**Lluis Masanes**, ICFO

*Entanglement and the three-dimensionality of the Bloch sphere *

We consider theories that satisfy: information causality, reversibility, local discriminability, all tight effects are measurable. A property of these theories is that binary systems (with two perfectly distinguishable states and no more) have state spaces with the shape of a unit ball (the Bloch ball) of arbitrary dimension. It turns out that for dimension different than three these systems cannot be entangled. Hence, the only theory with entanglement which satisfying the above assumptions is quantum theory.

**Valerio Scarani**, National University of Singapore

*3 >> 2*

Three-partite quantum systems exhibit interesting features that are absent in bipartite ones. Several instances are classics by now: the GHZ argument, the W state, the UPB bound entangled states, Svetlichny inequalities... In this talk, I shall discuss some on-going research projects that we are pursuing in my group (in collaboration, or in friendly competition, with other groups) and that involve three-partite entanglement or non-locality:

* Activation of non-locality in networks.

* Device-independent assessment of the entangling power of a measurement.

* Can one falsify all models of hidden communication with finite speed?

* Information causality in the three-partite scenario.

I shall conclude by a blind excursion into uncertainty relations and cryptography, which also shows 3>>2 albeit with a different meaning.

**Bob Coecke**, University of Oxford

*The logic of quantum mechanics - take II *

It is now exactly 75 years ago that John von Neumann denounced his own Hilbert space formalism: "I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.'' (sic) [1] His reason was that Hilbert space does not elucidate in any direct manner the key quantum behaviors. One year later, together with Birkhoff, they published "The logic of quantum mechanics". However, it is fair to say that this program was never successful nor does it have anything to do with logic. So what is logic? We will conceive logic in two manners: (1) Something which captures the mathematical content of language (cf 'and', 'or', 'no', 'if ... then' are captured by Boolean algebra); (2) something that can be encoded in a 'machine' and enables it to reason.

Recently we have proposed a new kind of 'logic of quantum mechanics' [4]. It follows Schrodinger in that the behavior of compound quantum systems, described by the tensor product [2, again 75 years ago], that captures the quantum behaviors. Over the past couple of years we have played the following game: how much quantum phenomena can be derived from 'composition + epsilon'. It turned out that epsilon can be taken to be 'very little', surely not involving anything like continuum, fields, vector spaces, but merely a 'two-dimensional space' of temporal composition (cf 'and then') and compoundness (cf 'while'), together with some very natural purely operational assertion. In a very short time, this radically different approach has produced a universal graphical language for quantum theory which helped to resolve some open problems.

Most importantly, it paved the way to automate quantum reasoning [5,6], and also enables to model meaning for natural languages [7,8]. That is, we are truly taking 'quantum logic' now! If time permits, we also discuss how this logical view has helped to solve concrete problems in quantum information.

[1] M Redei (1997) Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead). Stud Hist Phil Mod Phys 27, 493-510.

[2] G Birkhoff and J von Neumann (1936) The logic of quantum mechanics. Annals of Mathematics 37, 823843.

[3] E Schroedinger, (1935) Discussion of probability relations between separated systems. Proc Camb Phil Soc 31, 555-563; (1936) 32, 446-451.

[4] B Coecke (2010) Quantum picturalism. Contemporary Physics 51, 59-83. arXiv:0908.1787

[5] L Dixon, R Duncan, A Kissinger and A Merry. http://dream.inf.ed.ac.uk/projects/quantomatic/

[6] L Dixon and R Duncan (2009) Graphical reasoning in compact closed categories for quantum computation. Annals of Mathematics and Articial Intelligence 56, 2342.

[7] B Coecke, M Sadrzadeh & S Clark (2010) Linguistic Analysis 36. Mathematical foundations for a compositional distributional model of meaning. arXiv:1003.4394

[8] New scientist (11 Dec 2011) Quantum links let computers read.

**Scott Aaronson**, MIT

*The Territory Around BQP: Results and Open Problems*

In this talk, I'll survey various "foils" of BQP (Bounded-Error Quantum Polynomial-Time) that have been proposed: that is, changes to the quantum model of computation that make it either more or less powerful. Possible topics include: postselected quantum computing, quantum computing with nonlinear Schrodinger equation, quantum computing with non-unitary linear transformations, quantum computing with hidden variables, linear-optical quantum computing, quantum computing with restricted gate sets, quantum computing with separable mixed states, quantum computing over finite fields, and more depending on audience interest.

**Markus Mueller**, Perimeter Institute

*From operational axioms to quantum theory - and beyond? *

Usually, quantum theory (QT) is introduced by giving a list of abstract mathematical postulates, including the Hilbert space formalism and the Born rule. Even though the result is mathematically sound and in perfect agreement with experiment, there remains the question why this formalism is a natural choice, and how QT could possibly be modified in a consistent way. My talk is on recent work with Lluis Masanes, where we show that five simple operational axioms actually determine the formalism of QT uniquely. This is based to a large extent on Lucien Hardy's seminal work. We start with the framework of "general probabilistic theories", a simple, minimal mathematical description for outcome probabilities of measurements. Then, we use group theory and convex geometry to show that the state space of a bit must be a 3D (Bloch) ball, finally recovering the Hilbert space formalism. There will also be some speculation on how to find natural post-quantum theories by dropping one of the axioms.

**Marc Kaplan**, Université de Montréal

*Communication cost Vs Bell inequality violation *

In 1964, John Bell proved that independent measurements on entangled quantum states lead to correlations that cannot be reproduced using local hidden variables. The core of his proof is that such distributions violate some logical constraints known as Bell inequalities. This remarkable result establishes the non-locality of quantum physics. Bell's approach is purely qualitative. This naturally leads to the question of quantifying quantum physics' non-locality. We will specifically consider two quantities introduced for this purpose. The first one is the maximum amount of Bell inequality violation, and the second one is the communication cost of simulating quantum distributions. In this talk, we prove that these two quantities are strongly related: the logarithm of the first is upper bounded by the second. We prove this theorem in the more general context of non-signalling distributions. This generalization gives us two clear benefits. First, the rich structure of the underlying affine space provides us with a very strong intuition. Secondly, non-signalling distributions capture traditional communication complexity of boolean functions. In that case, our theorem is equivalent to the factorization norm lower bound of Linial and Shraibman, for which we give an elementary proof.

**Gilles Brassard**, Université de Montréal

*Is Information the Key?*

Consider the two great physical theories of the twentieth century: relativity and quantum mechanics. Einstein derived relativity from very simple principles. By contrast, the foundation of quantum mechanics is built on a set of rather strange, disjointed and ad hoc axioms, reflecting at best the history that led to discovering this new world order. The purpose of this talk is to argue that a better foundation for quantum mechanics lies within the teachings of quantum information science. The basic postulate is that the truly fundamental laws of Nature concern information, not waves or particles. For example, it is known that quantum key distribution is possible but quantum bit commitment is not and that nature is nonlocal but not as nonlocal as is imposed by causality. But should these statements be considered as theorems or axioms? It's time to pause and reflect on what is really fundamental and what are merely consequences. Could information be the key?

**Andreas Winter**, University of Bristol

*Non-contextual correlations in probabilistic models*

Non-contextuality is presented as an abstraction and at the same time generalisation of locality. Rather than in correlations, the underlying physical model leaves its signature in collections of expectation values, which are contrained by inequalities much like Bell's or Tsirelson's inequalities. These non-contextual inequalities reveal a deep connection to classic topics in graph theory, such as independence numbers, Lovasz numbers and other graph parameters. By considering the special case of bi-local experiments, we arrive at a semidefinite relaxation (and indeed a whole hierarchy of such relaxations) for the problem of determining the maximum quantum violation of a given Bell inequality.

**Alex Wilce**, Susquehanna University

*Symmetry, Self-Duality and the Jordan Structure of Quantum Theory*

This talk reviews recent and on-going work, much of it joint with Howard Barnum, on the origins of the Jordan-algebraic structure of finite-dimensional quantum theory. I begin by describing a simple recipe for constructing highly symmetrical probabilistic models, and discuss the ordered linear spaces generated by such models. I then consider the situation of a probabilistic theory consisting of a symmetric monoidal *-category of finite-dimensional such models: in this context, the state and effect cones are self-dual. Subject to a further "steering" axiom, they are also homogenous, and hence, by the Koecher-Vinberg Theorem, representable as the cones of formally real Jordan algebras. Finally, if the theory contains a single system with the structure of a qubit, then (by a result of H. Hanche-Olsen), each model in the category is the self-adjoint part of a C*-algebra.

**Renato Renner**, ETH Zurich

*How Fundamental is the Uncertainty Principle?*

According to quantum theory, it is impossible to prepare the state of a system such that the outcome of any projective measurement on the system can be predicted with certainty. This limitation of predictive power, which is known as the uncertainty principle, is one of the main distinguishing properties of quantum theory when compared to classical theories. In this talk, I will discuss the implications of this principle to foundational questions. In particular, I will consider the hypothesis that the uncertainty principle, rather than (only) telling us something about reality, may be seen as a manifestation of the limitations of our (classical) methods used to describe reality.

**Caslav Brukner**, University of Vienna

*Quantum correlations with no causal order*

Much of the recent progress in understanding quantum theory has been achieved within an operational approach. Within this context quantum mechanics is viewed as a theory for making probabilistic predictions for measurement outcomes following specified preparations. However, thus far some of the essential elements of the theory – space, time and causal structure – elude such an operational formulation and are assumed to be fixed. Is it possible to extend the operational approach to quantum mechanics such that the notions of an underlying spacetime or causal structure are not assumed? What new phenomenology can follow from such an approach? We

develop a framework for multipartite quantum correlations that does not presume these notions, but simply that experimenters in their local laboratories are free to perform arbitrary quantum operations. All known situations that respect definite causal order, including signalling and no-signalling correlations between space-like and time-like separated experiments, as well as probabilistic mixtures of these, can be expressed in this framework. Remarkably, we find quantum correlations which are neither causally ordered nor in a probabilistic mixture of definite causal orders. These correlations are shown to enable performing a communication task that is impossible if a fixed background time is assumed and the events are sufficiently localized in the time.

**Paolo Perinotti**, University of Pavia

*Quantum Theory as a Theory of Information Processing*

Quantum Theory can be derived from six operational axioms. We introduce the operational and probabilistic language that is used to formulate the principles. After the basic notions of system, state, effect and transformation are reviewed, the principles are stated, and their immediate consequences and interpretations are analyzed. Finally, some key results that represent milestones of the derivation are discussed, with particular focus on their implications on information processing and their relation with the standard quantum formalism. The global picture of the presentation highlights quantum theory as a particular operational language emerging from a background of information processing theories, thanks to the purification postulate that singles out the strictly quantum features of information.

**Gen Kimura**, Shibaura Institute of Technology

*On Basic Principles of General Probabilistic Theories*

We propose an operationally motivated definition of the physical equivalence of states in General Probabilistic Theories and consider the principle of the physical equivalence of pure states, which turns out to be equivalent to the symmetric structure of the state space. We further consider a principle of the decomposability with distinguishable pure states and give classification theorems of the state spaces for each principle, and derive the Bloch ball in 2 and 3 dimensional systems.

**Mauro D'Ariano**, University of Pavia

*A Quantum-Digital Universe*

David Deutsch re-formulated the Church-Turing thesis as a physical principle, asserting that "every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means". Such principle can be regarded as a new theoretical paradigm, whereby the entire Physics is emerging from a quantum computation. But for a theory to be a good one, it must explain a large class of phenomena based on few general principles. Taking as a general principle the topological homogeneity of the computational network with graph-dimension equal to the space-time dimension corresponds to replacing quantum field theory (QFT) with a numerable set of quantum systems in local interaction. This means to consider QFT as a kind of Fermi-scale "thermodynamic" limit of a deeper Planck-scale theory, with the quantum field replaced by a giant quantum computer.

In the talk, I will illustrate mechanisms of emergence of physics from the quantum computation in 1+1 dimensions. We will see that Dirac's is just the equation describing the free flow of information, leading to an informational definition of inertial mass and Planck constant. I will then illustrate the emergence mechanism of Minkowsian space-time from the computation, how the field Hamiltonian comes out, and how quantum fields are actually eliminated in favor of qubits. We will see that the digital nature of the field leads to an in-principle observable consequence in terms of a mass-dependent refraction index of vacuum, with the information becoming stationary at the Planck mass. Such refraction index of vacuum is a general phenomenon due to unitariety in the discrete, and can also help in solving the speed-of-light isotropy conundrum posed by digitalization of the field in more than 1 space dimensions. We will also see how the quantum nature of the processed information plays a crucial role in other practical informational issues, e.g. the possibility of driving the information in different directions, without the need of increasing the complexity of the circuit.

Finally I will briefly comment about gravity as emergent from the quantum computation, and the connection with Verlinde-Jacobson approach.

**Antonio Acín,** ICFO Barcelona

*Guess your neighbor input*

We present “guess your neighbor input” (GYNI), a multipartite nonlocal task in which each player must guess the input received by his neighbor. We show that quantum correlations do not perform better than classical ones at this task, for any prior distribution of the inputs. There exist, however, input distributions for which general no-signalling correlations can outperform classical and quantum correlations. Some of the Bell inequalities associated to our construction correspond to facets of the local polytope. We then discuss implications of this game in connection with recent attempts of deriving quantum correlations from information based principles, such as non-trivial communication complexity, information causality and Gleason’s theorem. Our results show that truly multipartite concepts are necessary to obtain the set of quantum correlations for an arbitrary number of parties.

**Ben Schumacher**, Kenyon College

*Almost quantum theory *

Modal quantum theory (MQT) is a discrete model that is similar in structure to ordinary quantum theory, but based on a finite field instead of complex amplitudes. Its interpretation involves only the "modal" concepts of possibility and impossibility rather than quantitative probabilities. Despite its very simple structure, MQT nevertheless includes many of the key features of actual quantum physics, including entanglement and nonclassical computation. In this talk we describe MQT and explore how modal and probabilistic theories are related. Under what circumstances can we assign probabilities to a given modal structure?

**Nicolas Brunner**, University of Bristol

*Data tables, dimension witnesses, and QKD*

We address the problem of testing the dimensionality of classical and quantum systems in a ‘black-box’ scenario. Imagine two uncharacterized devices. The first one allows an experimentalist to prepare a physical system in various ways. The second one allows the experimentalist to perform some measurement on the system. After collecting enough statistics, the experimentalist obtains a ‘data table’, featuring the probability distribution of the measurement outcomes for each choice of preparation (of the system) and of measurement. Here, we develop a general formalism to assess the minimal dimensionality of classical and quantum systems necessary to reproduce a given data table. To illustrate these ideas, we provide simple examples of classical and quantum ‘dimension witnesses’. In general quantum systems are more economical than classical ones in terms of dimensionality, in the sense that there exist data tables obtainable from quantum systems of dimension d which can only be generated from classical systems of dimension strictly greater than d. By drawing connections to communication complexity one can find data tables for which this classical/quantum separation is dramatic. Finally, these ideas can also be used to demonstrate security of one-way QKD in a semi-device-independent scenario, in which devices are uncharacterized, but only assumed to produce quantum systems of a given dimension.

**Tsuyoshi Ito**, Institute for Quantum Computing

*Nonlocal Games and Computational Complexity: A Survey *

A seminal work by Cleve, Høyer, Toner and Watrous (quant-ph/0404076) proposed a close connection between quantum nonlocality and computational complexity theory by considering nonlocal games and multi-prover interactive proof systems with entangled provers. It opened up the whole area of study of the computational nature of nonlocality. Since then, understanding nonlocality has been one of the major goals in computational complexity theory in the quantum setting. This talk gives a survey of this exciting area.

**Jonathan Barrett**, Royal Holloway

*Is the universe exponentially complicated? A no-go theorem for hidden variable interpretations of quantum theory.*

The quantum mechanical state vector is a complicated object. In particular, the amount of data that must be given in order to specify the state vector (even approximately) increases exponentially with the number of quantum systems. Does this mean that the universe is, in some sense, exponentially complicated? I argue that the answer is yes, if the state vector is a one-to-one description of some part of physical reality. This is the case according to both the Everett and Bohm interpretations. But another possibility is that the state vector merely represents information about an underlying reality. In this case, the exponential complexity of the state vector is no more disturbing that that of a classical probability distribution: specifying a probability distribution over N variables also requires an amount of data that is exponential in N. This leaves the following question: does there exist an interpretation of quantum theory such that (i) the state vector merely represents information and (ii) the underlying reality is simple to describe (i.e., not exponential)? Adapting recent results in communication complexity, I will show that the answer is no. Just as any realist interpretation of quantum theory must be non-locally-causal (by Bell's theorem), any realist interpretation must describe an exponentially complicated reality.

**Tony Short**, University of Cambridge

*Generalized entropies, information causality, and non-local games *

We will explore generalizations of the Shannon and von Neumann entropy to other probabilistic theories, and their connection to the principle of information causality. We will also investigate the link between information causality and non-local games, leading to a new quantum bound on computing the inner product non-locally.

**Dan Browne**, University College London

*Computation from correlations - in classical, quantum and generalised theories*

Operational theories [1], defined in terms of the actions and observations of an experimenter, have been extremely successful as foils to quantum mechanics, providing a generic framework in which families of theories may be compared and classified. One area of particular interest has been in the non-classical correlations (often referred to non-locality) which can arise in quantum (and generalized) theories, when measurements are space-like separated. In the context of non-locality, one usually considers the correlations in separated measurements on isolated systems. A similar setting arises in quantum computation theory, in measurement-based quantum computation, a model of computation of equivalent power to standard circuit model quantum computation. Measurements are made on isolated non-interacting quantum systems, and the non-classical correlations which arise embody (in some loose sense) the mechanism via which the computation is executed. These measurements are adaptive, meaning that bases are chosen dependent upon the outcome of prior measurements, but apart from this, the setting is essentially identical to a multi-party Bell non-locality experiment (e.g. [2]). In this talk I will review some recent work [3] in which Bell-type correlations are studied from the perspective of computation - in particular drawing parallels with measurement-based quantum computation. In particular, I shall give examples of results [3] which appear naturally in this setting, while being not so self-evident in more conventional approaches. Finally, I shall discuss approaches to and challenges in developing non-trivial models of correlation-based quantum computation in general operational theories.

[1] See e.g. H. Barnum, J. Barrett, M. Leifer and A. Wilce, Phys. Rev. Lett., 99, 240501 (2007).

[2] See e.g. R. F. Werner and M. M. Wolf, Phys. Rev. A 64, 032112 (2001), M. Zukowski, C. Brukner, Phys. Rev. Lett. 88 210401 (2002).

[3] M.J. Hoban and D.E. Browne, http://arxiv.org/abs/1102.1438, M.J. Hoban et al,http://arxiv.org/abs/1009.5213, J. Anders and D.E. Browne http://arxiv.org/abs/0805.1002.

**Sandu Popescu**, University of Bristol

*Dynamical quantum nonlocality*

That quantum mechanics is non-local, in the sense of Bell inequality violations and the associated entanglement effects, is by now well known. In my talk however I will argue that quantum mechanics contains also a completely different type of non-locality, that has received virtually no attention until now: It is the fact that the quantum equations of motion are nonlocal. The discovery of dynamic non-locality dates from before that of Bell’s inequalities – it is the nonlocality responsible for the Aharonov-Bohm effect. Its implications however transcend the context in which they were discovered and go directly to the core of quantum physics.

**Lucien Hardy**, Perimeter Institute

*Reformulating and reconstructing quantum theory*

I provide a reformulation of finite dimensional quantum theory in the circuit framework in terms of mathematical axioms, and a reconstruction of quantum theory from operational postulates. The mathematical axioms for quantum theory are the following:

[Axiom 1] Operations correspond to operators.

[Axiom 2] Every complete set of positive operators corresponds to a complete set of operations.

The following operational postulates are shown to be equivalent to these mathematical axioms:

[P1] Definiteness. Associated with any given pure state is a unique maximal effect giving probability equal to one. This maximal effect does not give probability equal to one for any other pure state.

[P2] Information locality. A maximal measurement on a composite system is affected if we perform maximal measurements on each of the components.

[P3] Tomographic locality. The state of a composite system can be determined from the statistics collected by making measurements on the components.

[P4] Compound permutatability. There exists a compound reversible transformation on any system effecting any given permutation of any given maximal set of distinguishable states for that system.

[P5] Preparability. Filters are non-mixing and non-flattening.

Hence, from these postulates we can reconstruct all the usual features of quantum theory: States are represented by positive operators, transformations by completely positive trace non-increasing maps, and effects by positive operators. The Born rule (i.e. the trace rule) for calculating probabilities also follows. See arXiv:1104.2066 for more details. These operational postulates are deeper than those I gave ten years ago in quant-ph/0101012.

**Charles Bennett**, IBM Research

*Quantum information, the ambiguity of the past, and the complexity of the present *

Entanglement provides a coherent view of the physical origin of randomness and the growth and decay of correlations, even in macroscopic systems exhibiting few traditional quantum hallmarks. It helps explain why the future is more uncertain than the past, and how correlations can become macroscopic and classical by being redundantly replicated throughout a system's environment. The most private information, exemplified by a quantum eraser experiment, exists only transiently: after the experiment is over no record remains anywhere in the universe of what "happened". At the other extreme is information that has been so widely replicated as to be infeasible to conceal and unlikely to be forgotten. But such conspicuous information is exceptional: a comparison of entropy flows into and out of the Earth with estimates of the planet's storage capacity leads to the conclusion that most macroscopic classical information---for example the pattern of drops in last week's rainfall---is impermanent, eventually becoming nearly as ambiguous, from a terrestrial perspective, as the transient result of a quantum eraser experiment. Finally we discuss prerequisites for a system to accumulate and maintain in its present state, as our world does, a complex and redundant record of at least some features of its past. Not all dynamics and initial conditions lead to this behavior, and in those that do, the behavior itself tends to be temporary, with the system losing its memory, and even its classical character, as it relaxes to thermal equilibrium.

**Roger Colbeck**, Perimeter Institute

*Randomness amplification*

I will discuss what we know about creating randomness within physics. Although quantum theory prescribes completely random outcomes to particular processes, could it be that within a yet-to-be-discovered post-quantum theory these outcomes are predictable? We have recently shown that this is not possible, using a very natural assumption. In the present talk, I will discuss some recent progress towards relaxing this assumption, providing arguably the strongest evidence yet for truly random processes in our world.

**Jonathan Oppenheim**, University of Cambridge

*Uncertainty, nonlocality & complementarity*

**Chris Fuchs**, Perimeter Institute

*Some Negative Remarks on Operational Approaches to Quantum Theory*

Over the last 10 years there has been an explosion of “operational reconstructions” of quantum theory. This is great stuff: For, through it, we come to see the myriad ways in which the quantum formalism can be chopped into primitives and, through clever toil, brought back together to form a smooth whole. An image of an IQ-Block puzzle comes to mind, http://www.prismenfernglas.de/iqblock_e.htm. There is no doubt that this is invaluable work, particularly for our understanding of the intricate connections between so many quantum information protocols. But to me, it seems to miss the mark for an ultimate understanding of quantum theory; I am left hungry. I still want to know what strange property of matter forces this formalism upon our information accounting. To play on something Einstein once wrote to Max Born, “The quantum reconstructions are certainly imposing. But an inner voice tells me that they are not yet the real thing. The reconstructions say a lot, but do not really bring us any closer to the secret of the 'old one’." In this talk, I hope to expand on these points and convey some sense of why I am fascinated with the problem of the symmetric informationally complete POVMs to an extent greater than axiomatic reconstructions.

**Howard Barnum**, University of New Mexico

*Composite systems and information processing*

The talk will focus primarily on recent work with Alexander Wilce in which we show that any locally tomographic composite of a qubit with any finite-dimensional homogeneous self-dual (equivalently Jordan-algebraic) system must be a standard finite-dimensional quantum (i.e. $C^*$-algebraic) system. I may touch on work in progress with collaborators on composites of arbitrary homogeneous self-dual systems. As motivation I will relate the properties of homogeneity and weak and strong self-duality to information processing phenomena, especially Schrooedingerian "steering" and teleportation (touching on earlier work with Wilce and Gaebler, as well as Barrett and Leifer). If time permits I will explain the relation between some category-theoretic notions coming from the approach of Abramsky and Coecke and Selinger, notably compactness and dagger-compactness, to weak self-duality (work with Ross Duncan and Wilce).

**Stephanie Wehner**, National University of Singapore

*Does ignorance of the whole imply ignorance of the parts?*

A central question in our understanding of the physical world is how our knowledge of the whole relates to our knowledge of the individual parts. One aspect of this question is the following: to what extent does ignorance about a whole preclude knowledge of at least one of its parts? Relying purely on classical intuition, one would certainly be inclined to conjecture that a strong ignorance of the whole cannot come without significant ignorance of at least one of its parts. Indeed, we show that this reasoning holds in any non-contextual hidden variable model (NC-HV). Curiously, however, such a conjecture is false in quantum theory: we provide an explicit example where a large ignorance about the whole can coexist with an almost perfect knowledge of each of its parts. More specifically, we provide a simple information-theoretic inequality satisfied in any NC-HV, but which can be arbitrarily violated by quantum mechanics. Our inequality has interesting implications for quantum cryptography.

**Giulio Chiribella**, Perimeter Institute

Opening Welcome