If a unitary transformation has a circuit representation with no directed path from input a to output b , then a does not influence b through the overall unitary. Conversely, if a unitary satisfies a number of no-influence conditions, it is natural to wonder whether a circuit decomposition exists in which all of them are represented by absences of paths. Such decompositions are known as causal decompositions; determining their existence in general is a central open problem in the study of causal structure in quantum theory. We present progress towards a general solution by considering the special case of unitary causal decompositions, i.e. decompositions in terms of unitary circuits in the traditional quantum circuit formalism that do not require the generalisation to 'extended' or 'routed' quantum circuits prompted by earlier research on this topic. We identify a combinatorial condition that characterises precisely those sets of no-influence constraints G for which any unitary transformation satisfying G admits a unitary causal decomposition representing the constraints. Our approach is systematic, grounded in lattice theory and finite-dimensional operator algebra, and offers hope for extensions to more general (e.g. routed unitary) causal decompositions in the future.
If a unitary transformation has a circuit representation with no directed path from input a to output b , then a does not influence b through the overall unitary. Conversely, if a unitary satisfies a number of no-influence conditions, it is natural to wonder whether a circuit decomposition exists in which all of them are represented by absences of paths. Such decompositions are known as causal decompositions; determining their existence in general is a central open problem in the study of causal structure in quantum theory. We present progress towards a general solution by considering the special case of unitary causal decompositions, i.e. decompositions in terms of unitary circuits in the traditional quantum circuit formalism that do not require the generalisation to 'extended' or 'routed' quantum circuits prompted by earlier research on this topic. We identify a combinatorial condition that characterises precisely those sets of no-influence constraints G for which any unitary transformation satisfying G admits a unitary causal decomposition representing the constraints. Our approach is systematic, grounded in lattice theory and finite-dimensional operator algebra, and offers hope for extensions to more general (e.g. routed unitary) causal decompositions in the future.
The causal structure of a unitary transformation is the set of relations of possible influence between any input subsystem and any output subsystem. We study whether such causal structure can be understood in terms of compositional structure of the unitary. Given a quantum circuit with no path from input system A to output system B , system A cannot influence system B . Conversely, given a unitary U with a no-influence relation from input A to output B , it follows from [B. Schumacher and M. D. Westmoreland, Quantum Information Processing 4 no. 1, (Feb, 2005)] that there exists a circuit decomposition of U with no path from A to B . However, as we argue, there are unitaries for which there does not exist a circuit decomposition that makes all causal constraints evident simultaneously . To address this, we introduce a new formalism of `extended circuit diagrams', which goes beyond what is expressible with quantum circuits, with the core new feature being the ability to represent direct sum structures in addition to sequential and tensor product composition. A causally faithful extended circuit decomposition, representing a unitary U , is then one for which there is a path from an input A to an output B if and only if there actually is influence from A to B in U . We derive causally faithful extended circuit decompositions for a large class of unitaries, where in each case, the decomposition is implied by the unitary's respective causal structure. We hypothesize that every finite-dimensional unitary transformation has a causally faithful extended circuit decomposition.
Quantum theory challenges classical causal explanations for physical phenomena, calling for a quantum generalisation of causal structure. This thesis advances the study of causal decompositions, a research programme that aims to bridge two prominent, yet conceptually distinct types of structural assumption found across the quantum causality literature. The first is in terms of the global dependency relations between input and output systems of quantum transformations, which in the unitary case we shall refer to as causal structure; the second is in terms of the compositional structure of quantum circuits or networks. Concretely, a causal decomposition is a circuit representation of a unitary transformation in which no-influence relations that hold between its inputs and outputs are made manifest in the absence of paths through the circuit connecting them. Applications include translating between different quantum causal modelling frameworks; equipping a framework based on unitary causal structure with a local, bottom-up description via generalised autonomous causal mechanisms; and relating notions of compatibility between quantum transformations and relativistic causality. Whether causal decompositions always exist remains an open problem. The contributions of this thesis are threefold. First, we develop an order-theoretic account of circuit syntax and demonstrate that every circuit with a given input-output connectivity can be rewritten into a one of a canonical form given by the concept lattice construction from lattice theory. This systematises the search for causal decompositions by providing the shape of the quantum circuit one should look for. Second, we establish a combinatorial condition that fully characterises when causal decompositions into unitary quantum circuits exist without requiring the generalisation to routed quantum circuits prompted by earlier research. This solves a well-defined subproblem of the broader quest, and our proof, an interplay of order theory and operator algebra, lays a foundation for future extensions. Third, we detail the application to relativistic causality. Within a multipartite framework generalising existing literature, we emphasise that the mere lack of superluminal signalling through a quantum channel is not sufficient to conclude its realisability by dynamics that are locally consistent with relativity theory. We conjecture that, on the other hand, a lack of superluminal causal influences through its unitary dilations is sufficient—at least up to arbitrarily small errors. We demonstrate this in various special cases and show that the general statement reduces to the existence of a weak version of causal decompositions.
We take an order-theoretic approach to circuit (string diagram) syntax, treating a circuit as a partial order with additional input-output structure. We define morphisms between circuits and prove a factorisation theorem showing that these can, in the finite case, be regarded as formalising a notion of syntactical circuit rewrites, with quotient maps in particular corresponding to gate composition. We then consider the connectivity of a circuit, expressed as a binary relation between its inputs and outputs, and characterise the concept lattice from formal concept analysis as the unique smallest circuit that admits morphisms from all other circuits with the same connectivity. This has significance for quantum causality, particularly to the study of causal decompositions of unitary transformations. We close by constructing the circuit characterised by the dual statement.
The concept of causal (non)separability describes whether the causal order between parties that perform local quantum operations is well-defined or indefinite. Causal (non)separability in the general multipartite setting was introduced and studied in [Oreshkov and Giarmatzi, New J. Phys. 18, 093020 (2016); Wechs, Abbott, and Branciard, New J. Phys. 21, 013027 (2019)]. We resolve an open problem from these earlier works by showing -- using a novel "coherent teleportation technique" -- that a sufficient condition for causal separability identified in [Wechs, Abbott, and Branciard, New J. Phys. 21, 013027 (2019)] is also necessary, and thus provides a complete characterisation of multipartite causal separability. A consequence of this result is that all causally separable processes admit a realisation as generalised quantum circuits in which the order between the operations is classically controlled, known as "quantum circuits with classical control of causal order".
We argue that the quantum-theoretical structures studied in several recent lines of research cannot be adequately described within the standard framework of quantum circuits. This is in particular the case whenever the combination of subsystems is described by a nontrivial blend of direct sums and tensor products of Hilbert spaces. We therefore propose an extension to the framework of quantum circuits, given by routed linear maps and routed quantum circuits . We prove that this new framework allows for a consistent and intuitive diagrammatic representation in terms of circuit diagrams, applicable to both pure and mixed quantum theory, and exemplify its use in several situations, including the superposition of quantum channels and the causal decompositions of unitaries. We show that our framework encompasses the `extended circuit diagrams' of Lorenz and Barrett [arXiv:2001.07774 (2020)], which we derive as a special case, endowing them with a sound semantics.
It is well-known that every n-qubit unitary can be decomposed into a product of controlled single-qubit unitaries. We show that the control conditions in such a decomposition naturally define a quasi-process function, an object that originally came up in the study of indefinite causal order. Here quasi-process functions yield a representation of every n-qubit unitary in a canonical form, namely, as a quasi-process function unitary (qPFU). We use this representation to show how quantum nonlocality without entanglement (QNLWE) can be understood in terms of unitaries that encode the trade-off between causal order and locality noted in R. Kunjwal and Ä. Baumeler, arXiv:2202.00440. Specifically, we obtain two sufficient (but not necessary) criteria---mutual exclusivity of control conditions (MECC) and Unambiguity---for a unitary to be a process function unitary (PFU), i.e., it admits a qPFU representation where the quasi-process function is a process function, an object that can model paradox-free classical causal loops. We also show the inequivalence of these criteria, neither implying the other. We then show that the circuit depth of MECC unitaries admits a characterization in terms of clique-partitioning problems on Hamming graphs and use this correspondence to provide bounds on the circuit depth of n-qubit MECC unitaries. Finally, we show how MECC unitaries provide a recipe for constructing a unitary purification of the associated process function, thereby providing a large class of unitary processes of interest in quantum causality. The generality of this causality-inspired representation of unitaries makes it a versatile tool for analyzing the structure of quantum circuits in other domains.