brazilian ministries MEC and MCTIC and agencies CNPq, CAPES, FAEPEX, FAPERJ, FAPESP, and INCT-IQ. A. 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