I am a Phd student at the Institute for Computing and Information Sciences of the Radboud University Nijmegen where I also did a Masters programme in Mathematical Physics. I was hired in 2016 as part of of the ERC advanced grant on Quantum Computation, Logic and Security. I am jointly supervised by Aleks Kissinger and Bart Jacobs.
My research focuses on the foundations of quantum theory. I am particularly interested in figuring out why nature is described by the laws of quantum mechanics. Related to that is also the question of why and when quantum computers are faster than regular classical computers.
I jointly teach the Masters course Quantum Processes and Computation with Aleks Kissinger at the Radboud University.
Abstract: We propose a definition of purity for positive linear maps between Euclidean Jordan Algebras (EJA) that generalizes the notion of Kraus rank one channels ($A↦B^*AB). We show that this definition of purity is closed under composition and taking adjoints and thus that the pure maps form a dagger category. This dagger yields a notion of positivity: maps of the form $f=g^\dagger∘g$ are called $\dagger$-positive. We show that such a $\dagger$-positive map $f$ is completely determined by $f(1)$ and equal to $Q_\sqrtf(1)$, the Jordan algebraic version of the sequential measurement map $A↦\sqrtf(1)A\sqrtf(1)$. The notion of $\dagger$-positivity therefore characterises the sequential product. These results hold for the general reason that the opposite category of EJAs with positive contractive linear maps is a $\dagger$-effectus. The notion of $\dagger$-effectus was introduced to prove similar results for the opposite category of normal completely positive contractive maps between von Neumann algebras.
Abstract: We study general physical systems with a notion of sequential measurement that satisfies a few simple and intuitive conditions. We show that, apart from two exceptional cases, the systems satisfying these conditions are exactly the real, complex or quaternionic quantum systems. Moreover, only the complex quantum systems remain if systems must be allowed to be composed together in a locally tomographic manner. In other words, quantum theory is the unique locally tomographic theory allowing sequential measurement.
Abstract: It has already been established that the properties required of an abstract sequential product as introduced by Gudder and Greechie are not enough to characterise the standard sequential product $a∘b = \sqrtab\sqrtb$ on an operator algebra. We give three additional properties, each of which characterises the standard sequential product on either a von Neumann algebra or a Euclidean Jordan algebra. These properties are (1) invariance under application of unital order isomorphisms, (2) symmetry of the sequential product with respect to a certain inner product, and (3) preservation of invertibility of the effects. To give these characterisations we first have to study convex sigma-sequential effect algebras. We show that these objects correspond to unit intervals of spectral order unit spaces with a homogeneous positive cone.
Abstract: There is an ongoing search for intuitive postulates of quantum theory from which its Hilbert space structure can be derived. The main contribution of this paper is the introduction of two postulates inspired by categorical logical notions from effectus theory, a framework in some ways similar to generalised probabilistic theories, but eschewing the familiar notions of real numbers and probabilities, which allows the description of more general theories. The postulates state the existence of certain physical filters that associate to each effect the subspace where it holds true. We show that when considering an operational probabilistic setting these relatively weak postulates lead to a spectral theorem and a duality between pure states and effects: each effect can be written as a probabilistic combination of perfectly distinguishable sharp effects in a unique way. In such a weak theory it is therefore already possible to define thermodynamic quantities like entropy. For these results we don't need any assumptions on the existence of pure states, or of sufficiently many reversible dynamics or even the existence of an invariant state. We finish the reconstruction of quantum theory by requiring three additional postulates: continuous symmetry, preservation of purity, and observability of energy.
Abstract: There is a long history of representing a quantum state using a quasi-probability distribution: a distribution allowing negative values. In this paper we extend such representations to deal with quantum channels. The result is a convex, strongly monoidal, functorial embedding of the category of trace preserving completely positive maps into the category of quasi-stochastic matrices. This establishes quantum theory as a subcategory of quasi-stochastic processes. Such an embedding is induced by a choice of minimal informationally complete POVM's. We show that any two such embeddings are naturally isomorphic. The embedding preserves the dagger structure of the categories if and only if the POVM's are symmetric, giving a new use of SIC-POVM's, objects that are of foundational interest in the QBism community. We also study general convex embeddings of quantum theory and prove a dichotomy that such an embedding is either trivial or faithful.
Abstract: We introduce a new universal, measurement-based model of quantum computation, which is potentially attractive for implementation on ion trap and superconducting qubit devices. It differs from the traditional one-way model in two ways: (1) the resource states which play the role of graph states are prepared via 2-qubit Mølmer-Sørensen interactions and (2) the model only requires non-rotated Pauli X and Z measurements to obtain universality. We exploit the fact that the relevant resource states and measurements have a simple representation in the ZX-calculus to give a short, graphical proof of universality.
Abstract: In this paper we introduce a new partial order on quantum states that considers which states can be achieved from others by updating on `agreeing' Bayesian evidence. We prove that this order can also be interpreted in terms of minimising worst case distuinguishability between states using the concept of quantum max-divergence. This order solves the problem of which states are optimal approximations to their more pure counterparts and it shows in an explicit way that a proposed quantum analogue of Bayes' rule leads to a Bayesian update that changes the state as little as possible when updating on positive evidence. We prove some structural properties of the order, specifically that the order preserves convex mixtures and tensor products of states and that it is a domain. The uniqueness of the order given these properties is discussed. Finally we extend this order on states to one on quantum channels using the Jamiolkowski isomorphism. This order turns the spaces of unital/non-unital trace-preserving quantum channels into domains that, unlike the regular order on channels, is not trivial for unital trace-preserving channels.
Abstract: In this paper we give an overview of partial orders on the space of probability distributions that carry a notion of information content and serve as a generalisation of the Bayesian order given in (Coecke and Martin, 2011). We investigate what constraints are necessary in order to get a unique notion of information content. These partial orders can be used to give an ordering on words in vector space models of natural language meaning relating to the contexts in which words are used, which is useful for a notion of entailment and word disambiguation. The construction used also points towards a way to create orderings on the space of density operators which allow a more fine-grained study of entailment. The partial orders in this paper are directed complete and form domains in the sense of domain theory.