Can causal relations be subject to quantum indefiniteness, similar to other physical properties? The process-matrix framework formalises this possibility: valid processes are defined by what local laboratories can implement, without assuming a global causal order. Standardly, the local labs are assumed to implement arbitrary quantum instruments. We ask what happens when symmetries restrict these local operations. Symmetry constraints, such as those arising from missing reference frames, superselection constraints, or the antiunitary symmetry defining real quantum theory, enlarge the admissible process cone. Do these extra processes generate genuinely new correlations? We prove a sharp dichotomy: no for any finite unitary symmetry, yes for real quantum theory. Recent work has shown that, under fixed and definite causal order, complex quantum theory is strictly richer than real quantum theory. Our work shows that this hierarchy is reversed under indefinite causal order: real quantum theory realizes strictly more process correlations than complex quantum theory.
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5 pages Main, 2 pages End Matter, 31 pages Appendix
Can causal relations be subject to quantum indefiniteness, similar to other physical properties? The process-matrix framework formalises this possibility: valid processes are defined by what local laboratories can implement, without assuming a global causal order. Standardly, the local labs are assumed to implement arbitrary quantum instruments. We ask what happens when symmetries restrict these local operations. Symmetry constraints, such as those arising from missing reference frames, superselection constraints, or the antiunitary symmetry defining real quantum theory, enlarge the admissible process cone. Do these extra processes generate genuinely new correlations? We prove a sharp dichotomy: no for any finite unitary symmetry, yes for real quantum theory. Recent work has shown that, under fixed and definite causal order, complex quantum theory is strictly richer than real quantum theory. Our work shows that this hierarchy is reversed under indefinite causal order: real quantum theory realizes strictly more process correlations than complex quantum theory.
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.
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.
It has been shown that it is theoretically possible for there to exist higher-order quantum processes in which the operations performed by separate parties cannot be ascribed a definite causal order. Some of these processes are believed to have a physical realization in standard quantum mechanics via coherent control of the times of the operations. A prominent example is the quantum SWITCH, which was recently demonstrated experimentally. However, the interpretation of such experiments as realizations of a process with indefinite causal structure as opposed to some form of simulation of such a process has remained controversial. Where exactly are the local operations of the parties in such an experiment? On what spaces do they act given that their times are indefinite? Can we probe them directly rather than assume what they ought to be based on heuristic considerations? How can we reconcile the claim that these operations really take place, each once as required, with the fact that the structure of the presumed process implies that they cannot be part of any acyclic circuit? Here, I offer a precise answer to these questions: the input and output systems of the operations in such a process are generally nontrivial subsystems of Hilbert spaces that are tensor products of Hilbert spaces associated with systems at different times---a fact that is directly experimentally verifiable. With respect to these time-delocalized subsystems, the structure of the process is one of a circuit with a causal cycle. This provides a rigorous sense in which processes with indefinite causal structure can be said to exist within the known quantum mechanics. I also identify a whole class of isometric processes, of which the quantum SWITCH is a special case, that admit a physical realization on time-delocalized subsystems. These results unveil a novel structure within quantum mechanics, which may have important implications for physics and information processing.
In a previous work, we showed that real quantum theory (RQT) cannot be experimentally distinguished from ordinary quantum theory (QT). In this paper, we investigate whether this indistinguishability survives two natural extensions of the setting: allowing separated measurements to be represented by commuting rather than tensor-product observables, and imposing superselection rules, as in fermionic information theory. We show that it does in both cases.
Our first result is a positive answer to the Tsirelson problem in RQT: commuting and tensor-product measurements generate the same finite-dimensional correlations. This is surprisingly not a trivial consequence of the standard complex proof, as it does not carry over to the reals. The proof obstruction becomes particularly relevant in the presence of superselection rules, which is why we first establish the equivalence in RQT before extending the framework to superselected RQT.
Our second result is a local, real quantum formulation of superselected theories. We extend the RQT framework to accommodate a representation of superselection over the reals and show, in particular, that it admits a fermionic version whose predictions are experimentally indistinguishable from those of fermionic information theory. More generally, the construction extends to all superselected QTs.
The fundamental nature of causality has been challenged by quantum theory. In particular, recent developments have shown that by relaxing the standard assumption that quantum events are embedded in a fixed background causal structure, a new kind of causal relations, indefinite, can emerge. These indefinite causal structures have been interpreted as causal loops allowing for the violation of a form of causality without paradoxes. On the other hand, quantum contextuality shows that any attempt to extend quantum theory such that classical logic holds in the extended domain fails due to the rise of inconsistent logical loops. There has been increasing evidence that it is a key feature in the understanding quantum paradoxes. This thesis aims at improving our understanding of indefinite causal orders and quantum paradoxes in order to address the question: “Could the non-fundamental nature of causality be related to a form of contextuality ?”The process matrix formalism is a mathematical framework which, in analogy withentanglement and the violation of Bell inequalities (nonlocality), offers new tools - witnesses of causal nonseparability and the violation of causal inequalities with noncausalcorrelations - to study and identify indefinite causal orders. In a bottom-up approach, wehave identified a large class of physically implementable processes, the quantum circuitswith quantum control of causal order (QC-QCs), that includes causally nonseparable processes such as the celebrated “quantum switch”. However these circuits cannot be used to violate causal inequalities. Nevertheless, in another work, I have proposed a significant relaxation of assumptions to certify the causal nonseparability of many processes, showing that it can be done in a semi-device-independent scenario with untrusted local operations and trusted quantum inputs. In particular, I have shown that the quantum switch can generate noncausal correlations in this kind of scenario.In order to refine my understanding of quantum contextuality, I have investigated theorigins of the Kochen-Specker theorem and I have analysed how the Hardy paradox entails a form of logical contextuality. I have identified that the two possible intermediatemeasurements involved in a pre-post-selection version of the paradox differ causally fromeach other. This analysis might be useful to study of the Frauchiger-Renner paradox, a“meta-version” of the Hardy paradox. On a more metaphysical aspect, I have argued fora “Gödelian hunch” from quantum theory, the idea that quantum paradoxes emerge froma lack of distinction between theoretical and metatheoretical objects.Finally, in order to establish a link between indefinite causal orders and contextuality,I have proposed a new causal game, in which the grandfather paradox is formalized interms of a logical inequality. Some process matrices can violate the causal inequalitywithout violating the logical one. I suggest that the emergence of logical inconsistenciesfrom the violation of a logical inequality might be interpreted as a form of contextuality.
Indefinite causal order extends the conventional circuit paradigm by allowing local quantum operations to be connected without a predetermined global order. Within the process-matrix framework, it is characterized by causal nonseparability, whereas extensible causal separability (ECS) requires that, under arbitrary input-ancilla extensions, a process admit a recursive decomposition into components, each compatible with a particular operation acting first. Quantum circuits with classical control of causal order (QC-CC), in which previous outcomes dynamically determine which operation acts next, are known to generate ECS processes, but whether every ECS process admits such a realization has remained a longstanding open problem. We settle it through a generalized teleportation construction that realizes every multipartite ECS process as a QC-CC. We further establish consistency between ECS definitions for trivial and nontrivial global past and future systems. These results provide ECS with an exact operational interpretation and identify QC-CC as the complete circuit structure underlying this class of processes.