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Within the field of causal inference, it is desirable to learn the structure of causal relationships holding between a system of variables from the correlations that these variables exhibit; a sub-problem of which is to certify whether or not a given causal hypothesis is compatible with the observed correlations. A particularly challenging setting...
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Quantum Fisher information is the principal tool used to give the ultimate precision bound on the estimation of parameters for quantum channels. In this work, we present analytical expressions for the quantum Fisher information with three noisy channels for the case where the channels are in superposition of causal orders. We found that the quantum...
Heisenberg's uncertainty principle implies fundamental constraints on what properties of a quantum system can we simultaneously learn. However, it typically assumes that we probe these properties via measurements at a single point in time. In contrast, inferring causal dependencies in complex processes often requires interactive experimentation - m...
Citations
... Ref. [2] established a bridge between quantum physicists and the classical causal inference community, paving the way for fresh insights on both fronts. For instance, it underscored the significance of causal compatibility inequalities (as opposed to equalities) for statisticians, and encouraged physicists to explore quantum advantages in more general causal structures [8,9,10,11,12,13] and to contribute to classical causal inference [14,15,16,17,18,19,20]. ...
... Furthermore, the statistical properties of a random subsequence of a random sequence are the same as those of the original sequence. Since the values of (X, Y ) in this protocol are freely chosen, 17 conditioning on the values of (X, Y ) has the effect of randomly picking a subsequence, and thus, we have f (c|xy) ∼ f (c). So, in the limit N → ∞, we have J/d N → 1. ...
... If Veronika obtains |f (c|xy) − f (c)| < ϵ, she writes down her verification result as "pass", and otherwise, she writes down her verification result as "fail". Neither of the results she writes down contain any information about 17 If instead, the values of (X, Y ) are not freely chosen, then J/d N will not approach 1 even when N → ∞ and thus, Veronika may not obtain the 'pass' outcome and can indeed find out that C and (X, Y ) are correlated. ...
Nonclassical causal modeling was developed in order to explain violations of Bell inequalities while adhering to relativistic causal structure and f a i t h f u l n e s s ---that is, avoiding fine-tuned causal explanations. Recently, a no-go theorem that can be viewed as being stronger than Bell's theorem has been derived, based on extensions of the Wigner's friend thought experiment: the Local Friendliness (LF) no-go theorem. Here we show that the LF no-go theorem poses formidable challenges for the field of causal modeling, even when nonclassical and/or cyclic causal explanations are considered. We first recast the LF inequalities, one of the key elements of the LF no-go theorem, as special cases of monogamy relations stemming from a statistical marginal problem. We then further recast LF inequalities as causal compatibility inequalities stemming from a n o n c l a s s i c a l causal marginal problem, for a causal structure implied by well-motivated causal-metaphysical assumptions. We find that the LF inequalities emerge from this causal structure even when one allows the latent causes of observed events to admit post-quantum descriptions, such as in a generalized probabilistic theory or in an even more exotic theory. We further prove that n o nonclassical causal model can explain violations of LF inequalities without violating the No Fine-Tuning principle. Finally, we note that these obstacles cannot be overcome even if one appeals to c y c l i c causal models, and we discuss potential directions for further extensions of the causal modeling framework.
... The availability of inequalities easily derived by reading the original causal structure can also be helpful in combination with the inflation method, in order to discard as many candidate causal structures as possible before the design of additional inflated graphs. The connection with other approaches [69][70][71][72][73][74] also deserves further investigation, ultimately to determine minimal sets of inequality constraints with equivalent inferential power. ...
The causal structure of a system imposes constraints on the joint probability distribution of variables that can be generated by the system. Archetypal constraints consist of conditional independencies between variables. However, particularly in the presence of hidden variables, many causal structures are compatible with the same set of independencies inferred from the marginal distributions of observed variables. Additional constraints allow further testing for the compatibility of data with specific causal structures. An existing family of causally informative inequalities compares the information about a set of target variables contained in a collection of variables, with a sum of the information contained in different groups defined as subsets of that collection. While procedures to identify the form of these groups-decomposition inequalities have been previously derived, we substantially enlarge the applicability of the framework. We derive groups-decomposition inequalities subject to weaker independence conditions, with weaker requirements in the configuration of the groups, and additionally allowing for conditioning sets. Furthermore, we show how constraints with higher inferential power may be derived with collections that include hidden variables, and then converted into testable constraints using data processing inequalities. For this purpose, we apply the standard data processing inequality of conditional mutual information and derive an analogous property for a measure of conditional unique information recently introduced to separate redundant, synergistic, and unique contributions to the information that a set of variables has about a target.
... IV E), the former conditions can be evaluated more efficiently. The latter condition involving supports (remarkably!) can be assessed using Fraser's algorithm [26], which generally requires higher computational overhead. All of these conditions can be utilized to prove the NON-ALGEBRAICNESS of a given DAG, via proving the classical observational inequivalence of said DAG with every latent-free graph, which has the same number of observed nodes. ...
... To exploit Theorem 10 in practice, we need an an algorithm capable of assessing whether or not a given support is compatible with a given DAG. Such algorithm was developed in Ref. [26], and is referred to here as Fraser's algorithm. We have implemented Fraser's algorithm in Python and scripted it to yield all the supports that are classically incompatible with a given DAG (for a certain assignment of the cardinalities of the observed variables). ...
... An example of DAG whose NON-ALGEBRAICNESS was first certified in Ref. [26] via the discovery of an incompatible support is presented in Fig. 10. By means of his eponymous algorithm for compatible supports, Fraser showed that the following support is classically incompatible with the DAG ...
The classical causal relations between a set of variables, some observed and some latent, can induce both equality constraints (typically conditional independencies) as well as inequality constraints (Instrumental and Bell inequalities being prototypical examples) on their compatible distribution over the observed variables. Enumerating a causal structure's implied inequality constraints is generally far more difficult than enumerating its equalities. Furthermore, only inequality constraints ever admit violation by quantum correlations. For both those reasons, it is important to classify causal scenarios into those which impose inequality constraints versus those which do not. Here we develop methods for detecting such scenarios by appealing to d separation, e separation, and incompatible supports. Many (perhaps all?) scenarios with exclusively equality constraints can be detected via a condition articulated by Henson, Lal, and Pusey (HLP). Considering all scenarios with up to four observed variables, which number in the thousands, we are able to resolve all but three causal scenarios, providing evidence that the HLP condition is, in fact, exhaustive.
Published by the American Physical Society 2024
... Although the latter condition subsumes the two former ones (as discussed in Section 4.5), the former conditions can be evaluated more efficiently. The latter condition involving supports (remarkably!) can be assessed using Fraser's algorithm [24], which generally requires higher computational overhead. All of these conditions can be utilized to prove the Non-Algebraicness of a given DAG, via proving the classical observational inequivalence of said DAG with every latent-free graph which has the same number of observed nodes. ...
... To exploit Theorem 40 [NALF Supports] in practice, we need an an algorithm capable of assessing whether or not a given support is compatible with a given DAG. Such algorithm was developed in Ref. [24], and refer to it here as Fraser's algorithm. We have implemented Fraser's algorithm in Python and scripted it to yield all the supports that are classically incompatible with a given DAG (for a certain assignment of the cardinalities of the observed variables). ...
... An example of DAG whose Non-Algebraicness was first certified in Ref. [24] via the discovery of an incompatible support is presented in Figure 10. By means of his eponymous algorithm for compatible supports, Fraser showed that the following support is classically incompatible with the DAG of Figure 1015 : Figure 10: A DAG with 4 observed nodes and 7 total nodes whose Non-Algebraicness can be shown by Fraser's algorithm for compatible supports. ...
The classical causal relations between a set of variables, some observed and some latent, can induce both equality constraints (typically conditional independences) as well as inequality constraints (Instrumental and Bell inequalities being prototypical examples) on their compatible distribution over the observed variables. Enumerating a causal structure's implied inequality constraints is generally far more difficult than enumerating its equalities. Furthermore, only inequality constraints ever admit violation by quantum correlations. For both those reasons, it is important to classify causal scenarios into those which impose inequality constraints versus those which do not. Here we develop methods for detecting such scenarios by appealing to d-separation, e-separation, and incompatible supports. Many (perhaps all?) scenarios with exclusively equality constraints can be detected via a condition articulated by Henson, Lal and Pusey (HLP). Considering all scenarios with up to 4 observed variables, which number in the thousands, we are able to resolve all but three causal scenarios, providing evidence that the HLP condition is, in fact, exhaustive.
... In the MArG projection of G (see Figure 7(b)) it clearly is possible to set these three variables to be equal, since c, d are both children of a. However, in fact this is not possible using the canonical DAG for G, which can be verified by applying the results of Wolfe et al. [2019] or Fraser [2020] (Elie Wolfe, personal communication.) A PROBABILISTIC RESULT Lemma A.1. ...
Directed acyclic graph models with hidden variables have been much studied, particularly in view of their computational efficiency and connection with causal methods. In this paper we provide the circumstances under which it is possible for two variables to be identically equal, while all other observed variables stay jointly independent of them and mutually of each other. We find that this is possible if and only if the two variables are `densely connected'; in other words, if applications of identifiable causal interventions on the graph cannot (non-trivially) separate them. As a consequence of this, we can also allow such pairs of random variables have any bivariate joint distribution that we choose. This has implications for model search, since it suggests that we can reduce to only consider graphs in which densely connected vertices are always joined by an edge.
