Quantum processes with indefinite causal structure emerge when we wonder which are the most general evolutions, allowed by quantum theory, of a set of local systems which are not assumed to be in any particular causal order. These processes can be described within the framework of higher-order quantum theory which, starting from considering maps from quantum transformations to quantum transformations, recursively constructs a hierarchy of quantum maps of increasingly higher order. In this work, we develop a formalism for quantum computation with indefinite causal structures; namely we characterize the computational structure of higher order quantum maps. Taking an axiomatic approach, the rules of this computation are identified as the most general compositions of higher order maps which are compatible with the mathematical structure of quantum theory. We provide a mathematical characterization of the admissible composition for arbitrary higher order quantum maps. We prove that these rules, which have a computational and information-theoretic nature, are determined by the more physical notion of the signalling relations between the quantum systems of the higher order quantum maps.

In physical experiments, reference frames are standardly modeled through a specific choice of coordinates used to describe the physical systems, but they themselves are not considered as such. However, any reference frame is a physical system that ultimately behaves according to quantum mechanics. We develop a framework for rotational (i.e., spin) quantum reference frames, with respect to which quantum systems with spin degrees of freedom are described. We give an explicit model for such frames as systems composed of three spin coherent states of angular momentum j and introduce the transformations between them by upgrading the Euler angles occurring in classical SO(3) spin transformations to quantum mechanical operators acting on the states of the reference frames. To ensure that an arbitrary rotation can be applied on the spin we take the limit of infinitely large j, in which case the angle operator possesses a continuous spectrum. In this poster, we further comment possible paths to generalize the notion of QRF for finite reference frames, and its connection to the framework of Penrose’s spin networks and some notion of causality within such framework.

Finding which causes are behind observed correlations in nature is at the core of the scientific discipline. Observed correlations are represented as the joint probability distribution of some random variables. One can ask whether certain causal relationships between variables are compatible with the observed correlations (e.g., if one variable could have a direct causal influence on another), or if there exist other hidden variables with an unknown joint distribution that could have such influence (e.g., hidden common-cause variables). Bayesian causal networks give the tools to formalise such questions. Each causal hypothesis can be encoded as a DAG (directed acyclic graph), with causal influence between variables being represented as directed edges. Whether some observed correlations are compatible with a particular DAG is known as the causal compatibility problem. In quantum theory, one cannot always assign a deterministic output to measurement results: the theory is inherently probabilistic. Thus, one needs to model measurement results as random variables. This naturally leads to causal analysis: which causal relationships can explain observed measurement statistics?
Roughly one can study compatibility of correlations with a *classical* cause-and-effect explanation or with a *quantum* cause-and-effect explanation, or DAGs that mix both. Both of these can be handled with a technique called quantum inflation. Our Python package can take as input any DAG with any assumption on the “quantumness” of each variable and then automatically implement the appropriate techniques for testing either causal compatibility, or also, for optimizing functions over correlations. Our package is the first open-source implementation of the quantum inflation technique, which in general has a high cost of implementation. Furthermore, we add extra improvements to the original technique.

Uncertainty relations express limits on the extent to which the outcomes of distinct measurements on a single state can be made jointly predictable. The existence of nontrivial uncertainty relations in quantum theory is generally considered to be a way in which it entails a departure from the classical worldview. However, this view is undermined by the fact that there exist operational theories which exhibit nontrivial uncertainty relations but which are consistent with the classical worldview insofar as they admit of a generalized-noncontextual ontological model. This prompts the question of what aspects of uncertainty relations, if any, cannot be realized in this way and so constitute evidence of genuine nonclassicality. We here consider uncertainty relations describing the tradeoff between the predictability of a pair of binary-outcome measurements (e.g., measurements of Pauli X and Pauli Z observables in quantum theory). We show that, for a class of theories satisfying a particular symmetry property, the functional form of this predictability tradeoff is constrained by noncontextuality to be below a linear curve. Because qubit quantum theory has the relevant symmetry property, the fact that it has a quadratic tradeoff between these predictabilities is a violation of this noncontextual bound, and therefore constitutes an example of how the functional form of an uncertainty relation can witness contextuality. We also deduce the implications for a selected group of operational foils to quantum theory and consider the generalization to three measurements.

We present a proposal for the definition of a time operator in a scalar QFT based on a generalization of the Newton-Wigner position operator. The key ingredient is the construction of quantum states on a timelike hyperplane Σ, defined by a constant value of the spatial coordinate x. Such states result not only from quantizing the standard propagating modes of the field but, crucially, also the evanescent ones thanks to the novel α-Kähler quantization introduced by two of the authors. In particular the time operator is defined as a map from a real-valued function on a timelike hyperplane to the space of self-adjoint operators on the Hilbert space of states on Σ. Correlation functions between states defined at different timelike hyperplanes Σ1 and Σ2 are computed and discussed.

In this work, I will describe an ongoing research project which has as goal connect foundations of quantum mechanics with advanced tools of theoretical physics such as CFT and AdS/CFT correspondence. The investigation of some fundamental questions from both perspectives through a quantum computation based on quantum gates over continuous-variable like trapped ions systems, for example. The smaller blocks to build a computation are the gates, in quantum computation: the quantum gates. The universe can be looked at through continuous variables using the coherent states – the formalism also used in quantum optics. Thus, we propose to do a generalization of quantum gates over continuous variables using Conformal Field Theory (CFT) -2d, specifically, vertex operators on a coherent basis and using effective field theory to do some correspondences between observables that are interested in the quantum gravity theory to systems based in a quantum laboratory.

The theoretical possibility of the existence of closed time-like curves (CTCs) has raised numerous debates among physicists and philosophers. In fact, such objects would allow an observer to travel back in her own past, and might lead to paradoxes such as the grandfather paradox – in which an effect suppresses its own cause – or the bootstrap paradox – in which an effect is its own cause. Quantum information analogies of CTCs have been suggested in order to circumvent these issues. Processes with indefinite causal order are an example of such causal resources allowing to “send information into the past without paradoxes” under free interventions. In this work, we give a deviceindependent definition of the grandfather and bootstrap antinomies using Boolean representations of their semantic counterparts, namely the Liar and Truth-Teller paradoxes. We present examples of a new kind of causal game that quantify a consistency principle in terms of a logical bound, in addition to discriminating causal and non-causal correlations with a causal bound. We show that the maximal probability of success with indefinite causal order lies between the causal and logical bounds, highlighting that noncausality does not imply logical inconsistency. On the one hand, the violation of a Liar-like logical bound might be interpreted as a form of meta-contextuality, as the outputs of each player are locally consistent but lead to inconsistencies when embedded globally. On the other hand, the violation of a Truth-Teller-like logical inequality might be interpreted as a witness of multiple worlds, as incompatible consistent events exist simultaneously.

Quantum correlations and quantum entanglement embody both our continued struggle towards a foundational understanding of quantum theory as well as its advantage over classical physics in various information processing tasks. Consequently, the problems of classifying (i) quantum states from more general (non-signalling) correlations, and (ii) entangled states within the set of all quantum states are at the heart of the subject of quantum information theory. My poster combines two recent results that shed new light on these problems, by exploiting a surprising connection with time in quantum theory: in arXiv:2204.11471 a solution to problem (i) is obtained for the bipartite case. I will sketch this characterisation, focussing on the new physical principle which singles out quantum theory: quantum states preserve time orientations—roughly, the unitary (or anti-unitary) evolution—in local subsystems. This is a genuine quantum phenomenon. Indeed, arXiv:2207.00024 shows that time orientations are intimately connected with quantum entanglement: a bipartite quantum state is separable if and only if it preserves arbitrary time orientations. As a variation of Peres’s entanglement criterion, this provides a solution to problem (ii).

Bell inequalities are rooted in a realist, causal worldview. Faced with experimentally observed violations of Bell inequalities, a causal explanation may be pursued based on violations of the locality assumption, the free choice assumption (sometimes called the measurement independence assumption) or the no-retro-causality assumption. We investigate the extent to which one of these assumptions needs to be relaxed for the other assumptions to hold at all costs by computing well-defined measures of violation. As a part of an ongoing project, we seek to extend our work in arXiv:2105.09037 to investigate the concept of retro-causality with the view to give a comprehensive quantitative treatment of violations of those three assumptions on an equal footing. This aims to advance the foundational debate and stimulate deeper discussion to what extent these three concepts are interchangeable in explaining observed correlations under a standard causal view of the world.

Causal diagrams have been mostly used as an explanatory tool in science. In this contribution, we explain how causal diagrams and causal inference – in particular, the d-separation rules – can be used to prove the validity of different postselection strategies for the certification of nonlocality. Postselecting statistics is known to possibly create fake correlations due to the selection bias, rendering nonlocality certification invalid. Here, we use causal diagrams to exclude the selection bias for different postselection strategies. We first understand and extend the widely-assumed fair-sampling assumption using causal diagrams, and see that it is also useful when demonstrating genuine multipartite nonlocality (arXiv:2207.09348). Next, we prove that certain collective postselection strategies are valid (i.e., do not open a postselection loophole) even for the certification of genuine multipartite nonlocality (arXiv:2104.10069). Finally, we show that a coincidence postselection (postselecting events for which each measurement party receives a single particle) is always valid if the number of particles is conserved. The results are applied to show that genuine N-partite nonlocality can be created from N independent particle sources.

Correlation does not imply causation—but then what does, especially in the quantum world? The discovery of causal relations is the basis of every scientific discipline, yet, a formal framework has been developed only recently. While it is now clear that to discover the causal relation between two classical events, one has to intervene on one and observe changes in the other, quantum causal relations are not that straightforward. As quantum devices increase in size and complexity, causal discovery tasks will be increasingly necessary, either as part of the algorithm or for the device’s characterisation protocol to know all the causes and effects and the presence of noise. We use a recently proposed framework for quantum causal models to develop computational and statistical tools for quantum causal discovery. We provide a suite of methods to discover the causal model of a quantum process. First, we improve an existing quantum causal discovery algorithm for better performance and write it as an open-source Python module. The algorithm inputs data from full process tomography and outputs a quantum causal model and the precise mechanisms behind the causal relations. Then, we develop witnesses of quantum causal orders that require far fewer measurements and output the underlying causal order. Lastly, we build Machine Learning techniques for causal discovery requiring even fewer measurements for a more experimentally-friendly method or to be used when the previous methods have reached their computational limit. This is the first complete toolkit for quantum causal discovery, taking into account experimental and computational limitations. It aims to provide the basis for our future quantum causal discovery needs as part of the quantum software and testing of future quantum devices.

By relating and ordering events, causality constitutes a pivotal feature of our world. However, different notions of causality exist, whose relation is not completely understood so far. In particular, we may consider both information-theoretic causality, covering the operational idea of information processing, and relativistic causality, linked to a light cone structure limiting signalling to the future. In this work, we improve on various results on the connection between both notions, as studied by V. Vilasini and R. Colbeck [arXiv:2109.12128, arXiv:2206.12887], in particular for questions of cyclicity. In the first part, we take an information-theoretic point of view, reviewing general, potentially cyclic or fine-tuned causal models. Here, the most general way of signalling is given by a concept of generalized affects relations, which use interventions on the model to uncover relations between nodes in these graphs. Building from their results, we study the properties of these affects relations and establish new ways to use them to characterize causal structures. Focusing on higher-order (HO) affects relations in particular, we can use knowledge of the absence of affects re-lations for causal inference. Further, we demonstrate a complete and constructive way to detect causal loops from a set of affects relations. In the second part, we embed these causal structures into a generic spacetime whose causal structure forms a partial order. Here, it was shown in [arXiv:2206.12887] that limiting signalling to the relativistic future does not suffice to generally rule out operationally detectable causal loops. In light of this, we propose additional stability conditions on the spacetime embedding and find that this can rule out a class of operationally detectable loops that cannot be ruled out by the principle of no-signalling (out- side the relativistic future) alone. We then propose a number of order-theoretic properties that we conjecture to hold in Minkowski spacetime with d ≥ 2 spatial dimensions. This would imply that in contrast to our result for generic spacetimes, in that Minkowski case, the no-signalling principle is indeed sufficient for ruling out this class of loops. Finally, we deduce novel restrictions for compatibility for certain HO affects relations.

Without a complete theory of quantum gravity, the question of how quantum fields and quantum particles behave in a superposition of spacetimes seems beyond the reach of theoretical and experimental investigations. Here we use an extension of the quantum reference frame formalism to address this question for the Klein-Gordon field residing on a superposition of conformally equivalent metrics. Based on the group structure of “quantum conformal transformations”, we construct an explicit quantum operator that can map states describing a quantum field on a superposition of spacetimes to states representing a quantum field with a superposition of masses on a Minkowski background. This constitutes an extended symmetry principle, namely invariance under quantum conformal transformations. The latter allows to build an understanding of superpositions of diffeomorphically non-equivalent spacetimes by relating them to a more intuitive superposition of quantum fields on curved spacetime. Furthermore, it can be used to import the phenomenon of particle production in curved spacetime to its conformally equivalent counterpart, thus revealing new features in modified Minkowski spacetime.

The study of indefinite causal order has seen rapid development, both theoretically and experimentally, in recent years. While classically, the causal order of two timelike separated events A and B is fixed – either A before B or B before A – this is no longer true in quantum theory. There, it is possible to encounter superpositions of causal orders. In light of recent work on quantum reference frames, which reveals that the superposition of locations, momenta, and other properties can depend on the choice of reference frame or coordinate system, the question arises whether this also holds true for superpositions of causal order. Here, we provide a negative answer to this question. First, we give an unambiguous definition of causal order between two events in terms of worldline coincidences and the proper time of a third particle. We demonstrate that it coincides with the operational notion employed in other works while contrasting it with the notion of causal structure. Then, we show that superpositions of causal order defined as such cannot be rendered definite even through the most general class of transformations – arbitrary and independent diffeomorphisms in each quantum branch. Finally, we discuss the implications of our result for possible formulations of a quantum equivalence principle.

A causal relationship can be described using the formalism of Generalised Bayesian Networks. This framework allows the depiction of cause and effect relations (causal scenarios) effectively using generalised directed acyclic graphs (GDAGs). A GDAG is considered “not interesting” if the classical correlations existing in it are just constrained by the observable conditional independencies in it. This implies that no non-classical correlations can ever be achieved in such ”non-interesting” scenarios, hence the name- “not interesting”. Henson, Lal and Pusey (HLP) have proposed a sufficient condition to check whether a causal scenario is “not interesting” but it’s not known whether their condition is a necessary one as well or not. Their condition does not hold for some GDAGs but for GDAGs up to 6 nodes, it’s known that the causal scenarios (or GDAGs) to which their condition does not apply are in fact “interesting”. The problem of identifying “interesting” causal scenarios for GDAGs of 7 nodes is nevertheless still an open one. We propose a new graphical theorem based on a graphical condition called “E-separation” to classify several of the GADGs of both 6 and 7 nodes for which HLP’s condition does not hold as “interesting”. In particular, for 7 nodes, there are 1,618,679 GDAGs to which HLP’s condition does not apply but using our theorem we are able to classify all of those but just 20 as “interesting”. It is important to identify GDAGs as “interesting” because it is crucial to understand the consequences of different types of causal structures, especially in the quantum setting. This will not only provide us with deep insights into quantum foundations such as how the stronger quantum correlations differ from the classical ones but also help us in advancing developments of new information processing protocols (such as quantum cryptography) whose performance and security level will depend crucially on the violations of the corresponding Bell-type inequalities. Amongst other things our work is also a contribution to the field of causal inference with hidden variables.

In this work, we investigate and generalize the quantum algorithm for accelerating causal inference based on the formulation presented in [G. Chiribella and D. Ebler, Nat. Comm., 10(1):1–8, 2019]. For the first time, we introduce a scalable gate-based model of the theoretical description of the formulation. We also introduce and elaborate on the gate-based construction of the oracle embedding the causal hypothesis and assessing the associated gate complexities. While implementing the quantum circuit on IBM’s open source quantum framework \texttt{qiskit} we uncover the following. $(1)$ The implementation aspects of a class of causal oracle, $(2)$ the gate and qubit complexity of the algorithm, $(3)$ the dependence of the hypothesis distinguishing error probability on the distance between the hypotheses being tested, and $(4)$ on the choice of the distance measure. The current technology constraints the scalability of the quantum systems, which in turn prevents us from embedding this causal reasoning within a broader application framework. By briefly reviewing the specifics of causal reasoning and some of the well-studied techniques; we discuss a correction factor introduced in the error probability based on our empirical results. Finally, we discuss some potential use cases in bioinformatics, artificial general intelligence and XAI pipelines [A. Lavin et.al, arXiv:2112.03235, (2021)].

In the quantization of gauge theories and quantum gravity, it is crucial to treat reference frames such as rods or clocks not as idealized external classical relata, but as internal quantum subsystems. In the Page-Wootters formalism, for example, evolution of a quantum system S is described by a stationary joint state of S and a quantum clock, where time-dependence of S arises from conditioning on the value of the clock. Here, we consider (possibly imperfect) internal quantum reference frames R for arbitrary compact symmetry groups, and show that there is an exact quantitative correspondence between the amount of entanglement in the invariant state on RS and the amount of asymmetry in the corresponding conditional state on S. Surprisingly, this duality holds exactly regardless of the choice of coherent state system used to condition on the reference frame. Averaging asymmetry over all conditional states, we obtain a simple representation-theoretic expression that admits the study of the quality of imperfect quantum reference frames, quantum speed limits for imperfect clocks, and typicality of asymmetry in a unified way. Our results shed light on the role of entanglement for establishing asymmetry in a fully symmetric quantum world.

In General Relativity, an event is a point on a manifold. In Quantum Mechanics, understanding what is an event is complicated, and one common view is the operational definition. In this view, an event is defined by the interaction between two systems.  This definition is used to characterize indefinite causal orders, where two agents A and B apply quantum operations on a target quantum system in an indefinite order. Such tasks are realized in optical experiments, but there are discussions on when these experiments are performing genuine processes with indefinite causal orders, or are being just a simulation of them. This discussion is closely related to the consistency of the operational definition of event. Moreover, this situation becomes even more complicated in quantum gravity scenarios, where an event could be defined taking reference on the proper time of the agents A and B, and few explicit examples are known. In this poster I will discuss many possible scenarios, including the optical experiments and when the agents are lying on a quantum spacetime, and also intermediate situations, in which the agents lie on a curved spacetime in an entangled state of different heights (arXiv:2012.03989), or when they are in an entangled state of different accelerations (arXiv:1712.02689). Moreover, I discuss different definitions of event and how they are related to the interpretation of each scenario.

The quantum switch, the canonical example of a process with indefinite causal order, has been claimed to provide various advantages over processes with definite causal orders in some particular tasks in the field of quantum metrology. In this work, we argue that some of these advantages in fact do not hold if a fairer comparison is made. To this end, we consider a framework that allows for a proper comparison between the performance, quantified by the quantum Fisher information, of different classes of indefinite causal order processes and that of causal strategies on a given metrological task. More generally, by considering the recently proposed classes of circuits with classical or quantum control of the causal order, we come up with different examples where processes with indefinite causal order offer (or not) an advantage over processes with definite causal order, qualifying the interest of indefinite causal order regarding quantum metrology. For a wide range of examples, the class of quantum circuits with quantum control of the causal order, which are known to be physically realizable, is shown to provide a strict advantage over causally ordered quantum circuits as well as the class of circuits with coherent control of the causal order, which includes processes such as the quantum switch with indefinite causal order.

Existing work on quantum causal structure assumes that one can perform arbitrary operations on the systems of interest. But this condition is often not met. Here, we extend the framework for quantum causal modelling to cases where a system can suffer sectorial constraints, that is, restrictions on the orthogonal subspaces of its Hilbert space that may be mapped to one another. Our framework 1. proves that a number of different intuitions about causal relations turn out to be equivalent; 2. shows that quantum causal structures in the presence of sectorial constraints can be represented with a directed graph; and 3. defines a fine-graining of the causal structure in which the individual sectors of a system bear causal relations, which provides a more detailed analysis than its coarse-grained counterpart. As an example, we apply our framework to purported photonic implementations of the quantum switch to show that while their coarse-grained causal structure is cyclic, their fine-grained causal structure is acyclic. We therefore conclude that these experiments realize indefinite causal order only in a weak sense. Notably, this is the first argument to this effect that is not rooted in the assumption that the causal relata must be localized in spacetime.

The Causaloid framework introduced by Hardy suggests a research program aimed at finding a theory of Quantum Gravity. On one side General Relativity (GR) while deterministic, features dynamic causal structures; on the other side Quantum Theory (QT) while having fixed causal structures, is probabilistic in nature. It is natural to then expect Quantum Gravity (QG) to house both of the radical aspects of GR and QT, and therefore incorporate indefinite causal structure. Through an Operational methodological approach, we study the Causaloid Framework that treats regions in a causally neutral manner and regions in a theory-independent manner by studying the correlations in the conditional probabilities through three levels of compression: Tomographic, Compositional and Meta. We study how models of definite and indefinite causal structures of classical and quantum theory (such as the process matrices) fit within this framework defined through a hierarchy of Meta Compression.

The existence of incompatible observables is a cornerstone of quantum mechanics and a valuable resource in quantum technologies. Here we introduce a measure of incompatibility, called the mutual eigenspace disturbance (MED), which quantifies the amount of disturbance induced by the measurement of a sharp observable on the eigenspaces of another. The MED is a faithful measure of incompatibility for sharp observables and provides a metric on the space of von Neumann measurements. It can be efficiently estimated by letting the measurements act in an indefinite order, using a setup known as the quantum switch. Thanks to these features, the MED can be used in quantum machine learning tasks, such as clustering quantum measurement devices based on their mutual compatibility. We demonstrate this application by providing an unsupervised algorithm that clusters unknown von Neumann measurements. Our algorithm is robust to noise can be used to identify groups of observers that share approximately the same measurement context.

The conventional representation of multipartite statistics in quantum information theory is through density operators, which implicitly treat the constituent subsystems as distinct degrees of freedom sharing the same temporal coordinate. But the pseudo-density operator (PDO) representation generalizes this to admit causal structures with subsystems associated with the same degrees of freedom at distinct time instants. Characterizing the set of possible PDO’s and classifying their associated causal structures are still open problems, even in the bipartite case. Here we tackle these problems for two-qubit PDO’s. We define the class $\mathcal T$ of PDO’s compatible with a temporally distributed causal structure, and study its relation to the set $\mathcal S$ of density operators. We define an efficiently computable witness (“atemporality witness”) for non-membership in $\mathcal T$. While all separable density operators are expected to be in $\mathcal T$, we find, somewhat surprisingly, that some entangled density operators are also in this class. Thus, within $\mathcal S$ atemporality seems to be a form of correlation stronger than entanglement, although a high enough presence of entanglement does appear to indicate atemporality.

Counterfactual questions and statements are intimately linked with discussions on causality. Lewis’ analysis of causality, for example, was based on counterfactuals. Counterfactual queries are described as the highest tier of causal questions in Judea Pearl’s book on (classical) causality, and are analysed in terms of what Pearl calls a structural causal model. However, counterfactuals have lacked a detailed interpretation when it comes to quantum causal models. Here we attempt to fill that gap by developing a systematic procedure to interpret and evaluate counterfactual questions and counterfactuals probabilities within the framework of quantum causal models. Our evaluation of counterfactual queries follows a three-step procedure of abduction, action and prediction, generalising Pearl’s semantics in classical causal models, but with some important distinctions. Following the quantum causal framework developed by Costa and Shrapnel, and Barrett, Lorenz, and Oreshkov, we define a quantum structural causal model in analogy to the classical structural causal model defined in [2]. We identify three types of counterfactual questions that one can answer in a quantum causality setup – passive, active, and active-observational counterfactuals – and show how the counterfactual probabilities can be evaluated in each case. We also show that Pearl’s classical structural model can be derived as a special case of the more general quantum structural causal model. This work completes the translation of Pearl’s causal hierarchy to the case of quantum causal models, and has potential applications in quantum machine learning and artificial intelligence.

The quantum switch is one of the few causally indefinite processes with a known physical interpretation; however, since it does not violate causal inequalities, a device-independent certification of its indefinite causal order has so far been absent. Here we provide such a certification by supplementing the standard causal inequality scenario with an additional spacelike-separated party and deriving a new set of inequalities from the assumptions of free randomness, definite causal order, and locality (in the sense of parameter independence). We then show that these are violated by correlations observed in the presence of a quantum switch entangled to a spacelike observer. This result highlights the need for an extension of the framework of causal inequalities that incorporates locality constraints.

Over the past decade, a number of quantum processes have been proposed which are logically consistent, yet feature a cyclic causal structure. However, there exists no general formal method to construct a process with an exotic causal structure in a way that ensures, and makes clear why, it is consistent. Here we provide such a method, given by an extended circuit formalism. This only requires directed graphs endowed with boolean matrices, which encode basic constraints on operations. Our framework (a) defines a set of elementary rules for checking the validity of any such graph, (b) provides a way of constructing consistent processes as a circuit from valid graphs, and (c) yields an intuitive interpretation of the causal relations within a process and an explanation of why they do not lead to inconsistencies. We display how several standard examples of exotic processes, including ones that violate causal inequalities, are among the class of processes that can be generated in this way; we conjecture that this class in fact includes all unitarily extendible processes.

We report on our results investigating bounds on temporal correlations achievable by classical and quantum systems under various discrete-time scenarios, with the assumption of a finite memory resource. This memory constraint imposes fundamental limitations on realizable correlations, which may be exploited to construct dimensional witnesses. In this poster, we report on our investigation on temporal correlations in the simplest measurement scenario, i.e., that of a physical system on which the same measurement is performed at different times, producing a sequence of dichotomic outcomes. We characterize the minimum memory requirements for sequences to be obtained deterministically, and numerically investigate the probabilistic behavior below this memory threshold, in both classical and quantum scenarios. A particular class of sequences is found to offer an upper-bound for all other sequences, which suggests a nontrivial universal upper-bound of 1/e for the classical probability of realization of any sequence below this memory threshold. We further present evidence that no such nontrivial bound exists in the quantum case.

Quantum supermaps are used to model holes into which deterministic quantum processes can be inserted. Whilst the concept of a supermap can be understood informally in terms of boxes, wires, and their composition rules, the mathematical formalization of supermaps currently refers to specific knowledge of probabilistic, linear, or compact closed structure of more general non-deterministic quantum processes. We prove that there is an alternative way to axiomatize quantum supermaps which references only the sequential and parallel composition rules for deterministic quantum processes. The axiomatization is given via a single simple definition of locally-applicable transformation, which can be stated for arbitrary symmetric monoidal categories, and so for arbitrary operational probabilistic theories. The definition can be rephrased in the language of category theory using the principle of naturality, and can be given an intuitive diagrammatic representation in terms of which all proofs are presented. In our main technical contribution, we use this diagrammatic representation to show that locally-applicable transformations on quantum channels are indeed in one-to-one correspondence with deterministic quantum supermaps. This alternative characterization of quantum supermaps is proven to hold for supermaps on arbitrary convex subsets of channels, including as a special case the supermaps on non-signaling channels used in the study of quantum causal structure.