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The three causal models consistent with the CI relation $(A\bot \bot B|C)$ ( A ⊥ ⊥ B ∣ C ) > .
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An active area of research in the fields of machine learning and statistics
is the development of causal discovery algorithms, the purpose of which is to
infer the causal relations that hold among a set of variables from the
correlations that these exhibit. We apply some of these algorithms to the
correlations that arise for entangled quantum syste...
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Citations
... Here no fine-tuning means that operational identities are reflected in the ontological model [7,[28][29][30], such as the same density operator having the same ontological representation, no matter how it is produced, also referred to as preparation noncontextuality [7]. Proofs of this result fall amongst others in the paradigm of generalised noncontextuality [7, 19-23, 25, 26, 31-34]. ...
In a recent paper [Carcassi, Oldofredi and Aidala, Found Phys 54, 14 (2024)] it is claimed that the whole Harrigan–Spekkens framework of ontological models is inconsistent with quantum theory. They show this by showing that all pure quantum states in -ontic models must be orthogonal. In this note, we identify some crucial assumptions that lack physical motivation in their argument to the extent that the main claim is incorrect.
... Furthermore, in accordance with the principles of causality [33,34], all variables in Eq. (6) are affected by the hidden variables λ 1 and/or λ 2 . Consequently, all optimization variables, including P 1 (λ 1 ), P 2 (λ 2 ), P A (a|λ 1 , λ 2 ), σ B1 λ1 , and σ B2 λ2 , are represented as outputs of the ANN. ...
Network quantum steering plays a pivotal role in quantum information science, enabling robust certification of quantum correlations in scenarios with asymmetric trust assumptions among network parties. Despite its significance, efficient methods for measuring network quantum steerability remain elusive. To tackle this issue, we develop a neural network-based method that can be generalized to arbitrary quantum networks, providing an effective framework for steerability analysis. Our method demonstrates remarkable accuracy and efficiency in single-source scenarios, specifically bipartite and multipartite steering scenarios, with numerical simulations involving isotropic states and noisy GHZ states showing consistent results with established findings. Furthermore, we demonstrate its utility in the bilocal steering scenario, where an untrusted central party shares two-qubit isotropic states of different visibilities, and , with trusted endpoint parties and performs a single Bell state measurement (BSM). Through explicit construction of network local hidden state (NLHS) model derived from numerical results and incorporation of the entanglement properties of network assemblages, we analytically demonstrate that the threshold for the existence of network steering is determined by the curve under the corresponding configuration.
... Hence, all latent-free pDAGs are algebraic. One example of a nonalgebraic pDAG is the causal structure appearing in Bell's theorem [1], because this structure implies causal-compatibility constraints that take the form of inequalities, namely, Bell inequalities [18]. As we will see later on, the classification of pDAGs under observational equivalence and dominance is highly pertinent to the question of which pDAGs are algebraic and which are nonalgebraic. ...
... It is worth noting that preferring causal models that satisfy the principle of faithfulness [30] (referred to as stability in Ref. [26] and no fine-tuning in Ref. [18]) is an instance of preferring classes lower in the observational partial order. ...
... In particular, with this analysis we reproduced one of the results of Ref. [18]: the observational equivalence class of the Bell mDAG ( Fig. 7.1(a)) consists of the 9 mDAGs whose associated pDAGs are consistent with a fixed nodal ordering depicted in Figure 24 of Ref. [18]. The Bell equivalence class is identified by d-separation alone, meaning that the particular pattern of d-separation relations presented by this observational equivalence class is not presented by any other class. ...
For two causal structures with the same set of visible variables, one is said to observationally dominate the other if the set of distributions over the visible variables realizable by the first contains the set of distributions over the visible variables realizable by the second. Knowing such dominance relations is useful for adjudicating between these structures given observational data. We here consider the problem of determining the partial order of equivalence classes of causal structures with latent variables relative to observational dominance. We provide a complete characterization of the dominance order in the case of three visible variables, and a partial characterization in the case of four visible variables. Our techniques also help to identify which observational equivalence classes have a set of realizable distributions that is characterized by nontrivial inequality constraints, analogous to Bell inequalities and instrumental inequalities. We find evidence that as one increases the number of visible variables, the equivalence classes satisfying nontrivial inequality constraints become ubiquitous. (Because such classes are the ones for which there can be a difference in the distributions that are quantumly and classically realizable, this implies that the potential for quantum-classical gaps is also ubiquitous.) Furthermore, we find evidence that constraint-based causal discovery algorithms that rely solely on conditional independence constraints have a significantly weaker distinguishing power among observational equivalence classes than algorithms that go beyond these (i.e., algorithms that also leverage nested Markov constraints and inequality constraints).
... Within a broader effort to understand the role of causal structure in quantum theory [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], a framework has recently emerged in which causal relations need not be fixed in advance [43][44][45] and can be discovered through experiments. The framework-often dubbed the "process matrix formalism"-also includes scenarios where causal relations are genuinely indefinite [46], with potential applications to quantum information processing [47][48][49][50][51] and fundamental models of quantum gravity [52][53][54][55]. ...
What does it mean for a causal structure to be `unknown'? Can we even talk about `repetitions' of an experiment without prior knowledge of causal relations? And under what conditions can we say that a set of processes with arbitrary, possibly indefinite, causal structure are independent and identically distributed? Similar questions for classical probabilities, quantum states, and quantum channels are beautifully answered by so-called "de Finetti theorems", which connect a simple and easy-to-justify condition – symmetry under exchange – with a very particular multipartite structure: a mixture of identical states/channels. Here we extend the result to processes with arbitrary causal structure, including indefinite causal order and multi-time, non-Markovian processes applicable to noisy quantum devices. The result also implies a new class of de Finetti theorems for quantum states subject to a large class of linear constraints, which can be of independent interest.
... Classical causal modelling [Pea09,SGS93], captures causal relationships among random variables and has been widely applied across fields such as machine learning, economics, and clinical trials [RCLH21,KH11,Pea09,Spi05,PL14,AHK20,LKK21]. However, as demonstrated by Bell's theorem [Bel64], this classical framework cannot account for quantum correlations without invoking fine-tuned mechanisms or modifications to the causal structure naturally associated with a Bell scenario [WS15]. This limitation has spurred the development of non-classical causal modelling frameworks that encompass quantum and broader operational theories [HLP14,BLO20], enabling the causal explanation of quantum correlations without invoking fine-tuning or adjustments to the operational causal structure. ...
... This theorem enables to read off conditional independences in correlations purely from the structure of the graph (causal structure), and is central to how we explain correlations (conditional dependencies) in terms of causal connections. Moreover, the d-separation theorem plays an integral role in causal discovery algorithms and inference across data-driven disciplines (see [SZ16]), in causal compatibility problems as well as certification of non-classical correlations in causal structures [WS15,HLP14]. ...
Causal modelling frameworks link observable correlations to causal explanations, which is a crucial aspect of science. These models represent causal relationships through directed graphs, with vertices and edges denoting systems and transformations within a theory. Most studies focus on acyclic causal graphs, where well-defined probability rules and powerful graph-theoretic properties like the d-separation theorem apply. However, understanding complex feedback processes and exotic fundamental scenarios with causal loops requires cyclic causal models, where such results do not generally hold. While progress has been made in classical cyclic causal models, challenges remain in uniquely fixing probability distributions and identifying graph-separation properties applicable in general cyclic models. In cyclic quantum scenarios, existing frameworks have focussed on a subset of possible cyclic causal scenarios, with graph-separation properties yet unexplored. This work proposes a framework applicable to all consistent quantum and classical cyclic causal models on finite-dimensional systems. We address these challenges by introducing a robust probability rule and a novel graph-separation property, p-separation, which we prove to be sound and complete for all such models. Our approach maps cyclic causal models to acyclic ones with post-selection, leveraging the post-selected quantum teleportation protocol. We characterize these protocols and their success probabilities along the way. We also establish connections between this formalism and other classical and quantum frameworks to inform a more unified perspective on causality. This provides a foundation for more general cyclic causal discovery algorithms and to systematically extend open problems and techniques from acyclic informational networks (e.g., certification of non-classicality) to cyclic causal structures and networks.
... More recently, notions of causality have begun to play an instructive role also in quantum physics. One of the best clues for deducing the precise manner in which quantum physics implies a departure from the principles underlying classical physics, Bell's theorem [2,3], admits a distinctly causal interpretation [4,5]. In the past decade, there has been much interest in the intersection of causality and quantum physics leading to a better understanding of quantum-classical gaps in an array of causal structures [6], the investigation of potential benefits for causal inference [7,8], and the formalization of the notion of an intrinsically quantum causal model [9][10][11]. ...
... This is because the latter article considered only unitaries for the channels mapping the latent systems to the visible systems, rather than allowing arbitrary operations as we do here. 4 In short, while we here consider state-preparability by Local Operations with 2-way Shared Entanglement, Ref. [28] considers state-preparability by Local Unitaries with 2-way Shared Entanglement (LU2WSE). Because LU2WSE ⊆ LO2WSE, any state shown to be preparable by LU2WSE is also preparable by LO2WSE, but it is unclear whether or not the opposite implication holds. ...
... But this implies that the protocol allows for the preparation |Ψ⟩ ABC without use of the shared randomness. 4 Ref. [28] did not allow for tracing out subsystems, which is why the two approaches are distinct in spite of the Stinespring dilation theorem. ...
Consider the problem of deciding, for a particular multipartite quantum state, whether or not it is realizable in a quantum network with a particular causal structure. This is a fully quantum version of what causal inference researchers refer to as the problem of causal discovery. In this work, we introduce a fully quantum version of the inflation technique for causal inference, which leverages the quantum marginal problem. We illustrate the utility of this method using a simple example: testing compatibility of tripartite quantum states with the quantum network known as the triangle scenario. We show, in particular, how the method yields a complete classification of pure three-qubit states into those that are and those that are not compatible with the triangle scenario. We also provide some illustrative examples involving mixed states and some where one or more of the systems is higher-dimensional. Finally, we examine the question of when the incompatibility of a multipartite quantum state with a causal structure can be inferred from the incompatibility of a joint probability distribution induced by implementing measurements on each subsystem.
... The causal structure is depicted using a directed acyclic graph (DAG), which can include both observed and latent (unobserved) nodes. Recently, the quantum information community has shown interest in this field, recognizing it as a highly precise framework for understanding and studying Bell nonlocality [2]. Physicists have innovated by allowing latent nodes to represent quantum systems. ...
... A natural question for future research would be to understand if other qualitatively distinct differences between correlation sets -that is, beyond just monogamy of nonlocality -can also be attributed to the role of intermediate latents. 2 We believe that all 64 variants of the quantum tetrahedron scenario are observationally distinct. ...
The use of graphical models to represent causal hypotheses has enabled revolutionary progress in the study of the foundations of quantum theory. Here we consider directed acyclic graphs each of which contains both nodes representing observed variables as well as nodes representing latent or hidden variables. When comparing distinct causal structure, a natural question to ask is if they can explain distinct sets of observable distributions or not. Statisticians have developed a great variety of tools for resolving such questions under the assumption that latent nodes be interpreted classically. Here we highlight how the change to a quantum interpretation of the latent nodes induces distinctions between causal scenarios that would be classically indistinguishable. We especially concentrate on quantum scenarios containing latent nodes with at least one latent parent, a.k.a. possessing intermediate latents. This initial survey demonstrates that many such quantum processes can be operationally distinguished by considerations related to monogamy of nonlocality, especially when computationally aided by a hierarchy of semidefinite relaxations which we tailor for the study such scenarios. We conclude by clarifying the challenges that prevent the generalization of this work, calling attention to open problems regarding observational (in)equivalence of quantum causal structures with intermediate latents.
... measurement in quantum mechanics (the "measurement problem") [52][53][54][55][56][57]: When is the value for the outcome of the measurement determined? The question is also about nonlocality [6,58,59]. Does the measurement at B cause the outcome forp A at A, or was that outcome determined prior? ...
The Einstein-Podolsky-Rosen (EPR) paradox was presented as an argument that quantum mechanics is an incomplete description of physical reality. However, the premises on which the argument is based are falsifiable by Bell experiments. In this paper, we examine the EPR paradox from the perspective of Schrodinger's reply to EPR. Schrodinger pointed out that the correlated states of the paradox enable the simultaneous measurement of and , one by direct, the other by indirect measurement. Schrodinger's analysis takes on a timely importance because a recent experiment realizes these correlations for macroscopic atomic systems. Different to the original argument, Schrodinger's analysis applies to the experiment at the time when the measurement settings have been fixed. In this context, a subset of local realistic assumptions (not negated by Bell's theorem) implies that x and p are simultaneously precisely defined. Hence, an alternative EPR argument can be presented that quantum mechanics is incomplete, based on a set of (arguably) nonfalsifiable premises. As systems are amplified, macroscopic realism can be invoked, and the premises are referred to as weak macroscopic realism (wMR). In this paper, we propose a realization of Schrodinger's gedanken experiment where field quadrature phase amplitudes and replace position and momentum. Assuming wMR, we derive a criterion for the incompleteness of quantum mechanics, showing that the criterion is feasible for current experiments. Questions raised by Schrodinger are resolved. By performing simulations based on an objective-field (Q-based) model for quantum mechanics, we illustrate the emergence on amplification of simultaneous predetermined values for and . The values can be regarded as weak elements of reality, along the lines of Bell's macroscopic beables.
... If one restricts attention to scenarios where the parties discard their output systems after making their local interventions (i.e., they do not communicate), antinomicity formally reduces to Bell nonlocality. 1 Intuitively, antinomicity captures the fact that some correlations can be so strong that any classical causal explanation of them-even one invoking cyclic causality [5]-would necessarily hit a roadblock: Namely, the underlying classical physics must entertain time-travel antinomies [9] that are hidden at the operational level via a statistical fine-tuning [10]. This is analogous to how some nonsignaling correlations can be so strong that any classical causal explanation of them must-under the assumption of free local interventions 2 -either allow a party to causally influence its own past ('retrocausality') [13] or it must invoke hidden superluminal causal influences ('nonlocality') [10]. ...
... 1 Intuitively, antinomicity captures the fact that some correlations can be so strong that any classical causal explanation of them-even one invoking cyclic causality [5]-would necessarily hit a roadblock: Namely, the underlying classical physics must entertain time-travel antinomies [9] that are hidden at the operational level via a statistical fine-tuning [10]. This is analogous to how some nonsignaling correlations can be so strong that any classical causal explanation of them must-under the assumption of free local interventions 2 -either allow a party to causally influence its own past ('retrocausality') [13] or it must invoke hidden superluminal causal influences ('nonlocality') [10]. Such influences, nevertheless, are not observed operationally because of fine-tuning [10]. ...
... This is analogous to how some nonsignaling correlations can be so strong that any classical causal explanation of them must-under the assumption of free local interventions 2 -either allow a party to causally influence its own past ('retrocausality') [13] or it must invoke hidden superluminal causal influences ('nonlocality') [10]. Such influences, nevertheless, are not observed operationally because of fine-tuning [10]. ...
Bell scenarios are multipartite scenarios that exclude any communication between parties. This constraint leads to a strict hierarchy of correlation sets in such scenarios, namely, classical, quantum, and nonsignaling. However, without any constraints on communication between the parties, they can realize arbitrary correlations by exchanging only classical systems. Here we consider a multipartite scenario where the parties can engage in at most a single round of communication, i.e., each party is allowed to receive a system once, implement any local intervention on it, and send out the resulting system once. While no global assumption about causal relations between parties is assumed in this scenario, we do make a causal assumption local to each party, i.e., the input received by it causally precedes the output it sends out. We then introduce antinomicity, a notion of nonclassicality for correlations in such scenarios, and prove the existence of a strict hierarchy of correlation sets classified by their antinomicity. Antinomicity serves as a generalization of Bell nonlocality: when all the parties discard their output systems (i.e., in a nonsignaling scenario), it is mathematically equivalent to Bell nonlocality. Like Bell nonlocality, it can be understood as an instance of fine-tuning, one that is necessary in any classical model of cyclic causation that avoids time-travel antinomies but allows antinomic correlations. Furthermore, antinomicity resolves a long-standing puzzle, i.e., the failure of causal inequality violations as device-independent witnesses of nonclassicality. Antinomicity implies causal inequality violations, but not conversely.
... The study of fine-tuning is critical for assessing the sensitivity of causal models, which gains deeper insights into the nature of physical processes. In causal models that describe quantum systems, fine-tuning is unavoidable to reproduce statistical independencies, such as those in nonlocal correlations (Wood and Spekkens, 2015). It highlights the limitations of causal models in explaining quantum behavior. ...
... This implies that probabilistic classical processes need carefully adjusted weights, since even small changes can lead to invalid measurements. As highlighted by previous work (Wood and Spekkens, 2015), such processes lack the stability needed for reliable realization, thereby limiting their applicability in practical scenarios. ...
Causality is one of the most fundamental notions in physics. Generalized probabilistic theories (GPTs) and the process matrix framework incorporate it in different forms. However, a direct connection between these frameworks remains unexplored. By demonstrating the duality between no-signaling principle and classical processes in tripartite classical systems, and extending some results to multipartite systems, we first establish a strong link between these two frameworks, which are two sides of the same coin. This provides an axiomatic approach to describe the measurement space within both box world and local theories. Furthermore, we describe a logically consistent 4-partite classical process acting as an extension of the quantum switch. By incorporating more than two control states, it allows both parallel and serial application of operations. We also provide a device-independent certification of its quantum variant in the form of an inequality.
