Quantum Foundations

In this talk I will review the construction of space starting purely from quantum mechanics and without assuming that the notion of space is attached to a preconceived notion of classical reality. I will show that if one start with the simplest notion of a quantum system encoded into the Heisenberg group algebra one naturally obtain a  notion of space that generalizes the usual notion of Euclidean space. Along the way I will try to illustrate how this notion is in a way going back to the roots of the discovery of QM by Heisenberg-Born-Jordan which never made it past the original papers and giving a critical reading of the subsequent interpretation of space and QM that were put forward by Schrodinger and von Neumann. The notion of space that emerge from quantum mechanics is naturally modular in the sense of Aharonov, it also naturally possess a built-in length scale and renders possible to assign a new notion of locality to non-local superpositions. I will illustrate how  such space can allow reconciliation of relativity with the presence of a fundamental scale.

I will show how to construct such spaces following the original analysis by Mackay and also show that such modular spaces possess a beautiful geometrical structure that generalizes Riemanian geometry to phase space. A geometry we have named Born geometry. I hope this will open wild speculations on the nature of locality in the presence of quantum mechanics and more broadly the nature of classical reality viewed from a quantum perspective.

Can we decompose the information of a composite system into terms arising from its parts and their interactions?
For a bipartite system (X,Y), the joint entropy can be written as an algebraic sum of three terms: the entropy of X alone, the entropy of Y alone, and the mutual information of X and Y, which comes with an opposite sign. This suggests a set-theoretical analogy: mutual information is a sort of "intersection", and joint entropy is a sort of "union".
The same picture cannot be generalized to three or more parts in a straightforward way, and the problem is still considered open. Is there a deep reason for why the set-theoretical analogy fails?
Category theory can give an alternative, conceptual point of view on the problem. As Shannon already noted, information appears to be related to symmetry. This suggests a natural lattice structure for information, which is compatible with a set-theoretical picture only for bipartite systems.
The categorical approach favors objects with a structure in place of just numbers to describe information quantities. We hope that this can clarify the mathematical structure underlying information theory, and leave it open to wider generalizations.

Non-abelian anyons have drawn much interest due to their suspected existence in two-dimensional condensed matter systems and for their potential applications in quantum computation. In particular, a quantum computation can in principle be realized by braiding and fusing certain non-abelian anyons. These operations are expected to be intrinsically robust due to their topological nature. Provided the system is kept at a
temperature T lower than the spectral gap, the density of thermal excitations is suppressed by an exponential Boltzman factor. In contrast to the topological protection however, this thermal protection is not scalable: thermal excitations do appear at constant density for any non-zero temperature and so their presence is unavoidable as the size of the computation increases. Thermally activated anyons can corrupt the encoded
data by braiding or fusing with the computational anyons.

In the present work, we generalize a fault-tolerant scheme introduced by Harrington for the toric-code to the setting of non-cyclic modular anyons. We prove that the quantum information encoded in the fusion space of a non-abelian anyon system can be preserved for arbitrarily long times with poly-log overhead. In particular, our model accounts for noise processes which lead to the creation of anyon pairs from the vacuum, anyon diffusion, anyon fusion as well as errors in topological charge measurements.
Related Arxiv #: arXiv:1607.02159

In order to solve the problem of quantum gravity, we first need to pose the problem. In this talk I will argue that the problem of quantum gravity arises already in the domain of quantum mechanics and the relativity principle. Specifically, the relativity principle implies that the concept of inertial motion should extend also to those systems that are in quantum superpositions of inertial motions. By contrast, relativistic quantum field theory only considers the point of view of classical observers in states of definite relative motion (i.e. the observers of a quantum field do not include inertial observers in quantum superpositions). The problem is that, if we extend the class of inertial observers to include quantum observers, the manifold of local events becomes ill-defined, as `locality' itself becomes an observer-relative property of an event. Thus, the Relativity Principle and the Superposition Principle are jointly opposed to the concept of a space-time manifold of local events, and our understanding of relativistic quantum theory needs to be revised before gravity even enters the picture.

I will explain and prove the statement of the title. The proof relies on a recent result of Slofstra in combinatorial group theory and the hypergraph approach to contextuality.

Based on http://arxiv.org/abs/1607.05870.

In quantum information, we frequently consider (for instance, whenever we talk about entanglement) a composite system consisting of two separated subsystems. A standard axiom of quantum mechanics states that a composite system can be modeled as the tensor product of the two subsystems. However, there is another less restrictive way to model a composite system, which is used in quantum field theory: we can require only that the algebras of observables for each subsystem commute within some larger subalgebra. For finite-dimensional systems, these two axioms are equivalent, but this is not necessarily true for infinite-dimensional systems. Tsirelson's question (which comes in several variants) asks whether the correlations arising from commuting-operator models can always be represented by tensor-product models. I will give examples of linear system non-local games which cannot be played perfectly with tensor-product strategies, but can be played perfectly with commuting-operator strategies, resolving (one version of) Tsirelson's question in the negative. From these examples, we can also derive other consequences for the theory of non-local games, such as the undecidability of determining whether a non-local game has a perfect quantum strategy. 

Bell's theorem shows that our intuitive understanding of causation must be overturned in light of quantum correlations. Nevertheless, quantum mechanics does not permit signalling and hence a notion of cause remains. Understanding this notion is not only important at a fundamental level, but also for technological applications such as key distribution and randomness expansion. It has recently been shown that a useful way to determine which classical causal structures give rise to a given set of correlations is to use entropy vectors. We consider the question of whether such vectors can lead to useful certificates of non-classicality. We find that for a family of causal structures that include the usual bipartite Bell structure they do not, in spite of the existence of non-classical correlations. Furthermore, we find that for many causal structures non-Shannon entropic inequalities give additional constraints on the sets of possible entropy vectors in the classical case. They hence lead to tighter approximations of the set of realisable entropy vectors, which enables a sharper distinction of different causal structures. Whether these improved characterisations are also valid for the quantum case remains an open problem whose resolution would have implications for the discrimination of classical and quantum causes.