Quantum Foundations

Cellular automata are a central notion for the formulation of physical laws in an abstract information-theoretical scenario, and lead in recent years to the reconstruction of free relativistic quantum field theory. In this talk we extend the notion of a Quantum Cellular Automaton to general Operational Probabilistic Theories. For this purpose, we construct infinite composite systems, illustrating the main features of their states, effects and transformations. We discuss the generalization of the concepts of homogeneity and locality, in an framework where space-time is not a primitive object. We show that homogeneity leads to a Cayley graph structure of the memory array, thus proving the universality of the connection between homogeneity and discrete groups. We conclude illustrating the special case of Fermionic cellular automata, discussing three relevant examples: Weyl and Dirac quantum walks, the Thirring automaton and the simplest families of automata on finite graphs.

In physics, every observation is made with respect to a frame of reference. Although reference frames are usually not considered as degrees of freedom, in all practical situations it is a physical system which constitutes a reference frame. Can a quantum system be considered as a reference frame and, if so, which description would it give of the world? The relational approach to physics suggests that all the features of a system —such as entanglement and superposition— are observer-dependent: what appears classical from our usual laboratory description might appear to be in a superposition, or entangled, from the point of view of such a quantum reference frame. In this work, we develop an operational framework for quantum theory to be applied within quantum reference frames. We find that, when reference frames are treated as quantum degrees of freedom, a more general transformation between reference frames has to be introduced. With this transformation we describe states, measurement, and dynamical evolution in different quantum reference frames, without appealing to an external, absolute reference frame. The transformation also leads to a generalisation of the notion of covariance of dynamical physical laws, which we explore in the case of ‘superposition of Galilean translations’ and ‘superposition of Galilean boosts’. In addition, we consider the situation when the reference frame moves in a ‘superposition of accelerations’, which leads us to extend the validity of the weak equivalence principle to quantum reference frames.

There has been a dissatisfaction with the postulates of quantum mechanics essentially since the moment that those postulates were first written down. Over the years since there have therefore been many attempts (some successful and some less so) to reconstruct quantum theory from various sets of postulates. The aim being to gain a deeper understanding of the theory by providing a conceptually clear underpinning from which the standard formalism can be derived. In this talk I present recent work with Carlo Maria Scandolo and Bob Coecke (arXiv:1802.00367) in which we reconstruct quantum theory from purely diagrammatic postulates within the process theory framework, showing that the conceptual bare-bones of quantum theory concerns the manner in which systems and processes compose.

In order to think about the foundations of physics it is important to understand the logical relationships among the physical principles that sustain the building. As part of these axioms of physics there is the core hypothesis that, how the Universe is partitioned into systems and subsystems is a subjective choice of the observer that should not affect the predictions of physics. Other foundational principles are the Postulates of Quantum Mechanics. However, we prove that these are not independent from the “independence of subsystem partitioning” hypothesis described above. In particular, we prove that the mathematical structure of quantum measurements and the formula for assigning outcome probabilities are implied by the mentioned hypothesis and the rest of quantum postulates. 

In quantum Shannon theory, the way information is encoded and decoded takes advantage of the laws of quantum mechanics, while the way communication channels are interlinked is assumed to be classical. In this talk, I will relax the assumption that quantum channels are combined classically, and I will show that, in certain situations, combining quantum channels in an indefinite causal order allows us to achieve tasks that are impossible in conventional quantum Shannon theory. Specifically, I will show a phenomenon of causal activation, whereby two identical copies of a completely depolarizing channel become able to transmit information when they are combined in a quantum superposition of two alternative orders. This finding runs counter to the intuition that if two communication channels are identical, using them in different orders should not make any difference.

In this talk, I will discuss two important cryptographic tasks in a post-quantum world. I will discuss the impossibility of bit-commitment and the possibility of physically unforgeable money in the framework of Generalized Probabilistic Theories. Our proofs rely crucially on cone programming, a natural generalization of semidefinite programming. This is based on work with John Selby, arXiv:1803.10279 and arXiv:1711.02662.