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Introduction: Objects in Mirror are Closer than They Appear

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Inequality 8 refers to the particular case of two input bits for Alice.

Introduction: Objects in Mirror are Closer than They Appear

As we prove in the Methods section, the following generalization is valid within the quantum theory for any number of input bits:. We further notice that the IC scenario is quite similar to the super-dense coding set-up 21 , where, however, in the latter case the message M is quantum. As proved in the Methods section, this implies that a similar inequality to 10 is a also valid for the super-dense coding scenario if one replaces the Shannon entropy H M by twice the von Neumann entropy 2 H M.

This generalizes the well-known phenomenon of super-dense coding 21 , which states that in the presence of shared entanglement, the exchange of one d -level system allows for one of d 2 -diferent messages to be communicated. Finally, to understand how much more powerful inequality 8 may be as a witness of post-quantum correlations, we perform a similar analysis to the one in ref. We consider the following section of the non-signalling polytope. The results are displayed in Fig. Our new inequality can witness, already on the single-copy level, the postquantumness of distributions that could not be detected before even in the limit of many copies.

A slice of the non-signalling polytope corresponding to the distribution The lower black-dashed line is an upper limit on quantum correlations obtained via the criterion in ref. The solid red, blue and orange curves correspond, respectively, to the boundaries obtained with the IC inequalities 8 , 6 and 7. Above each of the curves, the corresponding inequalities are violated. See Supplementary Note 1 for details of how the curves are computed. Quantum networks are ubiquitous in quantum information.

The basic scenario consists of a collection of entangled states that are distributed among several spatially separated parties to perform some informational task, for example, entanglement percolation 17 , entanglement swapping 44 or distributed computing 15 , A similar set-up is of relevance in classical causal inference, namely the inference of latent common ancestors 14 , As we will show next, just the topology of these quantum networks already imply non-trivial constraints on the correlations that can be obtained between the different parties.

We will consider the particular case where all the parties can be connected by at most bipartite states. We note, however, that our framework applies as well to the most general case and results along this line are presented in the Supplementary Notes 2 and 3. The problem can be restated as follows. Consider n observable variables that may be assumed to have no direct causal influence on each other as they are space-like separated. Given some observed correlations between them, the basic question is then: can the correlations between these n variables be explained by hidden common ancestors each connecting at most two of them?

In the case where the underlying hidden variables are classical for example, separable states , the entropic marginal cone associated with this DAG has been completely characterized in ref. Following the framework delineated before, we can prove that facets of this cone are also obtained if we replace the underlying classical variables by quantum states Supplementary Note 2. This implies that entropically quantum correlations respect the same type of monogamy relations as classical variables.

The natural question is how to generalize this result to more general common ancestor structures for arbitrary n. With this aim, we prove in the Methods section that the monogamy relation.

We also prove in the Supplementary Note 3 that this inequality is valid for general non-signalling theories, generalizing the result obtained in ref. In addition, we exhibit that for any non-trivial common ancestor structure there are entropic constraints even if we allow for general non-signalling theories. The inequality 12 can be seen as a kind of monogamy of correlations. The inequality 12 makes this intuition precise.

In this work, we have introduced a systematic algorithm for computing information—theoretic constraints arising from quantum causal structures. Moreover, we have demonstrated the versatility of the framework by applying it to a set of diverse examples from quantum foundations, quantum communication and the analysis of distributed architectures. In particular, our framework readily allows us to obtain a much stronger version of information causality.

These examples aside, we believe that the main contribution of this work is to highlight the power of systematically analysing entropic marginals. A number of future directions for research immediately suggests themselves. In particular, it will likely be fruitful to consider multipartite versions of information causality or other information—theoretical principles and to further look into the operational meaning of entropy inequality violations.

Given the inequality description of the entropic cone describing a causal structure, to obtain the description of an associated marginal scenario we need to eliminate from the set of inequalities all variables not contained in. After this elimination procedure, we obtain a new set of linear inequalities, constraints that correspond to facets of a convex cone, more precisely the marginal entropic cone characterizing the compatibility region of a certain causal structure 7.

This can be achieved via a FM elimination, a standard linear programming algorithm for eliminating variables from systems of inequalities The problem with the FM elimination is that it is a doubly exponential algorithm in the number of variables to be eliminated. As the number of variables in the causal structure of interest increases, typically this elimination becomes computationally intractable. While it can be computationally very demanding to obtain the full description of a marginal cone, to check whether a given candidate inequality is respected by a causal structure is relatively easy.

Consider that a given causal structure leads to a number N of possible entropies. These are organized in a n -dimensional vector h. In the quantum case, since not all subsets of variables may jointly coexist we will have typically that N is strictly smaller than 2 n. As explained in detail in the main text, for this entropy vector to be compatible with a given causal structure, a set of linear constraints must be fulfilled. Given the entropy vector h , any entropic-linear inequality can be written simply as the inner product , where is the associated vector to the inequality.

That is, to check the validity of a test inequality, one simply needs to solve the following linear programme:. In general, this linear programme only provides a sufficient but not necessary condition for the validity of an inequality. The reason for that is the existence of non-Shannon type inequalities, which are briefly discussed in the Supplementary Note 2. We provide in the following an analytical proof of the validity of the generalized IC inequality 10 for the quantum causal structure in Fig. Further details can be found in the Supplementary Note 1.

This concludes the proof. Note that this proof can be easily adapted to the case where the message M sent from Alice to Bob is a quantum state. In this case there are two differences. First, because the message is disturbed to create the guess Y i , we cannot assign an entropy to M and Y i simultaneously. In the following, we provide an analytical proof of the monogamy inequality 12 in the main text. Further details can be found in the Supplementary Note 2. For a Hilbert space , we denote the set of quantum states, that is, the set of positive semidefinite operators with trace one, on it by.

Theorem 1. Let be a six-partite quantum state on. We have therefore. In the third line of 26 we used strong subadditivity, and in the last line we used that the entropy of a classical state conditioned on a quantum state is positive.

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Structural realism has rapidly gained in popularity in recent years, but it has splintered into many distinct denominations, often underpinned by diverse. Editorial Reviews. From the Back Cover. Structural realism has rapidly gained in popularity in.

This proof can easily be generalized to the case of an arbitrary number of random variables resulting from a classical-quantum Bayesian network in which each parent has at most two children. How to cite this article: Chaves, R. Information—theoretic implications of quantum causal structures.

Pearl, J. Causality Cambridge Univ. Press Spirtes, P. Bell, J. On the Einstein—Podolsky—Rosen paradox. Physics 1 , — Wood, C. The lesson of causal discovery algorithms for quantum correlations: causal explanations of bell-inequality violations require fine-tuning.

Fritz, T. Beyond bell's theorem: correlation scenarios. New J. Entropic inequalities and marginal problems. IEEE Trans. Theory 59 , — Chaves, R.

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Causal structures from entropic information: geometry and novel scenarios. Beyond bell's theorem ii: scenarios with arbitrary causal structure. Henson, J. Theory-independent limits on correlations from generalised bayesian networks.

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Pienaar, J. A graph-separation theorem for quantum causal models. Yeung, R. Information technology—transmission, processing, and storage Springer Entropic approach to local realism and noncontextuality. A 85 , Entropic inequalities as a necessary and sufficient condition to noncontextuality and locality.

A 87 , Inferring latent structures via information inequalities. Van Meter, R.