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.
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.
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.
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 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.
One can theoretically conceive of processes in which quantum operations are composed cyclically in a way that is incompatible with a well-defined causal order. Some of these processes can be realized within standard quantum temporal evolutions on systems that are delocalized in time. In this Letter, we introduce the "circuit (super)operator," a tool that allows one to express how a (possibly cyclic) quantum circuit transforms under arbitrary changes of subsystem decompositions. This framework concisely captures the link between causally indefinite processes and their temporal realizations. We apply it to the quantum switch, a paradigmatic example of a causally indefinite process, and consider two distinct temporal realizations. These correspond to the "causal perspectives" or "causal reference frames" of the two parties involved, describing the process from their respective viewpoints. Surprisingly, we show that these causal perspectives cannot be related by a change of subsystem decomposition, i.e., they cannot be seen as two equivalent descriptions of the same process.
It is known that the classical framework of causal models is not general enough to allow for causal reasoning about quantum systems. While the framework has been generalized in a variety of different ways to the quantum case, much of this work leaves open whether causal concepts are fundamental to quantum theory, or only find application at an emergent level of classical devices and measurement outcomes. Here, we present a framework of quantum causal models, with causal relations defined in terms intrinsic to quantum theory, and the central object of study being the quantum process itself. Following Allen et al., Phys. Rev. X 7, 031021 (2017), the approach defines quantum causal relations in terms of unitary evolution, in a way analogous to an approach to classical causal models that assumes underlying determinism and situates causal relations in functional dependences between variables. We show that any unitary quantum circuit has a causal structure corresponding to a directed acyclic graph, and that when marginalising over local noise sources, the resulting quantum process satisfies a Markov condition with respect to the graph. We also prove a converse to this statement. We introduce an intrinsically quantum notion that plays a role analogous to the conditional independence of classical variables, and (generalizing a central theorem of the classical framework) show that d-separation is sound and complete for it in the quantum case. We present generalizations of the three rules of the classical do-calculus, in each case relating a property of the causal structure to a formal property of the quantum process, and to an operational statement concerning the outcomes of interventions. In addition, we introduce and derive similar results for classical split-node causal models, which are more closely analogous to quantum causal models than the classical causal models that are usually studied.
Quantum circuits form the standard framework of quantum computing, and have started playing the same role for some parts of quantum foundations. However, they cannot satisfactorily capture some possibilities offered by quantum theory, in particular coherent control and indefinite causal order. In this thesis, we therefore propose an extension to this framework, given by routed quantum circuits. Our extension is based on the addition of sectorial constraints, which specify the basic sectorial structure of the scenarios at hand, on top of the factorial structure described by the connectivity of circuits. This yields a consistent and scalable framework, applicable to pure and mixed quantum theory. It admits a sound and intuitive diagrammatic notation. We show that this framework allows to properly model all forms of coherent control of the application of a channel, for instance the superposition of a quantum information carrier's trajectory, or the 'extended circuit diagrams' recently introduced for the study of causal decompositions. We demonstrate how this leads to a redefinition of the task of coherently controlling a channel, and study in detail the necessary resources for this task. We further show that the scenarios featuring indefinite causal order also fit within our framework once feedback loops are introduced. The connectivity and sectorial constraints are then sufficient to specify the core behaviour of a process, and in particular to check that it is valid, i.e. that its cycles do not lead to any logical inconsistency. 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.