Recently, there has been substantial interest in studying the dynamics of quantum theory beyond that of states, in particular, the dynamics of channels, measurements, and higher-order transformations. Castro-Ruiz et al. pursues this using the process-matrix formalism, together with a definition of the possible dynamics of such process matrices, and focusing especially on the question of evolution of causal structures. One of its major conclusions is a strong theorem saying that within the formalism, under continuous and reversible transformations, the causal order between operations must be preserved. Our result here challenges that of Castro-Ruiz et al.: if one is to take into account a full picture of the physical evolution of operations within the standard quantum-mechanical formalism, then the conclusion of Castro-Ruiz et al. does not hold. That is, we show that under certain continuous and reversible dynamics, the causal order between operations is not necessarily preserved. We moreover identify and analyse the root of this apparent contradiction, specifically, that the commonly accepted and widely applied framework of higher-order processes, whilst mathematically sound, is not always appropriate for drawing conclusions on physical dynamics. Finally, we show how to reconcile the elements of the whole picture following the intuition based on entanglement processing by local operations and classical communication.
Recently, there has been substantial interest in studying the dynamics of quantum theory beyond that of states, in particular, the dynamics of channels, measurements, and higher-order transformations. Castro-Ruiz et al. pursues this using the process-matrix formalism, together with a definition of the possible dynamics of such process matrices, and focusing especially on the question of evolution of causal structures. One of its major conclusions is a strong theorem saying that within the formalism, under continuous and reversible transformations, the causal order between operations must be preserved. Our result here challenges that of Castro-Ruiz et al.: if one is to take into account a full picture of the physical evolution of operations within the standard quantum-mechanical formalism, then the conclusion of Castro-Ruiz et al. does not hold. That is, we show that under certain continuous and reversible dynamics, the causal order between operations is not necessarily preserved. We moreover identify and analyse the root of this apparent contradiction, specifically, that the commonly accepted and widely applied framework of higher-order processes, whilst mathematically sound, is not always appropriate for drawing conclusions on physical dynamics. Finally, we show how to reconcile the elements of the whole picture following the intuition based on entanglement processing by local operations and classical communication.
In the last few years, there has been increasing interest in quantum processes with indefinite causal order. Process matrices are a convenient framework to study such processes. Ref. [1] defines higher order transformations from process matrices to process matrices and shows that no continuous and reversible transformation can change the causal order of a process matrix. Ref. [2] argues, based on a set of examples, that there are situations where a process can change its causal order over time. Ref. [2] claims that the formalism of higher order transformations is not general enough to capture its examples. Here we show that this claim is incorrect. Moreover, a crucial example of Ref. [2] has already been explicitly considered in Ref. [1] and shown to be compatible with its results.
The process-matrix formalism describes quantum processes without assuming a fixed global causal order, but the physical meaning of indefinite causal order remains open. We address this question within histories theory, where temporal ordering and dynamical evolution are distinct structures. We show that the decoherence functional admits coherent superpositions of dynamics. In measurement settings, under suitable factorization conditions, these generate process matrices. Thus, we identify process-matrix indefinite causal order as an operationally restricted realization of the more general phenomenon of superposition of dynamics. Histories theory also admits a distinct kinematical notion of indefinite order, in which the ordering of physical events is itself a history observable. The resulting distinction between event order and intervention order clarifies the relation between indefinite causal order, quantum dynamics, and spacetime structure.
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.
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.
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.
Can quantum computation become more powerful when operations do not need to follow a definite causal order? We study this question in the process matrix formalism of Oreshkov et al. (Nature Communications, 2012), which allows quantum operations to be connected without a fixed global order. In the computational model, a quantum circuit succinctly describes a process matrix, and the algorithm is given access to its generated channel, with computational cost measured by the size of the generating circuit.
We give polynomial-time algorithms in two settings. First, general process matrix computation suffices to solve every problem in NP. The valid processes used by our algorithm satisfy the purification postulate of Araújo et al. (Quantum, 2017), and their generated channels are unitary, although their generating circuits are not. Second, requiring the process matrices themselves to be pure, we give a polynomial-time algorithm for unique-sink orientation, a general problem capturing several longstanding computational problems in optimization. Our unitary generators are constructed explicitly and efficiently. Assuming that unique-sink orientation has no quantum polynomial-time algorithm, our result shows that indefinite causal order can provide computational power beyond BQP within the process matrix formalism, even while preserving unitarity.