Natural deduction and coherence for weakly distributive categories

AbstractThis paper examines coherence for certain monoidal categories using techniques coming from the proof theory of linear logic, in particular making heavy use of the graphical techniques of proof nets. We define a two sided notion of proof net, suitable for categories like weakly distributive categories which have the two-tensor structure (times/par) of linear logic, but lack a negation operator. Representing morphisms in weakly distributive categories as such nets, we derive a coherence theorem for such categories. As part of this process, we develop a theory of expansion-reduction systems with equalities and a term calculus for proof nets, each of which is of independent interest. In the symmetric case the expansion-reduction system on the term calculus yields a decision procedure for the equality of maps for free weakly distributive categories.The main results of this paper are these. First we have proved coherence for the full theory of weakly distributive categories, extending similar results for monoidal categories to include the treatment of the tensor units. Second, we extend these coherence results to the full theory of ∗-autonomous categories — providing a decision procedure for the maps of free symmetric ∗-autonomous categories. Third, we derive a conservative extension result for the passage from weakly distributive categories to ∗-autonomous categories. We show strong categorical conservativity, in the sense that the unit of the adjunction between weakly distributive and ∗-autonomous categories is fully faithful

Quantum field theory on a growing lattice

We construct the classical and canonically quantized theories of a massless scalar field on a background lattice in which the number of points--and hence the number of modes--may grow in time. To obtain a well-defined theory certain restrictions must be imposed on the lattice. Growth-induced particle creation is studied in a two-dimensional example. The results suggest that local mode birth of this sort injects too much energy into the vacuum to be a viable model of cosmological mode birth.Comment: 28 pages, 2 figures; v.2: added comments on defining energy, and reference

Quantum Speedup and Categorical Distributivity

This paper studies one of the best known quantum algorithms - Shor's factorisation algorithm - via categorical distributivity. A key aim of the paper is to provide a minimal set of categorical requirements for key parts of the algorithm, in order to establish the most general setting in which the required operations may be performed efficiently. We demonstrate that Laplaza's theory of coherence for distributivity provides a purely categorical proof of the operational equivalence of two quantum circuits, with the notable property that one is exponentially more efficient than the other. This equivalence also exists in a wide range of categories. When applied to the category of finite dimensional Hilbert spaces, we recover the usual efficient implementation of the quantum oracles at the heart of both Shor's algorithm and quantum period-finding generally; however, it is also applicable in a much wider range of settings.Comment: 17 pages, 11 Figure

Constructing Differential Categories and Deconstructing Categories of Games

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A Logical Basis for Quantum Evolution and Entanglement

International audienceWe reconsider discrete quantum causal dynamics where quan-tum systems are viewed as discrete structures, namely directed acyclic graphs. In such a graph, events are considered as vertices and edges de-pict propagation between events. Evolution is described as happening between a special family of spacelike slices, which were referred to as locative slices. Such slices are not so large as to result in acausal influ-ences, but large enough to capture nonlocal correlations. In our logical interpretation, edges are assigned logical formulas in a spe-cial logical system, called BV, an instance of a deep inference system. We demonstrate that BV, with its mix of commutative and noncommutative connectives, is precisely the right logic for such analysis. We show that the commutative tensor encodes (possible) entanglement, and the non-commutative seq encodes causal precedence. With this interpretation, the locative slices are precisely the derivable strings of formulas. Several new technical results about BV are developed as part of this analysis