We show that quantum theory allows for transformations of black boxes that cannot be realized by inserting the input black boxes within a circuit in a predefined causal order. The simplest example of such a transformation is the classical switch of black boxes, where two input black boxes are arranged in two different orders conditionally on the value of a classical bit. The quantum version of this transformation\char22{}the quantum switch\char22{}produces an output circuit where the order of the connections is controlled by a quantum bit, which becomes entangled with the circuit structure. Simulating these transformations in a circuit with fixed causal structure requires either postselection or an extra query to the input black boxes.

We present a framework to treat quantum networks and all possible transformations thereof, including as special cases all possible manipulations of quantum states, measurements, and channels, such as, e.g., cloning, discrimination, estimation, and tomography. Our framework is based on the concepts of quantum comb—which describes all transformations achievable by a given quantum network—and link product—the operation of connecting two quantum networks. Quantum networks are treated both from a constructive point of view—based on connections of elementary circuits—and from an axiomatic one—based on a hierarchy of admissible quantum maps. In the axiomatic context a fundamental property is shown, which we call universality of quantum memory channels: any admissible transformation of quantum networks can be realized by a suitable sequence of memory channels. The open problem whether this property fails for some nonquantum theory, e.g., for no-signaling boxes, is posed.

We derive quantum theory from purely informational principles. Five elementary axioms\char22{}causality, perfect distinguishability, ideal compression, local distinguishability, and pure conditioning\char22{}define a broad class of theories of information processing that can be regarded as standard. One postulate\char22{}purification\char22{}singles out quantum theory within this class.

We investigate general probabilistic theories in which every mixed state has a purification, unique up to reversible channels on the purifying system. We show that the purification principle is equivalent to the existence of a reversible realization of every physical process, that is, to the fact that every physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded. From the purification principle we also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Jamio\l{}kowski isomorphism in quantum theory. Such an isomorphism allows one to prove most of the basic features of quantum theory, like, e.g., existence of pure bipartite states giving perfect correlations in independent experiments, no information without disturbance, no joint discrimination of all pure states, no cloning, teleportation, no programming, no bit commitment, complementarity between correctable channels and deletion channels, characterization of entanglement-breaking channels as measure-and-prepare channels, and others, without resorting to the mathematical framework of Hilbert spaces.

Quantum operations describe any state change allowed in quantum mechanics, including the evolution of an open system or the state change due to a measurement. We present a general method based on quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation. As input the method needs only a single entangled state. The feasibility of the technique for the electromagnetic field is shown, and the experimental setup is illustrated based on homodyne tomography of a twin beam.