# Publications

**Authors: **Camilo Miguel Signorelli, Selma Dündar-Coecke, Vincent Wang and Bob Coecke

**Year:** 2020

In physics, the analysis of the space representing states of physical systems often takes the form of a layer-cake of increasingly rich structure. In this paper, we propose an analogous hierarchy in the cognition of spacetime. Firstly, we explore the interplay between the objective physical properties of space-time and the subjective compositional modes of relational representations within the reasoner. Secondly, we discuss the compositional structure within and between layers. The existing evidence in the available literature is reviewed to end with some testable consequences of our proposal at the brain and behavioral level.

**Authors: **Stefano Gogioso, Maria E. Stasinou and Bob Coecke

**Year:** 2020

We present a compositional algebraic framework to describe the evolution of quantum fields in discretised spacetimes. We show how familiar notions from Relativity and quantum causality can be recovered in a purely order-theoretic way from the causal order of events in spacetime, with no direct mention of analysis or topology. We formulate theory-independent notions of fields over causal orders in a compositional, functorial way. We draw a strong connection to Algebraic Quantum Field Theory (AQFT), using a sheaf-theoretical approach in our definition of spaces of states over regions of spacetime. We introduce notions of symmetry and cellular automata, which we show to subsume existing definitions of Quantum Cellular Automata (QCA) from previous literature. Given the extreme flexibility of our constructions, we propose that our framework be used as the starting point for new developments in AQFT, QCA and more generally Quantum Field Theory.

**Authors: **Nicola Pinzani and Stefano Gogioso

**Year:** 2020

In this work, we give rigorous operational meaning to superposition of causal orders. This fits within a recent effort to understand how the standard operational perspective on quantum theory could be extended to include indefinite causality. The mainstream view, that of “process matrices”, takes a top-down approach to the problem, considering all causal correlations that are compatible with local quantum experiments. Conversely, we pursue a bottom-up approach, investigating how the concept of indefiniteness emerges from specific characteristics of generic operational theories. Specifically, we pin down the operational phenomenology of the notion of non-classical (e.g. “coherent”) control, which we then use to formalise a theory-independent notion of control (e.g. “superposition”) of causal orders. To validate our framework, we show how salient examples from the literature can be captured in our framework.

**Authors: **Cole Comfort and Aleks Kissinger

**Year:** 2021

Symplectic vector spaces are the phase space of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution — and more generally linear constraints on the evolution — of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled’ category of linear relations. More precisely, we show that it arises as a variation of Selinger’s CPM construction applied to linear relations, where the covariant orthogonal complement functor plays of the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens’ toy theory, and odd-prime-dimensional stabilizer quantum circuits.

**Authors: **Cole Comfort, Aleks Kissinger

**Year:** 2021

Symplectic vector spaces are the phase space of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution — and more generally linear constraints on the evolution — of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled’ category of linear relations. More precisely, we show that it arises as a variation of Selinger’s CPM construction applied to linear relations, where the covariant orthogonal complement functor plays of the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens’ toy theory, and odd-prime-dimensional stabilizer quantum circuits.

**Authors: **John H. Selby, Carlo Maria Scandolo, Bob Coecke

**Year:** 2021

We present a reconstruction of finite-dimensional quantum theory where all of the postulates are stated in diagrammatic terms, making them intuitive. Equivalently, they are stated in category-theoretic terms, making them mathematically appealing. Again equivalently, they are stated in process-theoretic terms, establishing that the conceptual backbone of quantum theory concerns the manner in which systems and processes compose. Aside from the diagrammatic form, the key novel aspect of this reconstruction is the introduction of a new postulate, symmetric purification. Unlike the ordinary purification postulate, symmetric purification applies equally well to classical theory as well as quantum theory. Therefore we first reconstruct the full process theoretic description of quantum theory, consisting of composite classical-quantum systems and their interactions, before restricting ourselves to just the `fully quantum’ systems as the final step. We propose two novel alternative manners of doing so, `no-leaking’ (roughly that information gain causes disturbance) and `purity of cups’ (roughly the existence of entangled states). Interestingly, these turn out to be equivalent in any process theory with cups & caps. Additionally, we show how the standard purification postulate can be seen as an immediate consequence of the symmetric purification postulate and purity of cups. Other tangential results concern the specific frameworks of generalised probabilistic theories (GPTs) and process theories (a.k.a. CQM). Firstly, we provide a diagrammatic presentation of GPTs, which, henceforth, can be subsumed under process theories. Secondly, we argue that the `sharp dagger’ is indeed the right choice of a dagger structure as this sharpness is vital to the reconstruction.

**Authors: ** Nick Ormrod, Augustin Vanrietvelde and Jonathan Barrett

**Year:** 2022

Existing work on quantum causal structure assumes that one can perform arbitrary operations on the systems of interest. But this condition is often not met. Here, we extend the framework for quantum causal modelling to cases where a system can suffer \textit{sectorial contraints}, that is, restrictions on the orthogonal subspaces of its Hilbert space that may be mapped to one another. Our framework (a) proves that a number of different intuitions about causal relations turn out to be equivalent; (b) shows that quantum causal structures in the presence of sectorial constraints can be represented with a directed graph; and (c) defines a fine-graining of the causal structure in which the individual sectors of a system bear causal relations, which provides a more detailed analysis than its coarse-grained counterpart. As an example, we apply our framework to purported photonic implementations of the quantum switch to show that while their coarse-grained causal structure is cyclic, their fine-grained causal structure is acyclic. We therefore conclude that these experiments realize indefinite causal order only in a weak sense. Notably, this is the first argument to this effect that is not rooted in the assumption that the causal relata must be localized in spacetime.

**Authors: ** Augustin Vanrietvelde, Nick Ormrod, Hlér Kristjánsson and Jonathan Barrett

**Year:** 2022

Over the past decade, a number of quantum processes have been proposed which are logically consistent, yet feature a cyclic causal structure. However, there exists no general formal method to construct a process with an exotic causal structure in a way that ensures, and makes clear why, it is consistent. Here we provide such a method, given by an exended circuit formalism. This only requires directed graphs endowed with boolean matrices, which encode basic constraints on operations. Our framework (a) defines a set of elementary rules for checking the validity of any such graph, (b) provides a way of constructing consistent processes as a circuit from valid graphs, and (c) yields an intuitive interpretation of the causal relations within a process and an explanation of why they do not lead to inconsistencies. We display how several standard examples of exotic processes, including ones that violate causal inequalities, are among the class of processes that can be generated in this way; we conjecture that this class in fact includes all unitarily extendible processes.

**Authors: ** Will Simmons and Aleks Kissinger

**Year:** 2022

The Caus[-] construction takes a compact closed category of basic processes and yields a *-autonomous category of higher-order processes obeying certain signalling/causality constraints, as dictated by the type system in the resulting category. This paper looks at instances where the base category C satisfies additional properties yielding an affine-linear structure on Caus[C] and a substantially richer internal logic. While the original construction only gave multiplicative linear logic, here we additionally obtain additives and a non-commutative, self-dual sequential product yielding a model of Guglielmi’s BV logic. Furthermore, we obtain a natural interpretation for the sequential product as “A can signal to B, but not vice-versa”, which sits as expected between the non-signalling tensor and the fully-signalling (i.e. unconstrained) par. Fixing matrices of positive numbers for C recovers the BV category structure of probabilistic coherence spaces identified by Blute, Panangaden, and Slavnov, restricted to normalised maps. On the other hand, fixing the category of completely positive maps gives an entirely new model of BV consisting of higher order quantum channels, encompassing recent work in the study of quantum and indefinite causal structures.

**Authors: ** Isaac Friend and Aleks Kissinger

**Year:** 2022

Causal identification is a type of causal inference problem concerned with recovering from observational data and qualitative assumptions the causal relationships generating the data, and hence the effects of hypothetical interventions. Though the topic is typically considered in the context of classical statistical models, recent years have seen great interest in extending causal inference techniques to quantum and generalised theories. A major obstacle to a theory of causal identification in the quantum setting is the question of what should play the role of “observational data,” as any means of extracting data at a certain locus will almost certainly disturb the system. Hence, one might think a priori that quantum measurements are already too much like interventions, so that the problem of causal identification trivialises. This is not the case. Fixing a limited class of quantum instruments (namely the class of all projective measurements) to play the role of “observations,” we note that as in the classical setting, there exist scenarios for which causal identification is not possible. We then present sufficient conditions for quantum causal identification, starting with an example of a quantum analogue of the well-known “front-door criterion” and finishing with a broader class of scenarios for which the effect of a single intervention is identifiable. These results arise from generalising the process-theoretic account of classical causal inference given by Jacobs, Kissinger, and Zanasi beyond the setting of Markov categories, and thereby treating the classical and quantum problems uniformly

**Published in**

*The Proceedings of the 19th International Conference on Quantum Physics and Logic (QPL 2022).*❱
**Authors: ** James Hefford and Aleks Kissinger

**Year:** 2022

The notion of a joint system, as captured by the monoidal (a.k.a. tensor) product, is fundamental to the compositional, process-theoretic approach to physical theories. Promonoidal categories generalise monoidal categories by replacing the functors normally used to form joint systems with profunctors. Intuitively, this allows the formation of joint systems which may not always give a system again, but instead a generalised system given by a presheaf. This extra freedom gives a new, richer notion of joint systems that can be applied to categorical formulations of spacetime. Whereas previous formulations have relied on partial monoidal structure that is only defined on pairs of independent (i.e. spacelike separated) systems, here we give a concrete formulation of spacetime where the notion of a joint system is defined for any pair of systems as a presheaf. The representable presheaves correspond precisely to those actual systems that arise from combining spacelike systems, whereas more general presheaves correspond to virtual systems which inherit some of the logical/compositional properties of their “actual” counterparts. We show that there are two ways of doing this, corresponding roughly to relativistic versions of conjunction and disjunction. The former endows the category of spacetime slices in a Lorentzian manifold with a promonoidal structure, whereas the latter augments this structure with an (even more) generalised way to combine systems that fails the interchange law.