What does it mean for a causal structure to be `unknown'? Can we even talk about `repetitions' of an experiment without prior knowledge of causal relations? And under what conditions can we say that a set of processes with arbitrary, possibly indefinite, causal structure are independent and identically distributed? Similar questions for classical probabilities, quantum states, and quantum channels are beautifully answered by so-called "de Finetti theorems", which connect a simple and easy-to-justify condition – symmetry under exchange – with a very particular multipartite structure: a mixture of identical states/channels. Here we extend the result to processes with arbitrary causal structure, including indefinite causal order and multi-time, non-Markovian processes applicable to noisy quantum devices. The result also implies a new class of de Finetti theorems for quantum states subject to a large class of linear constraints, which can be of independent interest.
What does it mean for a causal structure to be unknown'? Can we even talk about repetitions' of an experiment without prior knowledge of causal relations? And under what conditions can we say that a set of processes with arbitrary, possibly indefinite, causal structure are independent and identically distributed? Similar questions for classical probabilities, quantum states, and quantum channels are beautifully answered by so-called "de Finetti theorems", which connect a simple and easy-to-justify condition – symmetry under exchange – with a very particular multipartite structure: a mixture of identical states/channels. Here we extend the result to processes with arbitrary causal structure, including indefinite causal order and multi-time, non-Markovian processes applicable to noisy quantum devices. The result also implies a new class of de Finetti theorems for quantum states subject to a large class of linear constraints, which can be of independent interest.
In recent years, the investigation of causal relations in quantum theory has attracted a lot of interest. In particular, it has been found that it is possible to conceive of scenarios where some parties perform operations that are compatible with quantum theory locally, but that cannot be embedded into a global background causal structure. Such indefinite causal structures are of interest from a fundamental point of view, but also from the perspective of quantum information processing, since they do not fit into the usual paradigm of quantum circuits, which assumes a definite causal order. The main aim of this thesis is to study indefinite quantum causal relations involving more than two parties. Compared to the bipartite case, there are many new aspects and complications that arise in multipartite situations, which need to be clarified in order to fundamentally understand quantum causal structures, and to shed light on their implications and potential usefulness for quantum information processing.A suitable mathematical framework for the investigation of quantum causal relations is the process matrix formalism. In this framework, the notion of causal nonseparability was introduced in order to qualify scenarios that are incompatible with a definite causal order. In Chapter 2, we study how to generalise this concept to the multipartite case, how to characterise multipartite causally (non)separable quantum processes, and how to certify multipartite causal nonseparability. Another important topic is to determine which quantum scenarios with indefinite causal order are physically implementable, and how they can be realised concretely. In Chapter 3, we introduce two new classes of physically realisable multipartite quantum processes, and characterise them in terms of their process matrix descriptions. In particular, we define the class of quantum circuits with quantum control of causal order. The simplest example of such a circuit is the quantum switch, a protocol in which the order between two operations is controlled by a qubit in a superposition state, and which defines a causally nonseparable process. The class we introduce also contains more general examples of causally nonseparable processes with new features. We then show how the process matrix characterisation of this class allows us to search for new quantum information processing applications of causal nonseparability. In Chapter 4, we investigate a particular quantum communication effect in a scenario involving coherent control between two quantum channels. This leads us to a more general analysis of the notion of a coherently controlled channel, which involves certain subtilities. In Chapter 5, we turn to another problem that arises in multipartite scenarios, namely whether a given phenomenon is genuinely multipartite or not. More particularly, we study genuinely multipartite (non)causal correlations. In Chapter 6, we show that anomalous weak values are possible without post-selection. In Chapter 7, we show that certain tripartite process matrices, namely those that are unitarily extensible, have a realisation on so-called time-delocalised subsystems, i.e., quantum subsystems that are not associated with a definite time. The class of unitarily extensible tripartite process matrices is larger than the class of quantum circuits with quantum control, and in particular contains processes that violate so-called causal inequalities.
The idea that events are equipped with a partial causal order is central to our understanding of physics in the tested regimes: given two pointlike events A and B , either A is in the causal past of B , B is in the causal past of A , or A and B are space-like separated. Operationally, the meaning of these order relations corresponds to constraints on the possible correlations between experiments performed in the vicinities of the respective events: if A is in the causal past of B , an experimenter at A could signal to an experimenter at B but not the other way around, while if A and B are space-like separated, no signaling is possible in either direction. In the context of a concrete physical theory, the correlations compatible with a given causal configuration may obey further constraints. For instance, space-like correlations in quantum mechanics arise from local measurements on joint quantum states, while time-like correlations are established via quantum channels. Similarly to other variables, however, the causal order of a set of events could be random, and little is understood about the constraints that causality implies in this case. A main difficulty concerns the fact that the order of events can now generally depend on the operations performed at the locations of these events, since, for instance, an operation at A could influence the order in which B and C occur in A ’s future. So far, no formal theory of causality compatible with such dynamical causal order has been developed. Apart from being of fundamental interest in the context of inferring causal relations, such a theory is imperative for understanding recent suggestions that the causal order of events in quantum mechanics can be indefinite. Here, we develop such a theory in the general multipartite case. Starting from a background-independent definition of causality, we derive an iteratively formulated canonical decomposition of multipartite causal correlations. For a fixed number of settings and outcomes for each party, these correlations form a polytope whose facets define causal inequalities. The case of quantum correlations in this paradigm is captured by the process matrix formalism. We investigate the link between causality and the closely related notion of causal separability of quantum processes, which we here define rigorously in analogy with the link between Bell locality and separability of quantum states. We show that causality and causal separability are not equivalent in general by giving an example of a physically admissible tripartite quantum process that is causal but not causally separable. We also show that there are causally separable quantum processes that become non-causal if extended by supplying the parties with entangled ancillas. This motivates the concepts of extensibly causal and extensibly causally separable (ECS) processes, for which the respective property remains invariant under extension. We characterize the class of ECS quantum processes in the tripartite case via simple conditions on the form of the process matrix. We show that the processes realizable by classically controlled quantum circuits are ECS and conjecture that the reverse also holds.
Recently, the possible existence of quantum processes with indefinite causal order has been extensively discussed, in particular using the formalism of process matrices. Here we give a new perspective on this question, by establishing a direct connection to the theory of multi-time quantum states. Specifically, we show that process matrices are equivalent to a particular class of pre- and post-selected quantum states. This offers a new conceptual point of view to the nature of process matrices. Our results also provide an explicit recipe to experimentally implement any process matrix in a probabilistic way, and allow us to generalize some of the previously known properties of process matrices. Finally we raise the issue of the difference between the notions of indefinite temporal order and indefinite causal order, and show that one can have indefinite causal order even with definite temporal order.
Many fundamental and key objects in quantum mechanics are linear mappings between particular affine/linear spaces. This structure includes basic quantum elements such as states, measurements, channels, instruments, non-signalling channels and channels with memory, and also higher-order operations such as superchannels, quantum combs, n-time processes, testers, and process matrices which may not respect a definite causal order. Deducing and characterising their structural properties in terms of linear and semidefinite constraints is not only of foundational relevance, but plays an important role in enabling the numerical optimisation over sets of quantum objects and allowing simpler connections between different concepts and objects. Here, we provide a general framework to deduce these properties in a direct and easy to use way. While primarily guided by practical quantum mechanical considerations, we also extend our analysis to mappings between general linear/affine spaces and derive their properties, opening the possibility for analysing sets which are not explicitly forbidden by quantum theory, but are still not much explored. Together, these results yield versatile and readily applicable tools for all tasks that require the characterisation of linear transformations, in quantum mechanics and beyond. As an application of our methods, we discuss how the existence of indefinite causality naturally emerges in higher-order quantum transformations and provide a simple strategy for the characterisation of mappings that have to preserve properties in a 'complete' sense, i.e., when acting non-trivially only on parts of an input space.
Quantum mechanics has achieved unparalleled success as an operational theory, describing a wide range of experiments to remarkable accuracy. However, the physical foundations on which it rests remain as puzzling as they were a century ago, and a concise statement of the physical principles that underlie quantum mechanics is still outstanding. One promising approach holds that these principles should be formulated in terms of information, since many of the counter-intuitive effects in quantum theory concern questions such as what one can know about a quantum system and how the information encoded therein can be processed and distributed between parties. Another challenging feature of quantum theory is its incompatibility with the other cornerstone of modern physics, general relativity. In order to reconcile the two, one must identify and retain only the essential concepts and principles of each theory, and in the case of general relativity, causality has been identified as such a concept. This raises the question of how our classical understanding of causality must change when quantum theory is taken into account. In order to address these questions about information, knowledge and causality, I turn to the framework of causal models. In classical statistics, causal models explain the relations among a set of variables in terms of causal influences, which makes them a powerful tool for structuring our knowledge about complex systems and developing strategies for interacting with them. More importantly, the framework provides the conceptual underpinnings and mathematical methods for addressing questions about causation, information and knowledge in a rigorous manner. In this thesis, I develop a version of the classical causal models framework that is compatible with quantum theory and explore its physical implications. \n \n \nA first step is to define quantum versions of fundamental elements such as variables, conditionals and belief propagation rules. This allows one to consider the question of what one can come to know about a quantum variable from the point of view of causal modelling. The conditionals relating quantum variables are found to have a richer structure than their classical counterparts, which can be exploited for the task of discerning causal relations given limited data -- a central problem in classical causal modelling. The mathematical properties of quantum conditionals also establish a correspondence between various classes of two-party correlations, such as bound and distillable entanglement, and the types of causal structures that can give rise to them, which may become a useful tool for entanglement theory and quantum information processing. In the context of open quantum systems dynamics, quantum causal models provide a clear physical explanation of not completely positive maps and, more broadly, of non-Markovian quantum dynamics. Finally, I consider the possibility of non-classical effects in the way that different causal mechanisms are combined -- that is, non-classical causal structures. For the simple case of two causally ordered variables, I propose indicators that witness different classes of combinations of causal mechanisms, including a non-classical mixture, and describe an experiment realizing examples of the different classes. \n \n \nAs these results illustrate, the framework of quantum causal models provides both greater conceptual clarity and a comprehensive mathematical formalism for studying a diverse set of problems, ranging from foundational questions to applications in open systems dynamics and the exploration of non-classical causal structures, which are likely to feature in a future theory of quantum gravity. The material presented in this thesis is intended as a foundation and inspiration for further applications of quantum causal models.
One of the key ways in which quantum mechanics differs from relativity is that it requires a fixed background reference frame for spacetime. In fact, this appears to be one of the main conceptual obstacles to uniting the two theories. Additionally, a combination of the two theories is expected to yield non-classical, or "indefinite", causal structures. In this paper, we present a background-independent formulation of the process matrix formalism – a form of quantum mechanics that allows for indefinite causal structure – while retaining operationally well-defined measurement statistics. We do this by postulating an arbitrary probability distribution of measurement outcomes across discrete "chunks" of spacetime, which we think of as physical laboratories, and then requiring that this distribution be invariant under any permutation of laboratories. We find (a) that one still obtains nontrivial, indefinite causal structures with background independence, (b) that we lose the idea of local operations in distinct laboratories, but can recover it by encoding a reference frame into the physical states of our system, and (c) that permutation invariance imposes surprising symmetry constraints that, although formally similar to a superselection rule, cannot be interpreted as such.