BIRS Workshop Lecture Videos

Banff International Research Station Logo

BIRS Workshop Lecture Videos

Theoretical framework for Higher-Order Quantum Theory Bisio, Alessandro


The most general (probabilistic) transformation of a quantum state is described by a quantum oper- ation. Quantum operations can be axiomatically defined as the most general map which are compatible with the probabilistic structure of the theory, and produce a legitimate output when applied locally on one side of a bipartite input. These admissibility requirements characterise quantum operations as com- pletely positive trace non increasing linear maps. What happens if we now consider maps from quantum operations to quantum operations Can we give an axiomatic characterisation of these objects according to some generalised notion of admissibility What happens if we recursively iterate the construction and we define a full hierarchy of higher order maps Some special cases of higher order maps have been already studied in the literature. Causally ordered Quantum Networks, which encompass all conceivable quantum protocols, form a sub-hierarchy of maps which are endowed with a well ordered causal structure and they can be realised as quantum circuits. However, more general higher order maps may exhibit an indefinite causal structure which prevents a physical implementation as a quantum circuit. Non circuital higher order maps allow to accomplishing certain tasks that cannot be achieved by circuital maps, like the violation of causal inequalities, and can outperform circuital maps in certain quantum information pro- cessing tasks. The experimental realisation of non-circuital higher order maps has also been considered. Notwithstanding many results on the subject, a general mathematical framework is still missing. The aim of this contribution is to fill this critical gap by providing an axiomatic framework for higher order quantum theory. Higher order quantum theory is introduced axiomatically with a formulation based on the language of types of transformations. Complete positivity of higher order maps is derived from the general admissibility conditions instead of being postulated as in previous approaches. We will see that a complete mathematical characterisation of admissible maps is possible and that the set of admissible maps of a given type is in correspondence with a convex subset of the cone of positive operators. This result encompasses the analysis existing in the literature and gives them an axiomatic operational foundation. The present axioms for higher order quantum theory have an operational nature and do not refer to the specific mathematical structure of quantum theory. Therefore, with due care, our framework can be applied to general operational probabilistic theories.

Item Media

Item Citations and Data


Attribution-NonCommercial-NoDerivatives 4.0 International