Measurement incompatibility and steering are necessary and sufficient for operational contextuality

Contextuality is a signature of operational nonclassicality in the outcome statistics of an experiment. This notion of nonclassicality applies to a breadth of physical phenomena. Here, we establish its relation to two fundamental nonclassical entities in quantum theory; measurement incompatibility and steering. We show that each is necessary and sufficient the failure of operational contextuality. We exploit the established connection to contextuality to provide a novel approach to problems in measurement incompatibility and steering.


I. INTRODUCTION
The nonclassical nature of quantum theory has a variety of different manifestations. On the one hand, quantum theory postulates theoretical entities with properties that lack a counterpart in classical physics. On the other hand, the nonclassicality of quantum theory is also present on the observable level, i.e. in the outcome statistics of experiments. Evidently, if an experiment takes the reality of the quantum formalism for granted, every nonclassical entity of quantum theory can be experimentally detected. However, if the assumption of nature being quantum is dropped, the outcome statistics can frequently be reproduced with some classical model. Matters become more interesting when the nonclassicality of outcome statistics can be operationally determined in the spirit of device-independence [1]. That is, in experiments that demonstrate nonclassicality while making weak assumptions on the underlying physical nature.
The strongest form of operational inference is encountered in tests of Bell inequalities [2]. These experiments statistically analyse the correlations between the outcomes of measurements performed in space-like separated events. If the correlations violate a Bell inequality, it follows that the outcome statistics cannot be explained by any classical (local hidden variable) theory. Famously, by sharing entangled states and performing incompatible measurements that together steer the remote partner system, quantum theory can violate these inequalities and therefore provide an unequivocal demonstration of nonclassicality [3,4]. Surprisingly however, not all incompatible measurements, nor all steerable ensembles enable Bell inequality violations [5][6][7]. This motivates the question: is nonclassicality at the level of theoretical entities both necessary and sufficient for some form of operational nonclassicality?
We show that two fundamental theoretical entities in quantum theory always enable a proof of operational nonclassicality. Specifically, we consider the incompatibility of quantum measurements and the ability to steer another system by local measurements and classical communication (a feature of quantum theory originating from Schrödinger's remarks [8] on the Einstein-Podolsky-Rosen paradox [9]). These two features of quantum theory are thoroughly researched, see e.g. Refs [10][11][12][13] and Refs [14,15] respectively. We estab- * A. T. and R. U. contributed equally to this work. lish their relation to operational nonclassicality as provided by outcome statistics that demonstrate contextuality. As originally introduced by Bell, Kochen and Spekker [16,17], contextuality is a property of projective measurements in quantum theory. However, the concept has seen a generalisation that applies on the level of ontological models, and therefore to general operational theories used to model outcome statistics [18]. Briefly, contextuality is the impossibility of building an ontological model that ascribes equivalent physical representation to operationally indistinguishable physical procedures. Said impossibility is referred to as preparation (measurement) contextuality when the procedures correspond to ways of preparating (measuring) a system. Contextuality (in its broad sense) in quantum theory is closely related to e.g. advantages in quantum computation [19][20][21][22], advantages in particular communication tasks [23][24][25][26], quantum zero-error communication [27] and anomalous weak values [28].
This work has two main results. Firstly, we show that incompatibility of quantum measurements is necessary and sufficient for enabling a demonstration of nonclassicality in terms of preparation contextual outcome statistics. Secondly, we show that every no-signaling ensemble of states is steerable if and only if it enables a demonstration of nonclassicality in terms of preparation and measurement contextual outcome statistics. These results allow us to view problems in joint measurability and steering through the lens of contextuality. We exploit this to present a family of noncontextuality inequalities and provide numerical evidence that these are necessary and sufficient conditions for certifying the incompatibility of any set of binary qubit observables, and that they also constitute optimal tests of the steerability of a pair of qubits in a singlet state subject to noisy environments. Such applications also make possible experimental tests of measurement incompatibility and steerability based on operational contextuality.

II. PRELIMINARIES
We begin with a brief summary of contextuality (in the generalised sense of Ref [18]), measurement incompatibility and quantum steering.

A. Contextuality
Contextuality in an operational theory [18] is developed within the framework of ontological models [29]. An ontological model describes a preparation procedure P by an ontic state (hidden variable) λ with a distribution p(λ|P). When a measurement procedure M is applied, the ontological model determines the probability of an outcome b with some response function p(b|M, λ). Therefore, the outcome statistics reads A state (measurement) can be realised in as many ways as it can be decomposed into mixtures of other states (decomposed into element-wise mixtures or coarse-grainings of other measurements). Different ways of preparing the same state (performing the same measurement) are called contexts for ρ (M ). An ontological model is said to be preparation noncontextual if the ontic state distribution is independent of the context, i.e. if p(λ|P) = p(λ|ρ). Similarly, an ontological model is said to be measurement noncontextual if the response functions are context-independent, i.e. if p(b|M, λ) = p(b|M, λ). These notions embody the idea that if two laboratory procedures are indistinguishable, then they are also indistinguishable on the level of ontic states. We remark that in order to ensure that two procedures truly are indistinguishable, one needs to be able to perform measurements (prepare states) that span the measurement (state) space. In contrast, if outcome statistics cannot be reproduced with any preparation (measurement) noncontextual model, it is said to be preparation (measurement) contextual. See Ref [18] for a detailed discussion of operational contextuality.

B. Measurement incompatibility
Measurement incompatibility [10][11][12][13] is the impossibility of jointly measuring a set of (at least two) POVMs by employing only a single measurement and classical post-processing of its outcomes. More precisely, let {A a|x } be a set of POVMs, with a labelling the outcome and x labelling the measurement. The set is called compatible (jointly measurable) if there exists a POVM {G λ } which allows us to recover the set {A a|x } via some post-processing probability distribution p(a|x, λ); If such a model does not exist, the set {A a|x } is called incompatible (not jointly measurable). This extends the textbook concept of commutativity in the sense that mutually commuting POVMs are jointly measurable, but the converse does not hold in general. The converse holds, however, for textbook observables, i.e. projective measurements. It is worth noting that joint measurability can be characterised as the existence of a common Naimark dilation in which the projective measurements commute.

C. Steering
Steering [7] is a qualitative property of some entangled quantum states regarding the set of ensembles that can be remotely prepared with local measurements and classical communication. Specifically, one considers a pair of entangled systems in state ρ and performs a set of measurements {A a|x } on the first system. Given the choice of x, this renders the second system in the state ρ a|x = tr A A a|x ⊗ 1 1ρ / tr A a|x ⊗ 1 1ρ with probability p(a|x) = tr A a|x ⊗ 1 1ρ . It is important to underline the fact that classical communication is necessary for the steered party to be able to distinguish between different local states ρ a|x . These local states can be effectively described with a the set of unnormalised states (called an assemblage) {σ a|x } where σ a|x = tr A A a|x ⊗ 1 1ρ . Such assemblages are no-signaling, i.e. a σ a|x = a σ a|x . In this work all assemblages are assumed to be no-signaling. We remark that the Gisin-Hughston-Josza-Wootters theorem [30,31] ensures that every assemblage can be prepared by a distant party's local measurements (supported by classical communication) on a properly chosen entangled state. An assemblage is said to be unsteerable if it admits a so-called local hidden state model. Such models use (a, x) as information towards a post-processing p(a|x, λ) of a set of local states ρ λ appearing with probability p(λ) to explain the assemblage {σ a|x }. Hence, if the state is unsteerable, it can be written as If no local hidden state model is possible, the assemblage is called steerable.

III. MAIN RESULTS
We begin by proving a one-to-one relation between measurement incompatibility and preparation contextuality. Proof. Assume that the set of POVMs {A a|x } when applied to any quantum state ρ returns outcome statistics that is preparation noncontextual. We denote the set of preparation procedures (contexts) in which ρ can be prepared by P ρ . Then, using the label x to denote the measurement procedure, it holds that ∀P ∈ P ρ : p(a|x, P) = λ p(λ|ρ)p(a|x, λ).
For each λ, the object p(λ|ρ) is a convexity-preserving map from the space of quantum states to the interval [0, 1]. The Riesz representation theorem [13,32] asserts that such maps can be written as an inner product p(λ|ρ) = tr [G λ ρ] for some unique operator 0 ≤ G λ ≤ 1 1. Moreover, since ∀ρ : λ p(λ|ρ) = 1, it follows that λ G λ = 1 1. Inserting this into Eq. (4), we have We have recovered the outcome statistics obtained from measuring ρ with a compatible set of POVMs. Conversely, assume that {A a|x } is a compatible set of POVMs. Then, the statistics obtained from measuring any state ρ prepared with a procedure P is given by Eq. (5). By defining p(λ|ρ) = tr [G λ ρ], we recover the definition of outcome statistics being preparation noncontextual.
It is interesting to note that tests of preparation contextuality can be formulated as communication tasks between two separated parties, in which the receiver is kept oblivious about parts of the sender's input [23][24][25]. Such obliviousness corresponds to different contexts for the states. From theorem 1, we can therefore infer that: Corollary 1. Every set of incompatible measurements enables a quantum-over-classical advantage in a communication task.
We remark that the advantages of all incompatible sets of measurements have recently been shown in various measurement-device-independent communication tasks [33][34][35][36].
In a spirit similar to that of theorem 1, we prove a one-toone relation between steering and measurement contextuality. where (a, x) labels the preparation procedure. For every λ, the object p(b|M, λ) is a map from the space of POVMs to the space of probability distributions. Such maps are characterised by the works of Gleason [37] and Busch [38]. The   FIG. 1: The orange arrows illustrate theorem 1 and theorem 2. The upper two figures represent theoretical entities which respectively enable the lower two operational features when provided a correct catalyst preparation or measurement procedure. Conversely, the two operational features imply the respective theoretical entities regardless of the underlying (no-signaling, "NS") procedure. The grey arrow indicates the previously known relation that a set of measurements is incompatible if and only if it enables steering with a proper catalyst state [39,40]. It is worth noting that this connection can also be seen as a mapping between the problems in measurement incompatibility and steerability [41].
Gleason-Busch theorem asserts that p(b|M, λ) = tr [ρ λ M b ] for some unique state ρ λ . Inserting this into Eq. (6), we have Using Bayes' rule and the fact that x and λ are independent 1 , one straightforwardly finds that p(a|x)p(λ|a, x) = p(λ)p(a|x, λ). Inserting this in (7), we recover the outcome statistics obtained from applying M to an unsteerable assemblage (3).
Conversely, if the assemblage has a local hidden state model, then for every POVM the outcome statistics reads From Bayes rule and the independence of x and λ, we have that p(λ)p(a|x, λ)/p(a|x) = p(λ|a, x). Note that said independence implies preparation noncontextuality. Inserted into Eq. (8) we find the outcome statistics obtained in a measurement noncontextual model.
We have illustrated the theorems in Figure 1. Notice that theorem 1 and theorem 2 give a characterisation of the ontic variables using quantum theory. Whereas this characterisation is relevant for noncontextual models covering all states or measurements, it would be interesting to see whether such characterisation exists in the case of fragments of quantum theory, i.e. for noncontextual models covering subsets of states and measurements.
Also, it is worth noting that a number of works have (in different ways) shown that outcome statistics that violate a Bell inequality is a proof of preparation contextuality [18,23,24,42]. In Appendix A, we note that this fact follows immediately from ontological models and the no-signaling principle (see also Ref. [43] for similar result) 2 .

IV. NONCONTEXTUALITY INEQUALITIES FOR QUBIT MEASUREMENT INCOMPATIBILITY AND STEERING
We proceed to use the established connection to contextuality to address two relevant problems in measurement incompatibility and steering. I) Can one find a preparation noncontextuality inequality whose violation is both necessary and sufficient for certifying measurement incompatibility for interesting families of measurements? Note that despite theorem 1 this is a nontrivial matter since any single test of preparation contextuality is only a sufficient condition for measurement incompatibility (and the full characterisation of all tests of preparation contextuality is a demanding problem). II) Can one find a measurement noncontextuality inequality (in a nosignaling scenario) that for an interesting class of states optimally certifies their steerability? In analogy to the previous, this is nontrivial since any single test of measurement contextuality in a no-signaling scenario is only a sufficient condition for steering.
To answer these questions, we present a family of correlation inequalities (parameterised by an integer n ≥ 2) inspired by the works of Refs [25,45,46]. Consider a Bell-like (nosignaling) experiment in which two separated observers, Alice and Bob, share parts of a physical system. Alice (Bob) performs measurements labelled by her (his) uniformly random input x ∈ {0, 1} n−1 (y ∈ {1, . . . , n}). The outcome is denoted by a ∈ {0, 1} (b ∈ {0, 1}). Alice's measurement procedures are constrained by operational equivalences. That is, her outcome statistics always upholds suitable indistinguishability relations which enable us to consider the statistics of different contexts for her measurements. Specifically, for every bit-string r ∈ {0, 1} n , we require that the measurement procedures M r,0 and M r,1 corresponding to a uniform mixing of all (a, x) satisfying r ·x = 0 and r ·x = 1 respectively (wherex = (a, x + a)), are indistinguishable from each other. In quantum theory, this means that a,x|r·x=0 Note that whenever r has an even number of ones, this condition is always satisfied since M 0|x + M 1|x = 1 1. For odd strings r, Eq. (9) is a nontrivial constraint. Now, let Alice and Bob play a game in which they aim to maximise the probability of finding a + b =x y mod 2. When Alice is considered the sender of Bob's remotely prepared local states, we can consider the scenario as a test of preparation contextuality. In contrast, when Bob is considered the sender of Alice's remotely prepared local states, we can consider the scenario as a test of preparation and measurement contextuality. In case of either being noncontextual, the average success probability is bounded by The proof of this result is a simple modification of the arguments presented in Ref [25] and is discussed in Appendix B.
A violation of the inequality (10) means that Bob's measurements (which are unconstrained) are incompatible (by theorem 1) and that Alice's local assemblage (prepared by Bob) is steerable (by theorem 2). We now study the usefulness of the inequality (10) for certifying qubit measurement incompatibility and two-qubit steerability.
For the case of n = 2 the inequality (10) reduces to the Clauser-Horne-Shimony-Holt Bell inequality [4] for which it is known that all pairs of incompatible measurements enable a violation [47]. For n > 2 (specifically studying n = 3, . . . , 7) we have numerically obtained support (10000 examples for each n) for the following conjecture Conjecture 1. Every set of n incompatible two-outcome qubit measurements enable a proof of preparation contextuality by a violation of the inequality (10).
In Appendix C, we describe the numerical procedure employed to motivate this conjecture.
Consider now the case of steering. For simplicity, let Alice and Bob share the noisy singlet state 1]. What is the critical value of v = v n so that Bob can steer Alice using n projective measurements? Although this question is well-studied (see e.g. Refs [48][49][50]) an analytical formula is lacking. However Ref [48] presented nearly matching upper and lower bounds on v n for n = 2, . . . , 13 and n = 2, . . . , 5 respectively. Using our inequality (10), we have numerically implemented alternating convex searches to find an upper bound on the critical v n (below which we can no longer find a quantum violation). This returns Interestingly, these numbers coincide precisely with those presented in Ref [48] (up to the number of decimals presented in Ref [48]). This motivates the conjecture Conjecture 2. The inequality (10) is a tight steering inequality for the noisy singlet state under n projective measurements.
Finding a conclusive proof of conjectures 1 and 2 would be interesting. We remark that although the above considerations are straightforwardly analysed with a computer, the criterion (10) can be treated in a fully analytical manner.

V. DISCUSSION
We have shown that every set of incompatible measurements and every steerable assemblage can be operationally certified as nonclassical in a test of operational contextuality, and that the latter also imply the formers. A direct consequence is that problems of joint measurability and steering can be viewed through the lens of contextuality, as we examplified through our conjectures. In this sense, our results bridge the two research directions of quantum measurements and quantum steering with the line of research focused on quantum contextuality.
Moreover, since tests of operational contextuality only rely on weak characterisation of the experimental devices [51], our results can also be considered as semi device-independent cer-tificates of measurement incomaptibility and steering. Naturally, fully device-independent certificates are found by violating a Bell inequality. However, in addition to such tests being experimentally demanding, it is importantly also the case that not all incompatible measurements nor all steerable ensembles violate any Bell inequality [5][6][7]. This makes tests of operational contextuality relevant for practical considerations when no fully device-independent certificate is either possible or known.
exists eight three-bit strings of which four have at least two instances of 1 . Those are r = 011, r = 101, r = 110 and r = 111. For each r we have the relation in Eq. (B2). In case of, for example, r = 011 we find (B3) However, this is trivially satisfied since ∀x : M 0|x + M 1|x = 1 1. Similarly, one finds that the constraint (B2) is trivial also for r = 101 and r = 110. However, for r = 111 we obtain which is a nontrivial constraint.
Imagine now that instead of performing local measurements on a shared state, Alice directly prepares the wouldhave-been post-measurement states of Bob's system (labelled by the pair (a, x)) and sends them to Bob, who then measures the system and records b ∈ {0, 1}. This represents a prepare-and-measure scenario in which Alice has 2 n inputs (a, x) with some prior distribution p(a, x) = p(a|x)/2 n−1 . Alice's preparations are required to satisfy the operational equivalence which in quantum theory reads ∀r : Notice first that in the original scenario, every assemblage prepared by Alice on Bob's side can also be directly sent in this prepare-and-measure model; simply define ρ a,x = tr A M a|x ⊗ 1 1ρ / tr M a|x ⊗ 1 1ρ , and the prior distribution as p(a|x) = tr M a|x ⊗ 1 1ρ . Conversely, every ensemble that Alice can communicate to Bob in the prepare-and-measure scenario can also be realised in the original scenario via local measurements on an entangled state and classical communication. This follows from the Gisin-Hughston-Josza-Wootters theorem [30,31] and the fact that Eq. (B5) enforces a nosignaling-like preparation ensemble.
In Ref. [25] it was shown that when p(a|x) = 1/2, the considered prepare-and-measure scenario serves as the following test of preparation contextuality; the inequality 1 n2 n−1 a,x,y p(a|x)p(b = (a, x) y |a, x, y) ≤ n + 1 2n (B6) holds for every preparation noncontextual model. Moreover, it is a trivial modification of the arguemtns of Ref. [25] to show that the same bound holds regardless of the prior distribution p(a|x). Therefore, due to the between the prepareand-measure scenario and the original scenario, it also holds that in a preparation noncontextual model, in which we view Alice as effectively preparing the local states of Bob.
Moreover, in the original scenario, we can equally well consider Bob as the effective sender of Alice's local states. If we impose measurement noncontextuality, the response function of Alice takes no regard to the different contexts of her measurement (related to r). Note that preparation noncontextuality is still present due to Alice and Bob being no-signlaing.