Global Causal Structure Research Articles

Quantum space-time marginal problem: global causal structure from local causal information

Spatial and temporal quantum correlations can be unified in the framework of the pseudo-density operators (PDOs), and quantum causality between the involved events in an experiment is encoded in the corresponding PDO. We study the relationship between local causal information and global causal structure. A space-time marginal problem is proposed to infer global causal structures from given marginal causal structures where causal structures are represented by the reduced PDOs; we show that there almost always exists a solution in this case. By imposing the corresponding constraints on this solution set, we could obtain the required solutions for special classes of marginal problems, like a positive semidefinite marginal problem, separable marginal problem, etc. We introduce a space-time entropy and propose a method to determine the global causal structure based on the maximum entropy principle. The notion of quantum pseudo-channel (QPC) is also introduced and we demonstrate that the QPC marginal problem can be solved by transforming it into a PDO marginal problem via the channel-state duality.

Local causal structure learning for streaming features

Local causal structure learning (LCSL) is the process of discovering and distinguishing direct causes (parents) and direct effects (children) of a given target variable (T) without learning a global causal structure. However, state-of-the-art LCSL algorithms can only address static feature space, while ignoring the dynamic changes of feature space with streaming features. For high dimensional data, existing methods fail to efficiently distinguish causality between features and the target variable. To address these issues, we propose Local Causal Structure Learning for Streaming Features (LCSLSF). LCSLSF comprises two sequential steps, as it first dynamically mines causal features to construct an approximate Markov Blanket (aMB) of the target variable. It proceeds by filtering irrelevant and redundant features, retaining causal features as much as possible from streaming features. Second, it learns local causal structures by mining and splicing the V-structure of feature nodes in batches from aMB of the target variable. On benchmark Bayesian networks with the number of variables ranging from 20 to 724, LCSLSF achieves a significantly better performance than its rivals. We demonstrate its effectiveness by conducting a real-world case study on causal discovery in medical diagnostics. Code is released at https://github.com/youdianlong/LCSLSF.

On the visibility of singularities in general relativity and modified gravity theories

We investigate the global causal structure of the end state of a spherically symmetric marginally bound Lemaitre–Tolman–Bondi (LTB) (Lemaitre 1933 Ann. Soc. Sci. Brux. A 53 51, Tolman 1934 Proc. Natl Acad. Sci. USA 20 169–76, Bondi 1947 Mon. Not. Astron. Soc. 107 410) collapsing cloud (which is well studied in general relativity) in the framework of modified gravity having the generalized Lagrangian in the action. Here R is the Ricci scalar, and is a constant. By fixing the functional form of the metric components of the LTB spacetime, using up the available degree of freedom, we realize that the matching surface of the interior and the exterior metric are different for different values of α. This change in the matching surface can alter the causal property of the first central singularity. We depict this by showing a numerical example. Additionally, for a globally naked singularity to have physical relevance, a congruence of null geodesics should escape from such singularity to be visible to an asymptotic observer for infinite time. For this to happen, the first central singularity should be a nodal point. We here give a heuristic method to show that this singularity is a nodal point by considering the above class of theory of gravity, of which general relativity is a particular case.

Causal learner: A toolbox for causal structure and Markov blanket learning

Causal Learner is a toolbox for learning causal structure and Markov blanket (MB) from data. It integrates functions for generating simulated Bayesian network data, a set of state-of-the-art global causal structure learning algorithms, a set of state-of-the-art local causal structure learning algorithms, a set of state-of-the-art MB learning algorithms, and functions for evaluating algorithms. The data generation part of Causal Learner is written in R, and the rest of Causal Learner is written in MATLAB. Causal Learner aims to provide researchers and practitioners with an open-source platform for causal learning from data and for the development and evaluation of new causal learning algorithms. The Causal Learner project is available at http://bigdata.ahu.edu.cn/causal-learner.

Causal nonseparability and its implications for spatiotemporal relations

Quantum nonseparability is a central feature of quantum mechanics, and raises important philosophical questions. Interestingly, a particular theoretical development of quantum mechanics, called the process matrix formalism (PMF), features another kind of nonseparability, called causal nonseparability. The PMF appeals to the notion of quantum process, which is a generalisation of the concept of quantum state allowing to represent quantum-like correlations between quantum events over multiple parties without specifying a priori their spatiotemporal locations. Crucially, since the PMF makes no assumption about the global causal structure between quantum events, it allows for the existence of causally nonseparable quantum processes. Such processes are said to have an indefinite causal structure. This work aims at investigating the philosophical implications of causal nonseparability, especially for the notion of spatiotemporal relations. A preliminary discussion will first study the formal connection between quantum and causal nonseparability. It will be emphasised that, although quantum processes can be seen as a generalisation of density matrices, the conceptual distinction between the two notions yields significant differences between quantum and causal nonseparability. From there, it will be shown that, depending on the interpretative framework, causal nonseparability suggests some kind of indeterminacy of spatiotemporal relations. Namely, within a realist context, spatiotemporal relations can be epistemically or metaphysically indeterminate. Finally, it will be argued that, in spite of the disanalogies between standard and causal nonseparability, similar implications for spatial relations can already be defended in the context of standard quantum mechanics. This work highlights the potentially very fruitful explorations of the implications of quantum features on the conception of spacetime, keeping in mind that quantum and spacetime theories are expected to be unified in a future theory of quantum gravity.

Nonsingular spherically symmetric black-hole model with holonomy corrections

We present a covariant model of a spherically symmetric black hole with corrections motivated by loop quantum gravity. The effective modifications, parametrized by a positive constant $\ensuremath{\lambda}$, are implemented through a canonical transformation and a linear combination of the constraints of general relativity, in such a way that the theory remains free of anomalies and general relativity is recovered for $\ensuremath{\lambda}=0$. In addition, the corresponding metric is constructed in a fully covariant way to ensure that gauge transformations on phase space correspond to coordinate changes. The solution for each gauge choice provides a chart and corresponding line element of a spacetime solution whose geometry is unambiguously determined in terms of the parameter $\ensuremath{\lambda}$ and a constant of motion $m$. For positive values of $m$, the solution is asymptotically flat and contains a globally hyperbolic black-hole/white-hole region with a minimal spacelike hypersurface that replaces the Schwarzschild singularity. The corresponding exterior regions are isometric and, in particular, allow the computation of the Arnowitt-Deser-Misner mass. The procedure to obtain the global causal structure of the solution yields also its maximal analytic extension.

Massive holographic QFTs in de Sitter

We study strongly coupled mass-deformed-CFT on a fixed de Sitter spacetime in three dimensions via holography. We elucidate the global causal structure of the four-dimensional spacetime dual to the de Sitter invariant vacuum state. The conformal boundaries of de Sitter appear as spacelike defects sourced by the mass deformation, which extend into the bulk as curvature singularities in AdS. We compute all one- and two-point functions of the deformed-CFT stress tensor and a scalar operator order-by-order in the mass deformation for a simple holographic model. These correlation functions admit a spectral representation as a sum of simple poles corresponding to normalisable modes in the bulk.

An effective model for the quantum Schwarzschild black hole

We present an effective theory to describe the quantization of spherically symmetric vacuum motivated by loop quantum gravity. We include anomaly-free holonomy corrections through a canonical transformation and a linear combination of constraints of general relativity, such that the modified constraint algebra closes. The system is then provided with a fully covariant and unambiguous geometric description, independent of the gauge choice on the phase space. The resulting spacetime corresponds to a singularity-free (black-hole/white-hole) interior and two asymptotically flat exterior regions of equal mass. The interior region contains a minimal smooth spacelike surface that replaces the Schwarzschild singularity. We find the global causal structure and the maximal analytical extension. Both Minkowski and Schwarzschild spacetimes are directly recovered as particular limits of the model.

Nonsingular black hole in two-dimensional asymptotically flat spacetime

We study a special two-dimensional dilaton gravity with Lagrangian $\mathcal{L}=\frac{1}{2}\sqrt{-g}(\phi R+{\lambda^2}{\rm sech}^2\phi)$ where $\lambda$ is a parameter of dimension mass. This theory describes two-dimensional spacetimes that are asymptotically flat. Very interestingly, it has an exact solution for the metric, $\d s^2=-(c+\tanh \lambda x)\d t^2+ \d x^2/(c+\tanh {\lambda} x)$, parametrized by $c$. For $c\in(-1,1)$, the solution presents an event horizon but no singularity. Because of the kink profile for the metric components appearing in the solution, we refer to the spacetime described by the metric as a {\it gravitational domain wall} with the wall simply being the event horizon and separating two asymptotically Minkowskian spacetimes. The global causal structure for such an object is studied via coordinate extension and the thermodynamical quantities are computed. Only when $c\in(-1,0]$ are the entropy and energy non-negative.

On the definition and characterisation of multipartite causal (non)separability

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.

Causal structure of evaporating black holes

We offer a mathematically rigorous basis for the widely held suspicion that full black hole evaporation is in tension with predictability. Based on conditions expressing the global causal structure of evaporating black hole spacetimes, we prove two theorems in Lorentzian geometry showing that such spacetimes either fail to be causally simple or fail to be causally continuous. These theorems, when combined with recent results (Aké et al 2018 (arXiv:1808.04412 [gr-qc])) on the causal structure of spacetimes with timelike boundary, bear significantly on the question of whether these spacetimes permit for a predictable evolution.

Genuinely multipartite noncausality

The study of correlations with no definite causal order has revealed a rich structure emerging when more than two parties are involved. This motivates the consideration of multipartite "noncausal" correlations that cannot be realised even if noncausal resources are made available to a smaller number of parties. Here we formalise this notion: genuinely N-partite noncausal correlations are those that cannot be produced by grouping N parties into two or more subsets, where a causal order between the subsets exists. We prove that such correlations can be characterised as lying outside a polytope, whose vertices correspond to deterministic strategies and whose facets define what we call "2-causal" inequalities. We show that genuinely multipartite noncausal correlations arise within the process matrix formalism, where quantum mechanics holds locally but no global causal structure is assumed, although for some inequalities no violation was found. We further introduce two refined definitions that allow one to quantify, in different ways, to what extent noncausal correlations correspond to a genuinely multipartite resource.

Non-Causal Computation

Computation models such as circuits describe sequences of computation steps that are carried out one after the other. In other words, algorithm design is traditionally subject to the restriction imposed by a fixed causal order. We address a novel computing paradigm beyond quantum computing, replacing this assumption by mere logical consistency: We study non-causal circuits, where a fixed time structure within a gate is locally assumed whilst the global causal structure between the gates is dropped. We present examples of logically consistent non-causal circuits outperforming all causal ones; they imply that suppressing loops entirely is more restrictive than just avoiding the contradictions they can give rise to. That fact is already known for correlations as well as for communication, and we here extend it to computation.

On the Splitting Problem for Lorentzian Manifolds with an $${\mathbb {R}}$$ R -Action with Causal Orbits

We study the interplay between the global causal and geometric structures of a spacetime (M, g) and the features of a given smooth \({\mathbb {R}}\)-action \(\rho \) on M whose orbits are all causal curves, building on classic results about Lie group actions on manifolds described by Palais (Ann Math 73:295–323, 1961). Although the dynamics of such an action can be very hard to describe in general, simple restrictions on the causal structure of (M, g) can simplify this dynamics dramatically. In the first part of this paper, we prove that \(\rho \) is free and proper (so that M splits topologically) provided that (M, g) is strongly causal and \(\rho \) does not have what we call weakly ancestral pairs, a notion which admits a natural interpretation in terms of “cosmic censorship.” Accordingly, such condition holds automatically if (M, g) is globally hyperbolic. We also prove that M splits topologically if (M, g) is strongly causal and \(\rho \) is the flow of a complete conformal Killing null vector field. In the second part, we investigate the class of Brinkmann spacetimes, which can be regarded as null analogues of stationary spacetimes in which \(\rho \) is the flow of a complete parallel null vector field. Inspired by the geometric characterization of stationary spacetimes in terms of standard stationary ones (Javaloyes and Sanchez in Class Quantum Gravity 25:168001, 2008), we obtain an analogous geometric characterization of when a Brinkmann spacetime is isometric to a standard Brinkmann spacetime. This result naturally leads us to discuss a conjectural null analogue for Ricci-flat four-dimensional Brinkmann spacetimes of a celebrated rigidity theorem by Anderson (Ann Henri Poincare 1:977–994, 2000) and highlight its relation with a long-standing conjecture by Ehlers and Kundt (Gravitation: an introduction to current research. Wiley, New York, pp 49–101, 1962).

Global properties of physically interesting Lorentzian spacetimes

Under normal circumstances most members of the general relativity community focus almost exclusively on the local properties of spacetime, such as the locally Euclidean structure of the manifold and the Lorentzian signature of the metric tensor. When combined with the classical Einstein field equations this gives an extremely successful empirical model of classical gravity and classical matter — at least as long as one does not ask too many awkward questions about global issues, (such as global topology and global causal structure). We feel however that this is a tactical error — even without invoking full-fledged “quantum gravity” we know that the standard model of particle physics is also an extremely good representation of some parts of empirical reality; and we had better be able to carry over all the good features of the standard model of particle physics — at least into the realm of semi-classical quantum gravity. Doing so gives us some interesting global features that spacetime should possess: On physical grounds spacetime should be space-orientable, time-orientable, and spacetime-orientable, and it should possess a globally defined tetrad (vierbein, or in general a globally defined vielbein/[Formula: see text]-bein). So on physical grounds spacetime should be parallelizable. This strongly suggests that the metric is not the fundamental physical quantity; a very good case can be made for the tetrad being more fundamental than the metric. Furthermore, a globally-defined “almost complex structure” is almost unavoidable. Ideas along these lines have previously been mooted, but much is buried in the pre- arXiv literature and is either forgotten or inaccessible. We shall revisit these ideas taking a perspective very much based on empirical physical observation.

Indefinite causal structures for continuous-variable systems

In all our well-established theories, it is assumed that events are embedded in a global causal structure such that, for every pair of events, the causal order between them is always fixed. However, the possible interplay between quantum mechanics and general relativity may require a revision of this paradigm. The process matrix framework keeps the validity of quantum physics locally but does not assume the existence of a global causal order. It allows to describe causal structures corresponding to a quantum superposition of ‘A is before B’ and ‘B is before A’. So far, the framework has been developed only for finite-dimensional systems, and a straightforward generalization to infinite dimensions leads to singularities. Such generalization is necessary for continuous-variable systems and is a prerequisite for quantum fields on indefinite causal structures. Here we develop the process-matrix framework for continuous-variable systems. We encounter and solve the problems of singularities. Moreover, we study an example of a process in infinite dimensions, and we derive correlations exhibiting intereference due to processes in which A is before B and B is before A.

Multipartite causal correlations: Polytopes and inequalities

We consider the most general correlations that can be obtained by a group of parties whose causal relations are well-defined, although possibly probabilistic and dependent on past parties' operations. We show that, for any fixed number of parties and inputs and outputs for each party, the set of such correlations forms a convex polytope, whose vertices correspond to deterministic strategies, and whose (nontrivial) facets define so-called causal inequalities. We completely characterize the simplest tripartite polytope in terms of its facet inequalities, propose generalizations of some inequalities to scenarios with more parties, and show that our tripartite inequalities can be violated within the process matrix formalism, where quantum mechanics is locally valid but no global causal structure is assumed.

Witnesses of causal nonseparability: an introduction and a few case studies.

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.

Information-theoretic constraints on correlations with indefinite causal order

Reconstructions of quantum theory usually implicitly assume that experimental events are ordered within a global causal structure. The process matrix framework accommodates quantum correlations that violate an inequality verified by all causally ordered correlations. Using a generalized probabilistic framework, we propose three principles constraining bipartite correlations to the quantum bound. Our approach highlights the role of a measure of dependence other than mutual information for an information-theoretic reconstruction of causal structures in quantum theory.

Witnessing causal nonseparability

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.