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Universal bound on the cardinality of local hidden variables in networks
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Abstract
We present an algebraic description of the sets of local correlations in arbitrary networks, when the parties have finite inputs and outputs. We consider networks generalizing the usual Bell scenarios by the presence of multiple uncorrelated sources. We prove a finite upper bound on the cardinality of the value sets of the local hidden variables. Consequently, we find that the sets of local correlations are connected, closed and semialgebraic, and bounded by tight polynomial Bell-like inequalities.
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We provide a scheme for inferring causal relations from uncontrolled statistical data based on tools from computational algebraic geometry, in particular, the computation of Groebner bases. We focus on causal structures containing just two observed variables, each of which is binary. We consider the consequences of imposing different restrictions on the number and cardinality of latent variables and of assuming different functional dependences of the observed variables on the latent ones (in particular, the noise need not be additive). We provide an inductive scheme for classifying functional causal structures into distinct observational equivalence classes. For each observational equivalence class, we provide a procedure for deriving constraints on the joint distribution that are necessary and sufficient conditions for it to arise from a model in that class. We also demonstrate how this sort of approach provides a means of determining which causal parameters are identifiable and how to solve for these. Prospects for expanding the scope of our scheme, in particular to the problem of quantum causal inference, are also discussed.
Quantum correlations arising in Bell experiments, involving a physical source
that emits a quantum state to a number of observers, have been intensively
studied over the last decades. Much less is known about the nature of quantum
correlations arising in network structures beyond the Bell experiments. Such
networks can involve many independent sources emitting states to observers in
accordance with the network configuration. Here, we will study classical and
quantum correlations in a family of networks which can be regarded as
compositions of several independent multipartite Bell experiments connected
together through a central node. For such networks we present tight Bell-type
inequalities which are satisfied by all classical correlations. We study
properties of the violations of our inequalities by probability distributions
arising in quantum theory.
In a quantum network, distant observers sharing physical resources emitted by
independent sources can establish strong correlations, which defy any classical
explanation in terms of local variables. We discuss the characterization of
nonlocal correlations in such a situation, when compared to those that can be
generated in networks distributing independent local variables. We present an
iterative procedure for constructing Bell inequalities tailored for networks:
starting from a given network, and a corresponding Bell inequality, our
technique provides new Bell inequalities for a more complex network, involving
one additional source and one additional observer. The relevance of our method
is illustrated on a variety of networks, for which we demonstrate significant
quantum violations.
It is a recent realization that many of the concepts and tools of causal
discovery in machine learning are highly relevant to problems in quantum
information, in particular quantum nonlocality. The crucial ingredient in the
connection between both fields is the tool of Bayesian networks, a graphical
model used to reason about probabilistic causation. Indeed, Bell's theorem
concerns a particular kind of a Bayesian network and Bell inequalities are a
special case of linear constraints following from such models. It is thus
natural to look for generalized Bell scenarios involving more complex Bayesian
networks. The problem, however, relies on the fact that such generalized
scenarios are characterized by polynomial Bell inequalities and no current
method is available to derive them beyond very simple cases. In this work, we
make a significant step in that direction, providing a general and practical
method for the derivation of polynomial Bell inequalities in a wide class of
scenarios, applying it to a few cases of interest. We also show how our
construction naturally gives rise to a notion of non-signalling in generalized
networks.
Quantum theory is known to be nonlocal in the sense that separated parties can perform measurements on a shared quantum state to obtain correlated probability distributions, which cannot be achieved if the parties share only classical randomness. Here we find that the set of distributions compatible with sharing quantum states subject to some sufficiently restricted dimension is neither convex nor a superset of the classical distributions. We examine the relationship between quantum distributions associated with a dimensional constraint and classical distributions associated with limited shared randomness. We prove that quantum correlations are convex for certain finite dimension in certain Bell scenarios and that they sometimes offer a dimensional advantage in realizing local distributions. We also consider if there exist Bell scenarios where the set of quantum correlations is never convex with finite dimensionality.
Bayesian networks provide a powerful tool for reasoning about probabilistic
causation, used in many areas of science. They are, however, intrinsically
classical. In particular, Bayesian networks naturally yield the Bell
inequalities. Inspired by this connection, we generalise the formalism of
classical Bayesian networks in order to investigate non-classical correlations
in arbitrary causal structures. Our framework of `generalised Bayesian
networks' replaces latent variables with the resources of any generalised
probabilistic theory, most importantly quantum theory, but also, for example,
Popescu-Rohrlich boxes. We obtain three main sets of results. Firstly, we prove
that all of the observable conditional independences required by the classical
theory also hold in our generalisation; to obtain this, we extend the classical
d-separation theorem to our setting. Secondly, we find that the
theory-independent constraints on probabilities can go beyond these conditional
independences. For example we find that no probabilistic theory predicts
perfect correlation between three parties using only bipartite common causes.
Finally, we begin a classification of those causal structures, such as the Bell
scenario, that may yield a separation between classical, quantum and
general-probabilistic correlations.
The fields of quantum non-locality in physics, and causal discovery in
machine learning, both face the problem of deciding whether observed data is
compatible with a presumed causal relationship between the variables (for
example a local hidden variable model). Traditionally, Bell inequalities have
been used to describe the restrictions imposed by causal structures on marginal
distributions. However, some structures give rise to non-convex constraints on
the accessible data, and it has recently been noted that linear inequalities on
the observable entropies capture these situations more naturally. In this
paper, we show the versatility of the entropic approach by greatly expanding
the set of scenarios for which entropic constraints are known. For the first
time, we treat Bell scenarios involving multiple parties and multiple
observables per party. Going beyond the usual Bell setup, we exhibit
inequalities for scenarios with extra conditional independence assumptions, as
well as a limited amount of shared randomness between the parties. Many of our
results are based on a geometric observation: Bell polytopes for two-outcome
measurements can be naturally imbedded into the convex cone of attainable
marginal entropies. Thus, any entropic inequality can be translated into one
valid for probabilities. In some situations the converse also holds, which
provides us with a rich source of candidate entropic inequalities.
Bell's Theorem witnesses that the predictions of quantum theory cannot be
reproduced by theories of local hidden variables in which observers can choose
their measurements independently of the source. Working out an idea of
Branciard, Rosset, Gisin and Pironio, we consider scenarios which feature
several sources, but no choice of measurement for the observers. Every Bell
scenario can be mapped into such a \emph{correlation scenario}, and Bell's
Theorem then discards those local hidden variable theories in which the sources
are independent. However, most correlation scenarios do not arise from Bell
scenarios, and we describe examples of (quantum) nonlocality in some of these
scenarios, while posing many open problems along the way. Some of our scenarios
have been considered before by mathematicians in the context of causal
inference.
For any Bell locality scenario (or Kochen-Specker noncontextuality scenario),
the joint Shannon entropies of local (or noncontextual) models define a convex
cone for which the non-trivial facets are tight entropic Bell (or
contextuality) inequalities. In this paper we explore this entropic approach
and derive tight entropic inequalities for various scenarios. One advantage of
entropic inequalities is that they easily adapt to situations like bilocality
scenarios, which have additional independence requirements that are non-linear
on the level of probabilities, but linear on the level of entropies. Another
advantage is that, despite the nonlinearity, taking detection inefficiencies
into account turns out to be very simple. When joint measurements are conducted
by a single detector only, the detector efficiency for witnessing quantum
contextuality can be arbitrarily low.
In this paper we give sufficient conditions for a compactum in
to have Carath\'{e}odory number less than n+1, generalizing an old result of
Fenchel. Then we prove the corresponding versions of the colorful
Carath\'{e}odory theorem and give a Tverberg type theorem for families of
convex compacta.
A directed acyclic graph (DAG) partially represents the conditional independence structure among observations of a system if the local Markov condition holds, that is if every variable is independent of its non-descendants given its parents. In general, there is a whole class of DAGs that represents a given set of conditional independence relations. We are interested in properties of this class that can be derived from observations of a subsystem only. To this end, we prove an information-theoretic inequality that allows for the inference of common ancestors of observed parts in any DAG representing some unknown larger system. More explicitly, we show that a large amount of dependence in terms of mutual information among the observations implies the existence of a common ancestor that distributes this information. Within the causal interpretation of DAGs, our result can be seen as a quantitative extension of Reichenbach’s principle of common cause to more than two variables. Our conclusions are valid also for non-probabilistic observations, such as binary strings, since we state the proof for an axiomatized notion of “mutual information” that includes the stochastic as well as the algorithmic version.
Tests of local realism versus quantum mechanics based on Bell's inequality employ two entangled qubits. We investigate the general case of two entangled quantum systems defined in N-dimensional Hilbert spaces, or " quNits." Via a numerical linear optimization method we show that violations of local realism are stronger for two maximally entangled quNits ( 3</=N</=9) than for two qubits and that they increase with N. The two quNit measurements can be experimentally realized using entangled photons and unbiased multiport beam splitters.
We provide a classification of graphical models according to their representation as exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical (DAG) models and chain graphs with no hidden variables, including DAG models with several families of local distributions, are curved exponential families (CEFs) and graphical models with hidden variables are stratified exponential families (SEFs). A SEF is a finite union of CEFs of various dimensions satisfying some regularity conditions. The main results of this paper are that graphical models are SEFs and that many graphical models are not CEFs. That is, roughly speaking, graphical models when viewed as exponential families correspond to a set of smooth manifolds of various dimensions and usually not to a single smooth manifold. These results are discussed in the context of model selection. Keywords : Bayesian networks, graphical models, hidden variables, cur...
Bell inequalities bound the strength of classical correlations between
observers measuring on a shared physical system. However, studies of physical
correlations can be considered beyond the standard Bell scenario by networks of
observers sharing some configuration of many independent physical systems.
Here, we show how to construct Bell-type inequalities for correlations arising
in any network that is a tree i.e. it does not contain a cycle. This is
achieved by an iteration procedure that in each step allows one to add a branch
to the tree-structured network and construct a corresponding Bell-type
inequality. We explore our inequalities in several examples, in all of which we
demonstrate strong violations from quantum theory.
1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilbert's 17th Problem. Quadratic Forms.- 7. Real Spectrum.- 8. Nash Functions.- 9. Stratifications.- 10. Real Places.- 11. Topology of Real Algebraic Varieties.- 12. Algebraic Vector Bundles.- 13. Polynomial or Regular Mappings with Values in Spheres.- 14. Algebraic Models of C? Manifolds.- 15. Witt Rings in Real Algebraic Geometry.- Index of Notation.
The concept of bilocality was introduced to study the correlations which
arise in an entanglement swapping scenario, where one has two sources which can
naturally taken to be independent. This additional constraint leads to stricter
requirements than simply imposing locality, in the form of bilocality
inequalities. In this work we consider a natural generalisation of the
bilocality scenario, namely the star-network consisting of a single central
party surrounded by n edge parties, each of which shares an independent
source with the centre. We derive new inequalities which are satisfied by all
local correlations in this scenario, for the cases when the central party
performs (i) two dichotomic measurements (ii) a single Bell state measurement.
We demonstrate quantum violations of these inequalities and study both the
robustness to noise and to losses.
Bell's 1964 theorem, which states that the predictions of quantum theory
cannot be accounted for by any local theory, represents one of the most
profound developments in the foundations of physics. In the last two decades,
Bell's theorem has been a central theme of research from a variety of
perspectives, mainly motivated by quantum information science, where the
nonlocality of quantum theory underpins many of the advantages afforded by a
quantum processing of information. The focus of this review is to a large
extent oriented by these later developments. We review the main concepts and
tools which have been developed to describe and study the nonlocality of
quantum theory, and which have raised this topic to the status of a full
sub-field of quantum information science.
Entanglement swapping is a process by which two initially independent quantum systems can become entangled and generate nonlocal correlations. To characterize such correlations, we compare them to those predicted by bilocal models, where systems that are initially independent are described by uncorrelated states. We extend in this paper the analysis of bilocal correlations initiated in [ Phys. Rev. Lett. 104 170401 (2010)]. In particular, we derive new Bell-type inequalities based on the bilocality assumption in different scenarios, we study their possible quantum violations, and we analyze their resistance to experimental imperfections. The bilocality assumption, being stronger than Bell's standard local causality assumption, lowers the requirements for the demonstration of quantumness in entanglement-swapping experiments.
A family of polytopes, correlation polytopes, which arise naturally in the theory of probability and propositional logic, is defined. These polytopes are tightly connected to combinatorial problems in the foundations of quantum mechanics, and to the Ising spin model. Correlation polytopes exhibit a great deal of symmetry. Exponential size symmetry groups, which leave the polytope invariant and act transitively on its vertices, are defined. Using the symmetries, a large family of facets is determined. A conjecture concerning the full facet structure of correlation polytopes is formulated (the conjecture, however, implies that NP=co-NP).
Various complexity results are proved. It is shown that deciding membership in a correlation polytope is an NP-complete problem, and deciding facets is probably not even in NP. The relations between the polytope symmetries and its complexity are indicated.
Quantum systems that have never interacted can become nonlocally correlated through a process called entanglement swapping. To characterize nonlocality in this context, we introduce local models where quantum systems that are initially uncorrelated are described by uncorrelated local variables. This additional assumption leads to stronger tests of nonlocality. We show, in particular, that an entangled pair generated through entanglement swapping will already violate a Bell-type inequality for visibilities as low as 50% under our assumption.
A Bell inequality is derived for a state of n spin-1/2 particles which superposes two macroscopically distinct states. Quantum mechanics violates this inequality by an amount that grows exponentially with n.
A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields.
We computationally investigate the complete polytope of Bell inequalities for 2 particles with small numbers of possible measurements and outcomes. Our approach is limited by Pitowsky's connection of this problem to the computationally hard NP problem. Despite this, we find that there are very few relevant inequivalent inequalities for small numbers. For example, in the case with 3 possible 2-outcome measurements on each particle, there is just one new inequality. We describe mixed 2-qubit states which violate this inequality but not the CHSH. The new inequality also illustrates a sharing of bi-partite non-locality between three qubits: something not seen using the CHSH inequality. It also inspires us to discover a class of Bell inequalities with m possible n-outcome measurements on each particle. Comment: 6 pages, 1 figure, family of m-measurement n-outcome inequalities, and demonstration of non-locality sharing added
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