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.
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 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.
It was recently found that it is theoretically possible for there to exist quantum processes in which the order of the operations performed by separate parties cannot be ascribed a definite causal order. Some of these processes have been claimed 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, up until now, there has been no rigorous justification for the interpretation of such an experiment as a genuine realization of a process with indefinite causal order as opposed to a simulation of such a process. 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 different times---a fact that is directly experimentally verifiable. With respect to these time-delocalized subsystems, the structure of the process is that of a circuit with a cycle, which cannot be reduced to a probabilistic mixture of acyclic circuits. I further show that all bipartite processes that admit a certain type of isometric extension have a physical realization on such time-delocalized subsystems. These results unveil a novel structure within quantum mechanics, which may have important implications for physics and information processing.
In relational quantum dynamics, evolution emerges via the correlations between some system of interest and a clock system, which plays the role of a temporal reference frame. Their combined state satisfies a Wheeler-de Witt-like constraint equation, and therefore does not evolve, leading to a ``block universe'' picture. Here we investigate the interplay of two aspects, namely temporal localization and causal relations, when comparing emergent dynamics with respect to different choices of clock. We first explore the extent to which two clocks can agree on the temporal localization of events. Then, focussing on the operational notion of causality, we require a clearly defined notion of interventions, i.e. quantum operations, and consider two different approaches to modeling these operations within relational dynamics. The first considers their application via the choice of solutions to the constraint equation, i.e.~the choice of which ``history'' is considered. The second approach incorporates the operations into the constraint equation itself and thereby into its solutions, giving a dynamical picture of the interventions. From the perspective of a single clock, both approaches allow for a notion of operational causality in relational dynamics. However, for multiple clocks, only the second approach gives a consistent picture regarding causal relations, while necessarily manifesting some degree of temporal delocalization between frames. Moreover, this second approach, when considering certain cases of temporal delocalization, naturally describes scenarios with indefinite causal order, a well-known quantum feature of operational causality.
Quantum supermaps are transformations that map quantum operations to quantum operations.It is known that quantum supermaps which respect a definite, predefined causal order between their input operations correspond to fixed-order quantum circuits, also called quantum combs.A systematic understanding of the physical interpretation of more general types of quantum supermaps-in particular, those incompatible with a definite causal structure-is however lacking.In this paper, we identify two types of circuits that naturally generalize the fixed-order case and that likewise correspond to distinct classes of quantum supermaps, which we fully characterize.We first introduce "quantum circuits with classical control of causal order," in which the order of operations is still well defined, but not necessarily fixed in advance: it can, in particular, be established dynamically, in a classically controlled manner, as the circuit is being used.We then consider "quantum circuits with quantum control of causal order," in which the order of operations is controlled coherently.The supermaps described by these classes of circuits are physically realizable, and the latter encompasses all known examples of physically realizable processes with indefinite causal order, including the celebrated "quantum switch."Interestingly, it also contains other examples arising from the combination of dynamical and coherent control of causal order, and we detail explicitly one such process.Nevertheless, we show that quantum circuits with quantum control of causal order can only generate "causal" correlations, compatible with a well-defined causal order.We furthermore extend our considerations to probabilistic circuits that produce also classical outcomes, and we demonstrate by an example how the characterizations derived in this work allow us to identify advantages for quantum information processing tasks that could be demonstrated in practice.
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.
We study how to implement and transform frame perspectives for quantum processes in the process-matrix formalism. We argue that, for pure processes, the causal reference frames (CRF)and time-delocalized subsystems (TDS) formalisms should be understood as coordinate parametrizations of a single perspective-neutral higher-order object. A genuine perspective arises when one endows the process with additional frame data by choosing an operational foliation into circuit fragments (events). With this distinction, existing no-go results acquire a clear scope: they rule out unitary transformations that preserve time foliation, attempting to switch perspectives while keeping the fragment boundaries -- hence the global past/future partition -- fixed. Focusing on the quantum switch, we construct explicit maps that transform perspectives unitarily at the price of reshuffling the notions of past and future. We then show that unitary transformations between perspectives can also be achieved in a different way, namely by extending the process with subsystems that define quantum reference frames and provide a shared spatiotemporal scaffold. In this extended setting, complementary CRF/TDS perspectives become unitarily related while preserving global past and future. We discuss how this frame-perspectival approach informs the broader question of empirical realizability of abstract process matrices.