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It is sometimes pointed out as a curiosity that the state space of quantum theory and actual physical space seem related in a surprising way: not only is space three-dimensional and Euclidean, but so is the Bloch ball which describes quantum two-level systems. In the talk, I report on joint work with Lluis Masanes, where we show how this observation can be turned into a mathematical result: suppose that physics takes place in d spatial dimensions, and that some events happen probabilistically (dropping quantum theory and complex amplitudes altogether). Furthermore, suppose there are systems that in some sense behave as “binary units of direction information”, interacting via some continuous reversible time evolution. We prove that this uniquely determines d=3 and quantum theory, and that it allows observers to infer local spatial geometry from probability measurements.

One of the most important open problems in physics is to reconcile quantum mechanics with our classical intuition. In this talk we look at quantum foundations through the lens of mathematical foundations and uncover a deep connection between the two fields. We show that Cantorian set theory is based on classical concepts incompatible with quantum experiments. Specifically, we prove that Zermelo-Fraenkel axioms of set theory (and the background classical logic) imply a Bell-type inequality. Consequently, quantum experiments violating Bell inequality cannot be described in the framework of classical set theory. This suggests that a non-Cantorian set theory could be a better framework to capture the elusive nature of quantum world. Finally, we discuss several possible options for a future logico-mathematical framework compatible with quantum experiments. 

We establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here.    Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity.  Since our main results are expressed in a diagrammatic language (the one of dagger-compact categories) they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory.    The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.   This is joint work with Ross Duncan, Aleks Kissinger and Quanlong (Harny) Wang. Paper: arXiv:1203.4988 - LiCS'12 

I present our work on inferring causality in the classical world and encourage the audience to think about possible generalizations to the quantum world. Statistical dependences between observed quantities X and Y indicate a causal relation, but it is a priori not clear whether X caused Y or Y caused X or there is a common cause of both. It is widely believed that this can only be decided if either one is able to do interventions on the system, or if X and Y are part of a larger set of variables. In the latter case, conditional statistical independences contain some information on causal directions, formalized by the Causal Markov Condition on directed acyclic graphs. Contrary to this belief, we have shown that empirical joint distributions of just two variables often indicate the causal direction. The observed asymmetry between cause and effect is, on the one hand, related to the thermodynamic arrow of time. On the other hand, it can be derived from a new principle that we have postulated: the Algorithmic Causal Markov Condition, which relates Kolmogorov complexity to causality.  
Literature: [1] Janzing, Schoelkopf: Causal inference using the algorithmic Markov condition, IEEE TIT 2010. 
[2] Daniusis, Janzing,...: Inferring deterministic causal relations, UAI 2010. 
[3] Janzing: On the entropy production of time-series with uni-directional linearity.Journ. Stat. Phys. 2010.

We study the problem of reconstructing an unknown matrix M, of rank r and dimension d, using O(rd poly log d) Pauli measurements. This has applications to compressed sensing methods for quantum state tomography.  We give a solution to this problem based on the restricted isometry property (RIP), which improves on previous results using dual certificates. In particular, we show that almost all sets of O(rd log^6 d) Pauli measurements satisfy the rank-r RIP. This implies that M can be recovered from a fixed ("universal") set of Pauli measurements, using nuclear-norm minimization (e.g., the matrix Lasso), with nearly-optimal bounds on the error. Our proof uses Dudley's inequality for Gaussian processes, together with bounds on covering numbers obtained via entropy duality.

In the de Broglie-Bohm pilot-wave theory, an ensemble of fermions is not only described by a spinor,  but also by a distribution of position beables. If the distribution of positions is different from the one predicted by the Born rule, the ensemble is said to be in quantum non-equilibrium. Such ensembles, which can lead to an experimental discrimination between the pilot-wave theory and standard quantum mechanics, are thought to quickly relax to quantum equilibrium in most cases.   In this talk, I will look at the Majorana equation from the point of view of the pilot-wave theory and I will show that it predicts peculiar trajectories for the beables; they have to move luminally at all times and they usually undergo complex helical trajectories to give the illusion that their motion is subluminal. The nature of the Majorana trajectory suggests that relaxation to quantum equilibrium could only be partial and that quantum non-equilibrium could still survive at length scales below the Compton wavelength.  I investigate this claim, thanks to some numerical simulations of the temporal evolution of non-equilibrium distributions,  for three-dimensional confined systems governed by the Dirac and Majorana equations.