Completeness of Graphical Languages for Mixed States Quantum Mechanics There exist several graphical languages for quantum information processing, like quantum circuits, ZXCalculus, ZWCalculus, etc. Each of these languages forms a xe2x80xa0symmetric monoidal category (xe2x80xa0SMC) and comes with an interpretation functor to the xe2x80xa0SMC of (finite dimension) Hilbert spaces. In the recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics. We address the question of the extension of these languages beyond pure quantum mechanics, in order to reason on mixed states and general quantum operations, i.e. completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map which allows one to get rid of a quantum system, operation which is not allowed in pure quantum mechanics. We introduce a new construction, the discard construction, which transforms any xe2x80xa0symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal. Using this construction we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However this construction fails for some fringe cases like the CliffordT quantum mechanics, as the category does not have enough isometries.

A First Step to the Categorical Logic of Quantum Programs The longterm goal of our research is to develop a powerful quantum logic which is useful in the formal verification of quantum programs and protocols. In this paper we introduce the basic idea of our categorical logic of quantum programs (CLQP): It combines the logic of quantum programming (LQP) and categorical quantum mechanics (CQM) such that the advantages of both LQP and CQM are preserved while their disadvantages are overcome. We present the syntax, semantics and proof system of CLQP. As a proofofconcept, we apply CLQP to verify the correctness of Deutschxe2x80x99s algorithm and the concealing property of quantum bit commitment.