Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements Ravi Kunjwal Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada, N2L 2Y5. September 4, 2019 We develop a hypergraph-theoretic frame- Kochen-Specker functionals appearing in the work for Spekkens contextuality applied to CSW framework; when Corr = 1, the inequal- Kochen-Specker (KS) type scenarios that goes ities formally reduce to CSW type bounds on beyond the Cabello-Severini-Winter (CSW) R. Along the way, we also consider in detail framework. To do this, we add new the scope of our framework vis-`a-vis the CSW hypergraph-theoretic ingredients to the CSW framework, particularly the role of Specker’s framework. We then obtain noise-robust non- principle in the CSW framework, i.e., what the contextuality inequalities in this generalized principle means for an operational theory sat- framework by applying the assumption of isfying it and why we don’t impose it in our (Spekkens) noncontextuality to both prepara- framework. tions and measurements. The resulting frame- work goes beyond the CSW framework in both senses, conceptual and technical. On the con- ceptual level: 1) as in any treatment based on Contents the generalized notion of noncontextuality `a la Spekkens, we relax the assumption of outcome 1 Introduction2 determinism inherent to the Kochen-Specker theorem but retain measurement noncontex- 2 Spekkens framework5 tuality, besides introducing preparation non- 2.1 Operational theory . 5 contextuality, 2) we do not require the exclu- 2.2 Ontological model . 6 sivity principle – that pairwise exclusive mea- surement events must all be mutually exclu- 2.3 Representation of coarse-graining . 7 sive – as a fundamental constraint on mea- 2.3.1 Coarse-graining of measurements 7 surement events of interest in an experimen- 2.3.2 Coarse-graining of preparations 8 tal test of contextuality, given that this prop- 2.4 Joint measurability (or compatibility) 9 erty is not true of general quantum measure- ments, and 3) as a result, we do not need to 2.5 Noncontextuality . 9 presume that measurement events of interest 2.6 An example of Spekkens contextuality: are “sharp” (for any definition of sharpness), the fair coin flip inequality . 10 where this notion of sharpness is meant to im- 2.7 Connection to Bell scenarios . 12 ply the exclusivity principle. On the techni- cal level, we go beyond the CSW framework 3 Hypergraph approach to Kochen- in the following senses: 1) we introduce a Specker scenarios in the Spekkens source events hypergraph – besides the mea- framework 13 arXiv:1709.01098v4 [quant-ph] 3 Sep 2019 surement events hypergraph usually consid- 3.1 Measurements . 13 ered – and define a new operational quantity Corr that appears in our inequalities, 2) we de- 3.1.1 Classification of probabilistic fine a new hypergraph invariant – the weighted models . 14 max-predictability – that is necessary for our 3.1.2 Distinguishing two conse- analysis and appears in our inequalities, and 3) quences of Specker's principle: our noise-robust noncontextuality inequalities Structural Specker's principle quantify tradeoff relations between three oper- vs. Statistical Specker's principle 15 ational quantities – Corr, R, and p0 – only one 3.1.3 What does it mean for an oper- of which (namely, R) corresponds to the Bell- ational theory to satisfy struc- tural/statistical Specker's prin- Ravi Kunjwal: [email protected] ciple? . 16 Accepted in Quantum 2019-09-01, click title to verify. Published under CC-BY 4.0. 1 3.1.4 Remark on the classification of A.2 Robustness of Bell nonlocality vis-`a-vis probabilistic models: why we POVMs . 36 haven't defined \quantum mod- els" as those obtained from pro- B Ontological models without respecting jective measurements . 20 coarse-graining relations 37 3.1.5 Scope of this framework . 20 B.1 How to construct a \KS- 3.2 Sources . 21 noncontextual" ontological model of the KCBS experiment [47] without 4 A key hypergraph invariant: the coarse-graining relations . 37 weighted max-predictability 23 B.2 How to construct a \preparation and measurement noncontextual" ontologi- 5 Noise-robust noncontextuality inequali- cal model without coarse-graining rela- ties 24 tions . 37 5.1 Key notions from CSW . 24 5.2 Key notion not from CSW: C Trivial POVMs 38 source-measurement correlation, Corr 25 C.1 Bell-CHSH scenario . 38 5.3 Obtaining the noise-robust noncontex- C.2 CHSH-type contextuality scenario: 4- tuality inequalities . 25 cycle . 38 5.3.1 Expressing operational quanti- ties in ontological terms . 25 D The KS-uncolourable hypergraph Γ18 40 5.3.2 Derivation of the noncontextual tradeoff for any graph G .... 26 References 41 5.3.3 When is the noncontextual tradeoff violated? . 27 1 Introduction 5.4 Example: KCBS scenario . 27 To say that quantum theory is counterintuitive, or 6 Discussion 30 that it requires a revision of our classical intuitions, 6.1 Measurement-measurement cor- requires us to be mathematically precise in our def- relations vs. source-measurement inition of these classical intuitions. Once we have a correlations . 30 precise formulation of such classicality, we can begin 6.2 Can our noise-robust noncontextuality to investigate those features of quantum theory that inequalities be saturated by a noncon- power its nonclassicality, i.e., its departure from our textual ontological model? . 30 classical intuitions, and thus prove theorems about 6.2.1 The special case of facet- such nonclassicality. To the extent that a physical defining Bell-KS inequalities: theory is provisional, likely to be replaced by a better Corr=1 . 30 theory in the future, it also makes sense to articu- 6.2.2 The general case: Corr < 1 .. 30 late such notions of classicality in as operational a 6.3 Can trivial POVMs ever violate these manner as possible. By `operational', we refer to a noncontextuality inequalities? . 31 formulation of the theory that takes the operations { 6.3.1 The case p ∈ C(ΓG) ...... 31 preparations, measurements, transformations { that 6.3.2 The case p ∈ can be carried out in an experiment as primitives and ConvHull(G(ΓG)|ind) ...... 31 which specifies the manner in which these operations 6.3.3 The general case p ∈ G(ΓG) .. 31 combine to produce the data in the experiment. Such an operational formulation often suggests generaliza- 7 Conclusions 32 tions of the theory that can then be used to better understand its axiomatics [1{3]. At the same time, Acknowledgments 33 an operational formulation also lets us articulate our notions of nonclassicality in a manner that is experi- A Status of KS-contextuality as an experi- mentally testable and thus allows us to leverage this mentally testable notion of nonclassical- nonclassicality in applications of the theory. Indeed, ity for POVMs in quantum theory 33 a key area of research in quantum foundations and A.1 Limitations of KS-contextuality vis-`a- quantum information is the development of methods vis POVMs . 34 to assess nonclassicality in an experiment under min- A.1.1 KS-contextuality for POVMs in imal assumptions on the operational theory describ- the literature . 34 ing it. The paradigmatic example of this is the case of A.1.2 Classifying probabilistic mod- Bell's theorem and Bell experiments [4{11], where any els: restriction of quantum operational theory that is non-signalling between the models to PVMs . 35 different spacelike separated wings of the experiment Accepted in Quantum 2019-09-01, click title to verify. Published under CC-BY 4.0. 2 is allowed. The notion of classicality at play in Bell's 23, 25] is a formal unification of Bell scenarios theorem is the assumption of local causality: any non- with KS-contextuality scenarios, treating them on signalling theory that violates the assumption of local the same footing. Indeed, the perspective there is causality is said to exhibit nonclassicality by the lights to consider Bell scenarios as a special case of KS- of Bell's theorem. contextuality scenarios. What is lost in this math- More recently, much work [12{17] has been devoted ematical unification, however, is the fact that Bell- to obtaining constraints on operational statistics that locality and KS-noncontextuality have physically dis- follow from a generalized notion of noncontextuality tinct, if related, motivations. The physical situation proposed by Spekkens [18]. This notion of classicality that Bell's theorem refers to requires (at least) two [18] has its roots in the Kochen-Specker (KS) theo- spacelike separated labs (where local measurements rem [19], a no-go theorem that rules out the possibility are carried out) so that the assumption of local causal- 1 that a deterministic underlying ontological model [20] ity (or Bell-locality) can be applied. On the other could reproduce the operational statistics of (projec- hand, the physical situation that the Kochen-Specker tive) quantum measurements in a manner that sat- theorem refers to does not require spacelike separa- isfies the assumption of KS-noncontextuality. KS- tion as a necessary ingredient and one can there- noncontextuality is the notion of classicality at play fore consider experiments in a single lab. However, in the Kochen-Specker theorem. The Spekkens frame- the assumption of KS-noncontextuality entails out- work abandons the assumption of outcome determin- come determinism [18], something not required by 2 ism [18] { the idea that the ontic state of a system local causality in Bell scenarios. This difference fixes the outcome of any measurement deterministi- in the physical situation for the two kinds of ex- cally { that is intrinsic to KS-noncontextuality. It also periments is one of the reasons for generalizing KS- applies to general operational theories and extends noncontextuality to the notion of noncontextuality in the notion of noncontextuality to general experimen- the Spekkens framework [18] (so that outcome deter- tal procedures { preparations, transformations, and minism is not assumed) while leaving Bell's notion of measurements { rather than measurements alone.