A categorical semantics for causal structure

We present a categorical construction for modelling causal structures within a general class of process theories that include the theory of classical probabilistic processes as well as quantum theory. Unlike prior constructions within categorical quantum mechanics, the objects of this theory encode fine-grained causal relationships between subsystems and give a new method for expressing and deriving consequences for a broad class of causal structures. We show that this framework enables one to define families of processes which are consistent with arbitrary acyclic causal orderings. In particular, one can define one-way signalling (a.k.a. semi-causal) processes, non-signalling processes, and quantum $n$-combs. Furthermore, our framework is general enough to accommodate recently-proposed generalisations of classical and quantum theory where processes only need to have a fixed causal ordering locally, but globally allow indefinite causal ordering. To illustrate this point, we show that certain processes of this kind, such as the quantum switch, the process matrices of Oreshkov, Costa, and Brukner, and a classical three-party example due to Baumeler, Feix, and Wolf are all instances of a certain family of processes we refer to as $\textrm{SOC}_n$ in the appropriate category of higher-order causal processes. After defining these families of causal structures within our framework, we give derivations of their operational behaviour using simple, diagrammatic axioms.Comment: Extended version of a LICS 2017 paper with the same titl

The role of positivity and causality in interactions involving higher spin

It is shown that the recently introduced positivity and causality preserving string-local quantum field theory (SLFT) resolves most No-Go situations in higher spin problems. This includes in particular the Velo–Zwanziger causality problem which turns out to be related in an interesting way to the solution of zero mass Weinberg–Witten issue. In contrast to the indefinite metric and ghosts of gauge theory, SLFT uses only positivity-respecting physical degrees of freedom. The result is a fully Lorentz-covariant and causal string field theory in which light- or space-like linear strings transform covariant under Lorentz transformation. The cooperation of causality and quantum positivity in the presence of interacting particles leads to remarkable conceptual changes. It turns out that the presence of H-selfinteractions in the Higgs model is not the result of SSB on a postulated Mexican hat potential, but solely the consequence of the implementation of positivity and causality. These principles (and not the imposed gauge symmetry) account also for the Lie-algebra structure of the leading contributions of selfinteracting vector mesons. Second order consistency of selfinteracting vector mesons in SLFT requires the presence of H-particles; this, and not SSB, is the raison d'être for H. The basic conceptual and calculational tool of SLFT is the S-matrix. Its string-independence is a powerful restriction which determines the form of interaction densities in terms of the model-defining particle content and plays a fundamental role in the construction of pl observables and sl interpolating fields

Predictive success, partial truth and skeptical realism

Realists argue that mature theories enjoying predictive success are approximately and partially true, and that the parts of the theory necessary to this success are retained through theory-change and worthy of belief. I examine the paradigmatic case of the novel prediction of a white spot in the shadow of a circular object, drawn from Fresnel's wave theory of light by Poisson in 1819. It reveals two problems in this defence of realism: predictive success needs theoretical idealizations and fictions on the one hand, and may be obtained by using different parts of the same theory on the other hand. I maintain that these two problems are not limited to the case of the white spot, but common features of predictive success. It shows that the no-miracle argument by itself cannot prove more than a \textit{skeptical realism}, the claim that we cannot know which parts of theories are true. I conclude by examining if Hacking's manipulability arguments can be of any help to go beyond this position

Quantum Supermaps are Characterized by Locality

We provide a new characterisation of quantum supermaps in terms of an axiom that refers only to sequential and parallel composition. Consequently, we generalize quantum supermaps to arbitrary monoidal categories and operational probabilistic theories. We do so by providing a simple definition of locally-applicable transformation on a monoidal category. The definition can be rephrased in the language of category theory using the principle of naturality, and can be given an intuitive diagrammatic representation in terms of which all proofs are presented. In our main technical contribution, we use this diagrammatic representation to show that locally-applicable transformations on quantum channels are in one-to-one correspondence with deterministic quantum supermaps. This alternative characterization of quantum supermaps is proven to work for more general multiple-input supermaps such as the quantum switch and on arbitrary normal convex spaces of quantum channels such as those defined by satisfaction of signaling constraints

Predictive success, partial truth and skeptical realism

Realists argue that mature theories enjoying predictive success are approximately and partially true, and that the parts of the theory necessary to this success are retained through theory-change and worthy of belief. I examine the paradigmatic case of the novel prediction of a white spot in the shadow of a circular object, drawn from Fresnel's wave theory of light by Poisson in 1819. It reveals two problems in this defence of realism: predictive success needs theoretical idealizations and fictions on the one hand, and may be obtained by using different parts of the same theory on the other hand. I maintain that these two problems are not limited to the case of the white spot, but common features of predictive success. It shows that the no-miracle argument by itself cannot prove more than a \textit{skeptical realism}, the claim that we cannot know which parts of theories are true. I conclude by examining if Hacking's manipulability arguments can be of any help to go beyond this position

Free Polycategories for Unitary Supermaps of Arbitrary Dimension

We provide a construction for holes into which morphisms of abstract symmetric monoidal categories can be inserted, termed the polyslot construction pslot[C], and identify a sub-class srep[C] of polyslots that are single-party representable. These constructions strengthen a previously introduced notion of locally-applicable transformation used to characterize quantum supermaps in a way that is sufficient to re-construct unitary supermaps directly from the monoidal structure of the category of unitaries. Both constructions furthermore freely reconstruct the enriched polycategorical semantics for quantum supermaps which allows to compose supermaps in sequence and in parallel whilst forbidding the creation of time-loops. By freely constructing key compositional features of supermaps, and characterizing supermaps in the finite-dimensional case, polyslots are proposed as a suitable generalization of unitary-supermaps to infinite dimensions and are shown to include canonical examples such as the quantum switch. Beyond specific applications to quantum-relevant categories, a general class of categorical structures termed path-contraction groupoids are defined on which the srep[C] and pslot[C] constructions are shown to coincide