Universal and complete graphical languages have been successfully designed for pure state quantum mechanics, corresponding to linear maps between Hilbert spaces, and mixed states quantum mechanics, corresponding to completely positive superoperators. In this paper, we go one step further and present a universal and complete graphical language for Hermiticity-preserving superoperators. Such a language opens the possibility of diagrammatic compositional investigations of antilinear transformations featured in various physical situations, such as the Choi-Jamio lkowski isomorphism, spin-flip, or entanglement witnesses. Our construction relies on an extension of the ZW-calculus exhibiting a normal form for Hermitian matrices.for any two states |ψ and |φ . In 1931, Wigner proved that the symmetries of quantum theory should be either unitary or antiunitary (see the appendix to Chapter 20 of [48]), meaning that the general form of T in Eq. ( 1) iswhere U is a linear operator respecting U † U = I = U U † , and |ψ is the complex conjugation of |ψ . This abstract point of view turned out to be extremely useful for all aspects of quantum theory. On the one hand, the study of symmetries has been a very fruitful method for simplifying problems encountered at all scales, from solid-state physics to high-energy physics, with notable uses in atomic physics and quantum information sciences. On the other hand, symmetries have been a guiding principle for the construction of theories such as the standard model and, by extension, Yang-Mills theory (see e.g. [45,46]).So far, physicists and quantum computer scientists have been mainly concerned with unitary transformations, the ubiquitous ingredient in the study of dynamics, and little attention has been devoted to antiunitary transformations. There is a clear reason why: unitary transformations represent what can be realised and tested in a (closed) lab. In contrast, one year after his theorem, Wigner showed that antiunitaries are typically involved in the case of a time reversal [47] (see, for example, §11.4.2 of [41] for a source in English), something experimentalists cannot achieve (at least within the current theoretical framework of physics). Still, some quantum computer scientists have started investigating the possible computational advantages processes involving time reversal could provide. We mention in particular the complexity-theoretic study of [1] and the categorical semantics of [39]. However, little has been done on the computational power of antiunitaries in general, although some results suggest possible advantages in quantum information [8,22,36], and that they may be simulable [5,19,40]. Such investigations are actually made difficult by the lack of a clearly defined computational framework because it would require going beyond the usual mathematics underlying quantum circuits. Rigorously put, antiuniatries are associated with positive but not completely positive mappings. Hence, they are outside the framework of quantum computing with open systems, and are assume...

No abstract