**Samson Abramsky**, Oxford University

*Towards a mathematical theory of contextuality*

The study of contextuality has progressed from sepcific examples to the beginnings of a general theory, which is both of foundational interest, and applicable to the use of contextuality as a resource in quantum information processing, among other things.

There are several different formulations, and a number of general results. We shall describe some recent progress on a number of topics, including: characterizations of the quantum resources required to achieve the various levels in the contextuality hierarchy; characterization of All-versus-Nothing arguments for stabilizer states; a quantum monad encapsulating quantum advantage in constraint systems and non-local games, and its correspondence to quantum witnesses for state-independent strong contextuality; and a quantitative measure for contextuality.

We shall also discuss some problems and objectives for future work.

**Stephen Bartlett**, University of Sydney

*Contextuality and quantum simulation*

Simulating quantum systems on a classical computer is typically hard. But why? And how hard is it? One standard approach to simulating quantum systems is to use phase space methods with Monte Carlo sampling of trajectories: an approach that has many similarities with the framework of ontological models. Specifically, a noncontextual ontological model of a quantum process provides an explicit efficient classical simulation of that process. Contextuality can be viewed as a potential obstruction to efficient classical simulation. I'll present some new results showing that, by appropriately incorporating contextuality into the ontological models framework, one can construct a new class of classical simulation methods in which the 'amount' of contextuality appears as a measure of the inefficiency (run time) of the simulator.

**Juan Bermejo-Vega**, Free University of Berlin

*Contextuality as a resource for quantum computation: the trouble with qubits*

Co-authors: Robert Raussendorf, Dan E. Browne, Nicolas Delfosse, and Cihan Okay

What are the physical mechanisms that power quantum computation? Recently, quantum contextuality has been shown to be a resource in the model of quantum computation via (magic) state injection (QCSI) in two restricted scenarios: namely, for QCSI models that use either “quopits” (odd-prime dimensional qudits) [1] or “rebits” (two-level systems with real wavefunctions) [2]. Yet, is contextuality a resource for QCSI for models that employ the most fundamental units of quantum information, i.e., regular qubit systems? In this talk I address this question and show how to circumvent an a priori technical obstruction towards characterizing qubit QCSI schemes, namely, the phenomenon of state-independent contextuality. I will establish that contextuality of magic states (with respect to Pauli observables) as a necessary resource for a large class of quantum computation via state-injection schemes on qubits [3,4]. I will further illustrate this result on a concrete example related to measurement-based quantum computation.

[1] M. Howard, J. Wallman, V. Veitch, and J. Emerson, Contextuality supplies the magic for quantum computation, Nature 510, 351 (2014), arXiv:1401.4174.

[2] N. Delfosse, P. Allard Guerin, J. Bian, and R. Raussendorf, Wigner function negativity and contextuality in quantum computation on rebits, Phys. Rev. X 5, 021003 (2015), arXiv:1409.5170.

[3] J. Bermejo-Vega, N. Delfosse, D. E. Browne, C. Okay, R. Raussendorf, Contextuality as a resource for qubit quantum computation (2016), arXiv:1610.08529.

[4] R. Raussendorf, D. E. Browne, N. Delfosse, C. Okay, and J. Bermejo-Vega, Contextuality and Wigner function negativity in qubit quantum computation, Phys. Rev. A 95, 052334, arXiv:1511.08506.

**Dan Browne**, University College London

*Contextuality and non-contextuality in (qudit) quantum computation *

Co-authors: Nicolas Delfosse, Cihan Okay, Juan Bermejo-Vega, Robert Raussendorf and Lorenzo Catani

What non-classical phenomena are necessary for the non-classicality of quantum computing?

In this talk, I will review recent progress in answering this question, introducing two forms of quantum computation, measurement-based quantum computing (MBQC), and fault tolerant quantum computing with state injection (QCSI) where a link can be explicitly made.

Focussing then of odd dimensional qudits, where stabilizer states and Clifford circuits (the core capabilities of many fault tolerant quantum computation schemes) are known to be compatible with a non-contextual hidden variable model HVM, I shall show that a generalisation of Spekken’s toy model [1] provides an explicit HVM for these theories in all odd-dimensions [2]. Then linking this to Wigner function representations I will present a simple proof [3] of the equivalence between contextuality and negativity of the Wigner function. In both cases, we are able to build on and generalise earlier seminal works [1,4] which studied prime (but not compound) odd dimensional systems.

[1] Robert Spekkens, Quasi-quantization: classical statistical theories with an epistemic restriction, https://arxiv.org/abs/1409.5041

[2] Lorenzo Catano and Dan E. Browne, Spekkens' toy model in all dimensions and its relationship with stabilizer quantum mechanics, https://arxiv.org/abs/1701.07801 (to appear in New Journal of Physics)

[3] Equivalence between contextuality and negativity of the Wigner function for qudits, Nicolas Delfosse, Cihan Okay, Juan Bermejo-Vega, Dan E. Browne, Robert Raussendorf, https://arxiv.org/abs/1610.07093

[4] Contextuality supplies the ‘magic’ for quantum computation, Nature 510, 351 (2014)

**Adan Cabello**, Universidad de Sevilla

*What do we learn about quantum theory from Kochen-Specker quantum contextuality?*

Intentionally, I define quantum contextuality as the quantum violation of inequalities involving correlations between the outcomes of compatible {\em sharp} measurements, as defined in the framework of general probabilistic theories, and satisfied by ontological models where the assumption of outcome noncontextuality for sharp measurements is made, as it is the case for the hidden variable theories considered by Bell, Kochen and Specker. These noncontextuality inequalities (specifically, some of them called tight) provide necessary and sufficient conditions for the existence of joint probability distribution. The purpose of my talk is twofold.

Firstly, explaining the reasons why focusing on this particular definition of contextuality teaches us much more about what is quantum theory than focusing on any other proposed notion of non-classicality, including other definitions of contextuality.

Secondly, introducing an alternative to the existing ways of dealing with the inevitable finite precision, imperfect compatibility, and unsharpness of the measurements in actual experiments testing contextuality on quantum systems. I will show that any experiment based on the assumptions that the measurements can have infinite precision, perfect compatibility and sharpness can be converted into a bipartite Bell inequality experiment in which none of these assumptions is needed. The interest of the method resides in that it does not only apply to state-independent experiments based on Kochen-Specker sets, in which the conversion from contextuality into nonlocality was already known, but to any state-dependent or state-independent violation of a noncontextuality inequality. The method provides a one-to-one correspondence between the initial state and the measurements tested in the contextuality experiment and the measurements used in the resulting Bell inequality test.

**Eric Cavalcanti**, Griffith University

*Nonlocality and contextuality as fine-tuning*

Nonlocality and contextuality are at the root of conceptual puzzles in quantum mechanics, and are key resources for quantum advantage in information-processing tasks. Bell nonlocality is best understood as the incompatibility between quantum correlations and the classical theory of causality, applied to relativistic causal structure. Contextuality, on the other hand, is on a more controversial foundation. In this work, I provide a common conceptual ground between nonlocality and contextuality as violations of classical causality. First, I generalise a recent work by Wood and Spekkens, who showed that all causal models for certain Bell-inequality violations require fine-tuning of its causal parameters -- regardless of the underlying causal structure. Here I show this result holds without two of the original assumptions, applies to all (bipartite) cases of Bell nonlocality, and remarkably, does not require any assumption related to "free choice", unlike all other derivations of Bell inequalities. As a consequence, it can be applied to contextuality scenarios: all causal models for violations of a Kochen-Specker-contextuality inequality (involving two measurements per context) require fine-tuning. Thus the quantum violation of classical causality goes beyond the case of space-like separated systems, and manifests already in scenarios involving single systems.

**Giulio Chiribella**, University of Hong Kong

*A physical picture for quantum contextuality*

Quantum theory is contextual, but is not the most contextual point in the space of all conceivable theories. A natural question is what physical principles, if any, are responsible for the exact amount of contextuality that we observe in nature. I propose that quantum contextuality may arise from the balance between two desiderata: the requirement that every measurement can be realized from a repeatable and minimally disturbing measurement and the requirement that every state can be generated from a pure state. Both requirements are incarnations of the same overarching principle, namely that empirical observations must be compatible with a deeper level of description characterized by maximal knowledge.

References for this talk:

G. Chiribella and X. Yuan, Measurement sharpness cuts nonlocality and contextuality in every physical theory, https://arxiv.org/abs/1404.3348

G. Chiribella and X. Yuan, Bridging the gap between general probabilistic theories and the device-independent framework for nonlocality and contextuality, Information and Computation 250, 15 (2016).

**Otfried ****Gühne**, University of Siegen

*Contextuality and Temporal Correlations in Quantum Mechanics*

Experimental tests of contextuality often make use of sequential measurements on single quantum systems. In this talk I will explore the temporal correlations that can arise, if a sequence of measurements on a single quantum systems is made. First, I will discuss the complexity of such correlations and the difficulty to simulate them classically. Second, I will present methods to characterize temporal correlations, allowing to compute the maximal violation of contextuality inequalities in quantum mechanics. Finally, I will discuss how contextuality tests can be implemented in continuous variable systems, using the concept of modular variables.

**Teik Heinosaari**, University of Turku

*Revisiting quantum incompatibility*

Traditionally, two quantum observables are called incompatible if they cannot be measured jointly. The existence of incompatible observables gives rise to e.g steering and measurement uncertainty relations. Incompatibility can be defined not only for observables but also for channels. It can also be defined in any general probabilistic theory. In this talk I will demonstrate how the general notion of incompatibility can be seen as a unifying concept behind several features of quantum theory. I will also comment on the special nature of incompatibility in quantum theory compared to some other probabilistic theories.

**Pawel Horodecki**, Gdansk University of Technology

*On two quantum-information processing applications of quantum contextuality*

We shall present two applications of quantum contextuality. The first one [1] concerns the classical noisy channel where the receiver (Bob) is supposed to guess whether the message was altered by the noise or not. The sender (Alice) is allowed to help him via a public channel in a way, that is oblivious to some variable correlated to the message. We show that, no matter how large message Alice sends, Bob is unable to make a perfect guess. However bounded dimension quantum message allows him to complete the task with 100 % success and the latter is possible due to application of quantum contextuality.

Second application [2] exploits quantum contextuality for randomness amplification robust against no-signaling eavesdropper.

We identify the role of some subgraphs - necessary and sufficient element of any KS proof of quantum contextuality - for a construction of Hardy-type paradox. We show that the latter is a very convenient Tool for randomness amplification against post-quantum adversary.

This gives a method to get some randomness amplification protocol from quantum contextuality primitives.

[1] D. Saha, P. Horodecki, M. Pawlowski, ,,Quantum contextuality can help in 1-way communication”; (2017), to be submitted.

[2] R. Ramanathan et al. ,,Randomness amplification via Hardy paradox and quantum contextuality”; (2017) in preparation.

**Angela Karanjai**, University of Sydney

*Contextuality, the PBR theorem and their effects on simulation of quantum systems*

This talk will be about constraints on any model which reproduces the qubit stabilizer sub-theory. We show that the minimum number of classical bits required to specify the state of an n-qubit system must scale as ~ n(n-3)/2 in any model that does not contradict the predictions of the quantum stabilizer sub-theory. The Gottesman-Knill algorithm, which is a strong simulation algorithm is in fact, very close to this bound as it scales at ~n(2n+1). This is a result of state-independent contextuality which puts a lower bound on the minimum number of states a model requires in order to reproduce the statistics of the qubit stabilizer sub-theory.

**Simon Kochen**, Princeton University

*Quantum Mechanics in a New Key*

We formulate a general principle that supplants a Boolean sigma-algebra of intrinsic properties of a classical system by a sigma-complex (a union of sigma-algebras) of extrinsic properties of a quantum system that are elicited by interactions with other systems. We apply the classical physics definitions of observables, states, combined systems, symmetries, and dynamics to extrinsic properties to derive the standard quantum formalism, including the Schrodinger equation and the von Neumann-Luder's Projection Rule. This reconstruction of quantum mechanics is then used to discuss and resolve dilemmas of the orthodox interpretation.

**Ravi Kunjwal,** Perimeter Institute

*How to go from the KS theorem to experimentally testable noncontextuality inequalities*

The purpose of this talk is twofold: one, to acquaint the wider community working mostly on Bell-Kochen-Specker contextuality with recent work on Spekkens’ contextuality that quantitatively demonstrates the sense in which Bell-Kochen-Specker contextuality is subsumed within Spekkens’ approach, and two, to argue that one can test for contextuality without appealing to a notion of sharpness which can needlessly restrict the scope of operational theories that could be considered as candidate explanations of experimental data. Testing contextuality in Spekkens’ approach therefore extends the range of experimental scenarios in which contextuality can be witnessed, and refines what it means to witness contextuality in the presence of inevitable noise in KS-type experiments. We will see this for both KS-uncolourability based logical contradiction type proofs of the KS theorem a la Kochen-Specker and statistical proofs on KS-colourable scenarios a la KCBS or Yu-Oh. While Bell-KS contextuality can be mathematically understood as an instance of the classical marginal problem, the same is not true of Spekkens' contextuality. The latter reduces to the classical marginal problem only under very specific conditions, being more general otherwise. All in all, we will argue that all you really need is Leibniz, i.e. identity of indiscernables, to make sense of contextuality in the most general context.

**Jan-Åke Larsson**, Linkopings University

*Maximal noncontextuality; and qubit contextuality as a resource for Quantum Computation *

Testing noncontextuality inequalities in experiment is possible as long as the marginal probabilities of a measurement outcome does not change with the context, because then the assumption of noncontextuality is reasonable. It is much more difficult to motivate the assumption when the marginals change. In this talk, the notion of noncontextuality will be extended to cover changing marginals, which in turn results in adjustments of the relevant noncontextuality inequalities. It is an open question whether qubit contextuality can be thought to be a resource for quantum computation. Using an extended variant of Spekkens' toy theory, still noncontextual, we have constructed and built a physical realization that is capable of running three quantum computational algorithms. Our realization efficiently runs Deutsch-Jozsa and Simon's algorithm with zero error, and Shor's algorithm with smaller error than any other state-of-the-art realization. This suggests that there is no strong evidence for qubit contextuality being a resource for quantum computation.

**Matthew Leifer**, Chapman University

*Aharonov vs. Spekkens round II: Contextuality in Pre- and Post-Selection Paradoxes*

Yakir Aharonov and collaborators have proposed a number of seemingly counter-intuitive effects involving pre- and post-selected quantum systems, but there has been controversy over the degree to which these effects are "guenuinely quantum" with some authors arguing that they have classical analogues. In this talk, I review progress in showing that many of these effects cannot occur within a noncontextual ontological model. This includes anomalous weak values, and logical pre- and post-selection paradoxes such as the three box paradox. The proofs highlight some important features of these effects that are not usually emphasized, such as the non-orthogonality of pre- and post-selection vectors, and the precise amount of disturbance to the ontic state that can be caused by a measurement in a noncontextual model.

**Shane Mansfield**, University of Edinburgh

*The contextual fraction as a measure of contextuality*

We consider the contextual fraction as a quantitative measure of contextuality of empirical models, i.e. tables of probabilities of measurement outcomes in an experimental scenario. It provides a general way to compare the degree of contextuality across measurement scenarios; it bears a precise relationship to violations of Bell inequalities; its value, and a witnessing inequality, can be computed using linear programming; it is monotone with respect to the "free" operations of a resource theory for contextuality; and it measures quantifiable advantages in informatic tasks, such as games and a form of measurement based quantum computing.

**Michael Mazurek**, Institue for Quantum Computing

*Experimental state and measurement tomography for generalised probabilistic theories: bounding deviations from quantum theory via noncontextuality inequality violations*

In order to perform foundational experiments testing the correctness of quantum mechanics, one requires data analysis tools that do not assume quantum theory. We introduce a quantum-free tomography technique that fits experimental data to a set of states and measurement effects in a generalised probabilistic theory (GPT). (This is in contrast to quantum tomography, which fits data to sets of density operators and POVM elements.) We perform an experiment on the polarization degree of freedom of single photons, and find GPT descriptions of the states and measurements in our experiment. We gather data for a large number of preparation and measurement procedures in order to map out the spaces of allowed GPT states and measurement effects, and we bound their possible deviation from quantum theory. Our GPT tomography method allows us to bound the extent to which nature might be more or less contextual than quantum theory, as measured by the maximum achievable violation of a particular noncontextuality inequality. We find that the maximal violation is confined to lie between 1.2±0.1% less than and 1.3±0.1% greater than the quantum prediction.

Coauthors: Matthew Pusey, Robert Spekkens, Kevin Resch

**Ana Belen Sainz**, Perimeter Institute

*Kochen-Specker contextuality: a hypergraph approach with operational equivalences*

Most work on contextuality so far has focused on specific examples and concrete proofs of the Kochen-Specker theorem, while general definitions and theorems about contextuality are sparse. For example, it is commonly believed that nonlocality is a special case of contextuality, but what exactly does this mean? In this work, that builds on the graph-theoretic approach of Cabello, Severini and Winter, we develop a hypergraph approach to study Kochen-Specker contextuality and Bell nonlocality in a unified manner. In this talk I will further focus on the relation between some sets of probabilistic models and graph invariants, and discuss principles to characterise quantum predictions.

**Jamie Sikora,** Centre for Quantum Technologies

*Large non-contextuality inequality violations via parity-oblivious random access codes*

**William Slofstra**, Institute for Quantum Computing

*Group theory and contextuality*

Mermin-Peres-type contextuality scenarios are easy to write down, but can we determine when a scenario has a quantum model? It turns out that contextuality scenarios of this type can encode the word problem of an arbitrary finitely-presented group. As a result, determining whether a scenario has a (possibly infinite-dimensional) quantum model is undecidable in general. We can also use this connection with group theory to construct scenarios with interesting properties, including examples of scenarios which do not have finite-dimensional models, but do have infinite-dimensional models. In this talk, I will give an overview of these results, along with some stronger results which imply that it is undecidable to determine if a contextuality scenario has a finite-dimensional model.

**Mordecai Waegell,** Chapman University

*Confined Contextuality and Weak Measurement*

It is shown that by using both pre-selection and post-selection, the explicit disagreement between the predictions of quantum mechanics and noncontextual hidden variable theories (NCHVTs) can be confined to a specific measurement context. If the contradictory values assigned by the NCHVT are interpreted as facts about nature during the time interval between the pre-selection and post-selection, then we obtain the class of logical pre- and post-selection (PPS) paradoxes, which include the 3-box paradox, the quantum Cheshire Cat, and the quantum pigeonhole effect. We argue here that this interpretation is dubious because the only physical experiment that can probe the observables in the contradictory context --- a weak measurement, reveals weak values that manifestly disagree with some values assigned by the NCHVT. Furthermore, we show that projectors with anomalous weak values are necessary for the existence of any PPS paradox, and for particular cases where the contextuality is confined within a Bell-Kochen-Specker set, the algebraic structure of the set forces some weak values in the contradictory context to be anomalous. Using these facts, we derive a contextuality witness observable for an entire context, whose weak value is positive for any non-contradictory NCHVT assignment to that context.

**Joel Wallman**, Institute for Quantum Computing

*Quasiprobability representations and contextuality*