It was recently realised that quantum theory allows for so-called causally nonseparable processes, which are incompatible with any definite causal order. This was first suggested on a rather abstract level by the formalism of process matrices, an extension of the quantum formalism which only assumes that quantum theory holds locally in some observers' laboratories, but does not impose a global causal structure; it was then shown, on a more practical level, that the quantum switch-a new, already implementable resource for quantum computation that goes beyond causally ordered circuits-provided precisely a physical example of a causally nonseparable process. To demonstrate that a given process is causally nonseparable, the concept of witnesses of causal nonseparability was introduced. Here we present a shorter introduction to this concept, and concentrate on some explicit examples-by considering in particular different noise models for the quantum switch-to show how to construct and use such witnesses in practice.
It was recently realised that quantum theory allows for so-called causally nonseparable processes, which are incompatible with any definite causal order. This was first suggested on a rather abstract level by the formalism of process matrices, an extension of the quantum formalism which only assumes that quantum theory holds locally in some observers' laboratories, but does not impose a global causal structure; it was then shown, on a more practical level, that the quantum switch-a new, already implementable resource for quantum computation that goes beyond causally ordered circuits-provided precisely a physical example of a causally nonseparable process. To demonstrate that a given process is causally nonseparable, the concept of witnesses of causal nonseparability was introduced. Here we present a shorter introduction to this concept, and concentrate on some explicit examples-by considering in particular different noise models for the quantum switch-to show how to construct and use such witnesses in practice.
In quantum mechanics, quantum nonseparability is at the core of philosophical debates regarding its meaning. In the context of the process matrix formalism, causal nonseparability characterises quantum processes (connecting the inputs and outputs of different local quantum operations) that are incompatible with any definite causal structure among interacting parties. One talks about indefinite causal orders. A famous example of causally nonseparable processes is called the quantum switch (QS). It is extensively studied in the literature in virtue of its simple architecture and its various implementations in laboratories.The present work will discuss the possible interpretations of the QS's causal nonseparability under the assumption that causal nonseparability is seen as pointing towards novel objective features of nature.It was first defended that a scientific realist approach towards the process matrix formalism, which is an operational theory generalising quantum mechanics, was as much legitimate as any possible antirealist reading, contrary to certain views found in the literature. The reason is that operational formalisms are ontologically and epistemically neutral.From there, the theoretical concepts of interest, namely causal nonseparability and its model-independent counterpart called noncausality, were analysed in more details, in order to highlight in what sense they are distinct from the standard notions of quantum nonseparability and nonlocality in quantum mechanics.The discussion then focused on noncausality. It was argued that noncausality has a strong connection with a notion of temporal nonlocality. In the same way that Bell nonlocality is given different underlying explanations depending on the details of the chosen quantum mechanics' account, noncausality pointing towards temporal nonlocality can be given a variety of underlying descriptions depending on the exact way to interpret the process matrix formalism.The last chapter focused precisely on this particular point, namely on the various ways to understand process matrices and causal nonseparability in a realist context. In order to explore the potential impact of causal nonseparability on spacetime,we shifted from the notion of (indefinite) causal structure to (indefinite) spatiotemporal ones. This shift was allowed under a set of reasonable assumptions regarding the properties of a physical spacetime manifold and the connection between an operational and relativistic notions of causal relations.While different readings were suggested for indefiniteness of spatiotemporal relations, we insisted in particular on an objective understanding appealing to the concept of metaphysical indeterminacy. It was argued that such an approach could prove useful in a more general theoretical context such as quantum gravity, while being already partly supported in standard quantum mechanics.
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.
Our common understanding of the physical world deeply relies on the notion that events are ordered with respect to some time parameter, with past events serving as causes for future ones. Nonetheless, it was recently found that it is possible to formulate quantum mechanics without any reference to a global time or causal structure. The resulting framework includes new kinds of quantum resources that allow performing tasks—in particular, the violation of causal inequalities —which are impossible for events ordered according to a global causal order. However, no physical implementation of such resources is known. Here we show that a recently demonstrated resource for quantum computation—the quantum switch —is a genuine example of ‘indefinite causal order’. We do this by introducing a new tool—the causal witness —which can detect the causal nonseparability of any quantum resource that is incompatible with a definite causal order. We show however that the quantum switch does not violate any causal inequality.
Causal nonseparability refers to processes where events take place in a coherent superposition of different causal orders.These may be the key resource for experimental violations of causal inequalities and have been recently identified as resources for concrete information-theoretic tasks.Here, we take a step forward by deriving a complete operational framework for causal nonseparability as a resource.Our first contribution is a formal definition for the specific notion of quantum control of causal orders, a stronger form of causal nonseparability-with the celebrated quantum switch as best-known example-where the causal orders of events for a target system are coherently controlled by a control system.We then build a resource theory-for both generic causal nonseparability as well as quantum control of causal orders-with a physically motivated class of free operations, based on process-matrix concatenations.We present the framework explicitly in the mindset with a control register.However, our machinery is totally versatile, being directly applicable also to scenarios with a target register alone.Moreover, an important subclass of our operations is free not only with respect to causal nonseparability and quantum control of causal orders but it also preserves the very causal structure of causal processes.Hence, our treatment contains, as a built-in feature, the basis of a resource theory of quantum causal networks too.As applications, first, we establish a simple sufficient condition for pure-process free convertibility.This imposes a hierarchy of quantum control of causal orders with the quantum switch at the top.Second, we prove that causal-nonseparability distillation exists.More precisely, we show how to convert multiple copies of a process with arbitrarily little causal nonseparability into fewer copies of a quantum switch.Our findings reveal conceptually new, unexpected phenomena, with both fundamental and practical implications.
The concept of causal nonseparability has been recently introduced, in opposition to that of causal separability, to qualify physical processes that locally abide by the laws of quantum theory, but cannot be embedded in a well-defined global causal structure. While the definition is unambiguous in the bipartite case, its generalisation to the multipartite case is not so straightforward. Two seemingly different generalisations have been proposed, one for a restricted tripartite scenario and one for the general multipartite case. Here we compare the two, showing that they are in fact inequivalent. We propose our own definition of causal (non)separability for the general case, which—although a priori subtly different—turns out to be equivalent to the concept of ‘extensible causal (non)separability’ introduced before, and which we argue is a more natural definition for general multipartite scenarios. We then derive necessary, as well as sufficient conditions to characterise causally (non)separable processes in practice. These allow one to devise practical tests, by generalising the tool of witnesses of causal nonseparability.
While the standard formulation of quantum theory assumes a fixed background causal structure, one can relax this assumption within the so-called process matrix framework. Remarkably, some processes, termed causally nonseparable, are incompatible with a definite causal order. We explore a form of certification of causal nonseparability in a semi-device-independent scenario where the involved parties receive trusted quantum inputs, but whose operations are otherwise uncharacterized. Defining the notion of causally nonseparable distributed measurements, we show that certain causally nonseparable processes that cannot violate any causal inequality, including the canonical example of the quantum switch, can generate noncausal correlations in such a scenario. Moreover, by imposing some further natural structure to the untrusted operations, we show that all bipartite causally nonseparable process matrices can be certified with trusted quantum inputs.