We derive new tight bipartite Bell inequalities for various scenarios. A bipartite Bell scenario $(X,Y,A,B)$ is defined by the numbers of settings and outcomes per party, $X$, $A$ and $Y$, $B$ for Alice and Bob, respectively. We derive the complete set of facets of the local polytopes of $(6,3,2,2)$, $(3,3,3,2)$, $(3,2,3,3)$, and $(2,2,3,5)$. We provide extensive lists of facets for $(2,2,4,4)$, $(3,3,4,2)$ and $(4,3,3,2)$. For each inequality we compute the maximum quantum violation, the resistance to noise, and the minimal symmetric detection efficiency required to close the detection loophole, for qubits, qutrits and ququarts. Based on these results, we identify scenarios which perform better in terms of visibility, resistance to noise, or both, when compared to CHSH. Such scenarios could find important applications in quantum communication.
Relatively few families of Bell inequalities have previously been identified. Some examples are the trivial, CHSH, I_{mm22}, and CGLMP inequalities. This paper presents a large number of new families of tight Bell inequalities for the case of many observables. For example, 44,368,793 inequivalent tight Bell inequalities other than CHSH are obtained for the case of 2 parties each with 10 2-valued observables. This is accomplished by first establishing a relationship between the Bell inequalities and the facets of the cut polytope, a well studied object in polyhedral combinatorics. We then prove…
The correlations that admit a local hidden-variable model are described by a family of polytopes, whose facets are the Bell inequalities. The CHSH inequality is the simplest such Bell inequality and is a facet of every Bell polytope. We investigate for which Bell polytopes the CHSH inequality is also the unique (non-trivial) facet. We prove that the CHSH inequality is the unique facet for all bipartite polytopes where at least one party has a binary choice of dichotomic measurements, irrespective of the number of measurement settings and outcomes for the other party. Based on numerical results…
Bell showed 50 years ago that quantum theory is nonlocal via his celebrated inequalities, turning the issue of quantum nonlocality from a matter of taste into a matter of test. Years later, Hardy proposed a test for nonlocality without inequality, which is a kind of "something-versus-nothing" argument. Hardy's test for $n$ particles induces an $n$-partite Bell's inequality with two dichotomic local measurements for each observer, which has been shown to be violated by all entangled pure states. Our first result is to show that the Bell-Hardy inequality arising form Hardy's nonlocality test is …
The characterization of the set of quantum correlations is a problem of fundamental importance in quantum information. The question whether every proper (tight) Bell inequality is violated in Quantum theory is an intriguing one in this regard. Here, we make significant progress in answering this question, by showing that every tight Bell inequality is violated by 'Almost Quantum' correlations, a semi-definite programming relaxation of the set of quantum correlations. As a consequence, we show that many (classes of) Bell inequalities including two-party correlation Bell inequalities and multi-o…
We establish a relation between the two-party Bell inequalities for two-valued measurements and a high-dimensional convex polytope called the cut polytope in polyhedral combinatorics. Using this relation, we propose a method, triangular elimination, to derive tight Bell inequalities from facets of the cut polytope. This method gives two hundred million inequivalent tight Bell inequalities from currently known results on the cut polytope. In addition, this method gives general formulas which represent families of infinitely many Bell inequalities. These results can be used to examine general pr…
A common problem in Bell type experiments is the well-known detection loophole: if the detection efficiencies are not perfect and if one simply post-selects the conclusive events, one might observe a violation of a Bell inequality, even though a local model could have explained the experimental results. In this paper, we analyze the set of all post-selected correlations that can be explained by a local model, and show that it forms a polytope, larger than the Bell local polytope. We characterize the facets of this post-selected local polytope in the CHSH scenario, where two parties have binary…
We explore the polytope structures for genuine entanglement, biseparability, full biseparability and Bell inequality of multi-qubit GHZ diagonal states. We first show that biseparable GHZ diagonal states make hypersimplices inside the simplices consisting of all GHZ diagonal states. Next, we consider full biseparability which is equivalent to positive partial transpose for GHZ diagonal states, and show that they make the convex hulls of simplices and cubes. We also visualize which part of the simplex violates multipartite Bell inequality. Finally, we compute precise volumes for genuine entangl…
A Bell inequality defined for a specific experimental configuration can always be extended to a situation involving more observers, measurement settings or measurement outcomes. In this article, such "liftings" of Bell inequalities are studied. It is shown that if the original inequality defines a facet of the polytope of local joint outcome probabilities then the lifted one also defines a facet of the more complex polytope.
In the well-studied (2,2,2) Bell experiment consisting of two parties, two measurement settings per party, and two possible outcomes per setting, it is known that if the experiment obeys no-signaling constraints, then the set of admissible experimental probability distributions is fully characterized as the convex hull of 24 distributions: 8 Popescu-Rohrlich (PR) boxes and 16 local deterministic distributions. Here, we refine this result to show that in the (2,2,2) case, any nonlocal nonsignaling distribution can always be uniquely expressed as a convex combination of exactly one PR box and (u…
We present a detailed investigation of minimum detection efficiencies, below which locality cannot be violated by any quantum system of any dimension in bipartite Bell experiments. Lower bounds on these minimum detection efficiencies are determined with the help of linear programming techniques. Our approach is based on the observation that any possible bipartite quantum correlation originating from a quantum state in an arbitrary dimensional Hilbert space is sandwiched between two probability polytopes, namely the local (Bell) polytope and a corresponding nonlocal no-signaling polytope. Numer…
The no-signaling polytope associated to a Bell scenario with three parties, two inputs, and two outputs is found to have 53856 extremal points, belonging to 46 inequivalent classes. We provide a classification of these points according to various definitions of multipartite non-locality and briefly discuss other issues like the interconversion between extremal points seen as a resource and the relation of the extremal points to Bell-type inequalities.
Finding all Bell inequalities for a given number of parties, measurement settings, and measurement outcomes is in general a computationally hard task. We show that all Bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full Bell polytope. As an illustration of our method, we generate 238885 new Bell inequalities and 1085 new Svetlichny inequalities. We find, in particular, facet inequalities for Bell experiments involving two parties and two measurement settings that are not of the Collins-Gisin-Linden-…
Classical Pitowsky correlation polytopes are reviewed with particular emphasis on the Minkowski-Weyl representation theorem. The inequalities representing the faces of polytopes are Boole's ``conditions of possible experience.'' Many of these inequalities have been discussed in the context of Bell's inequalities. We introduce CddIF, a Mathematica package created as an interface between Mathematica and the cdd program by Komei Fukuda, which represents a highly efficient method to solve the hull problem for general classical correlation polytopes.
We study the faces of the set of quantum correlations, i.e., the Bell and noncontextuality inequalities without any quantum violation. First, we investigate the question whether every proper (tight) Bell inequality for two parties, other than the trivial ones from positivity, normalization and no-signaling can be violated by quantum correlations, i.e., whether the classical Bell polytope or the smaller correlation polytope share any facets with their respective quantum sets. To do this, we develop a recently derived bound on the quantum value of linear games based on the norms of game matrices…
In this paper we consider the possible correlations between two parties using local machines and shared randomness with an additional amount of classical communication. This is a continuation of the work initiated by Bacon and Toner in Ref. [\textit{Phys. Rev. Lett.} \textbf{90}, 157904 (2003)] who characterized the correlation polytope for $2\times 2$ measurement settings with binary outcomes plus one bit of communication. Here, we derive a complete set of Bell Inequalities for $3\times 2$ measurement settings and a shared bit of communication. When the communication direction is fixed, nine …
The Grishukhin inequality Gr_7 is a facet of CutP_7, the cut polytope on seven points, which is ``sporadic'' in the sense that its proper generalization has not been known. In this paper, we extend Gr_7 to an inequality I(G,H) valid for CutP_{n+1} where G and H are graphs with n nodes satisfying certain conditions, and prove a necessary and sufficient condition for I(G,H) to be a facet. This result combined with the triangular elimination theorem of Avis, Imai, Ito and Sasaki settles Collins and Gisin's conjecture in quantum theory affirmatively: the I_{mm22} Bell inequality is a facet of the …
The Bell and Leggett-Garg tests offer operational ways to demonstrate that non-classical behavior manifests itself in quantum systems, and experimentalists have implemented these protocols to show that classical worldviews such as local realism and macrorealism are false, respectively. Previous theoretical research has exposed important connections between more general Bell inequalities and polyhedral combinatorics. We show here that general Leggett-Garg inequalities are closely related to the cut polytope of the complete graph, a geometric object well-studied in combinatorics. Building on tha…
Three extensions and reinterpretations of nonclassical probabilities are reviewed. (i) We propose to generalize the probability axiom of quantum mechanics to self-adjoint positive operators of trace one. Furthermore, we discuss the Cartesian and polar decomposition of arbitrary normal operators and the possibility to operationalize the corresponding observables. Thereby we review and emphasize the use of observables which maximally represent the context. (ii) In the second part, we discuss Pitowsky polytopes for automaton logic as well as for generalized urn models and evaluate methods to find…
Instituto de Telecomunicações（葡萄牙电信研究所）；Departamento de Engenharia Electrotécnica e de Computadores, Instituto Superior Técnico（里斯本技术学院电气与计算机工程系）；Group of Applied Physics, University of Geneva（日内瓦大学应用物理小组）；Constructor University（Constructor大学）
The Bell inequalities stand at the cornerstone of the developments of quantum theory on both the foundational and applied side. The discussion started as a way to test whether the quantum description of reality is complete or not, but it developed in such a way that a new research area stemmed from it, namely quantum information. Far from being and exhausted topic, in the present paper we present a constructive and geometrically intuitive description of the local polytope and its facets in a bipartite Bell scenario with two dichotomic measurements per party.
Bell correlation inequalities for two sites and 2+n or 3+3 two-way measurements ("dichotomic observables") are considered. In the 2+n case, any facet of the classical experience polytope is defined by a CHSH inequality involving only two pairs of the observables. In the 3+3 case, contrary to earlier results, the action of the symmetry group reduces the set of all Bell inequalities to just 3 orbits, only one of them being "new" (not known from the 2+2 case). A detailed calculation for the singlet state of two qubits reveals the configurations of a maximal violation for this class of inequalitie…
Tight Bell inequalities are facets of Pitowsky's correlation polytope and are usually obtained from its extreme points by solving the hull problem. Here we present an alternative method based on a combination of algebraic results on extensions of measures and variable elimination methods, e.g., the Fourier-Motzkin method. Our method is shown to overcome some of the computational difficulties associated with the hull problem in some non-trivial cases. Moreover, it provides an explanation for the arising of only a finite number of families of Bell inequalities in measurement scenarios where one …
In this paper we prove that the inequality introduced by Collins, Gisin, Linden, Massar and Popescu is tight, or in other words, it is a facet of the convex polytope generated by all local-realistic joint probabilities of d outcomes. This means that this inequality is optimal. We also show that, for correlation functions generalized to deal with three-outcome measurements, the satisfyability of this inequality is a necessary and sufficient condition for the existence of a local-realistic model accounting for them.
We present Bell inequalities for graph states with high violation of local realism. In particular, we show that there is a two-setting Bell inequality for every nontrivial graph state which is violated by the state at least by a factor of two. These inequalities are facets of the convex polytope containing the many-body correlations consistent with local hidden variable models. We first present a method which assigns a Bell inequality for each graph vertex. Then for some families of graph states composite Bell inequalities can be constructed with a violation of local realism increasing exponen…
For the Bell scenario with two parties and two binary observables per party, it is known that the no-signaling polytope is the polyhedral dual (polar) of the Bell polytope. Computational evidence suggests that this duality also holds for three parties. Using ideas of Werner, Wolf, Żukowski and Brukner, we prove this for any number of parties by describing a simple linear bijection mapping (tight) Bell inequalities to (extremal) no-signaling boxes and vice versa. Furthermore, a symmetry-based technique for extending Bell inequalities (resp. no-signaling boxes) with two binary observables from n…
The famous Clauser-Horne-Shimony-Holt (CHSH) inequality certifies a quantum violation, by a factor $\sqrt{2}$, of correlations predicted by the classical view of the world in the simplest possible nontrivial measurement setup (two systems with two dichotomic measurements each). In such setting, this is the largest possible violation, which is known as the \emph{Tsirelson bound}. In this paper we calculate the exact values of quantum violations for the other Bell correlation inequalities that appear in the setups involving up to four measurements; they are all smaller than $\sqrt{2}$. While var…
We present a family of Bell inequalities for three parties and arbitrarily many outcomes, which can be seen as a natural generalization of the Mermin Bell inequality. For a small number of outcomes, we verify that our inequalities define facets of the polytope of local correlations. We investigate the quantum violations of these inequalities, in particular with respect to the Hilbert space dimension. We provide strong evidence that the maximal quantum violation can only be reached using systems with local Hilbert space dimension exceeding the number of measurement outcomes. This suggests that …
Bipartite and tripartite EPR-Bell type systems are examined via joint quasi-probability distributions where elementary probabilities are permitted to be negative. It is shown that such distributions exist only when the no-signalling condition is satisfied. A characteristic measure, the probability mass, is introduced and, via its minimization, limits the number of quasi-distributions describing a given marginal probability distribution. The minimized probability mass is shown to be an alternative way to characterize non-local systems. Non-signalling polytopes for two, three, and four settings …
We show that some two-party Bell inequalities with two-valued observables are stronger than the CHSH inequality for 3 \otimes 3 isotropic states in the sense that they are violated by some isotropic states in the 3 \otimes 3 system that do not violate the CHSH inequality. These Bell inequalities are obtained by applying triangular elimination to the list of known facet inequalities of the cut polytope on nine points. This gives a partial solution to an open problem posed by Collins and Gisin. The results of numerical optimization suggest that they are candidates for being stronger than the I_3…
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 b…
Cyclic systems of dichotomous random variables have played a prominent role in contextuality research, describing such experimental paradigms as the Klyachko-Can-Binicoglu-Shumovky, Einstein-Podolsky-Rosen-Bell, and Leggett-Garg ones in physics, as well as conjoint binary choices in human decision making. Here, we understand contextuality within the framework of the Contextuality-by-Default (CbD) theory, based on the notion of probabilistic couplings satisfying certain constraints. CbD allows us to drop the commonly made assumption that systems of random variables are consistently connected. C…
Facet inequalities play an important role in detecting the nonlocality of a quantum state. The number of such inequalities depends on the Bell test scenario. With the increase in the number of parties, measurement outcomes, or/and the number of measurement settings, there are more nontrivial facet inequalities. For several Bell scenarios, by involving two dichotomic measurement settings for two parties and one dichotomic measurement by other parties, we show that the local polytope has only one nontrivial facet. For three parties, we have three variants of this inequality, depending upon which…
The problem of computing the local hidden variable (LHV) value of a Bell inequality plays a central role in the study of quantum nonlocality. In particular, this problem is the first step towards characterizing the LHV polytope of a given scenario. In this work, we establish a relation between the LHV value of bipartite Bell inequalities and the mathematical notion of excess of a matrix. Inspired by the well developed theory of excess, we derive several results that directly impact the field of quantum nonlocality. We show infinite families of bipartite Bell inequalities for which the LHV valu…
Quantum Nonlocality and Device-Independent Randomness are Robust to Noisy Signaling Channels
Inria（法国国家信息与自动化研究所）；Univ. Grenoble Alpes, CNRS（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心）；University of Bristol（布里斯托大学）；University of York（约克大学）；ENS de Lyon, UCBL, LIP（里昂高等师范学院，里昂第一大学，计算机科学实验室）
Bell inequalities are derived for any number of observers, any number of alternative setups for each one of them, and any number of distinct outcomes for each experiment. It is shown that if a physical system consists of several distant subsystems, and if the results of tests performed on the latter are determined by local variables with objective values, then the joint probabilities for triggering any given set of distant detectors are convex combinations of a finite number of Boolean arrays, whose components are either 0 or 1 according to a simple rule. This convexity property is both necess…
Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\text{MP}_β$ obtained from the Mermin scenario, parametrized by a function $β$ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\text{MP}_0$ and $\text{MP}_1$ depending on the parity of $β$. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of $\text{MP}_0$ turn out to be deterministic. This result provides a new topological p…
We explore correlation polytopes to derive a set of all Boole-Bell type conditions of possible classical experience which are both maximal and complete. These are compared with the respective quantum expressions for the Greenberger-Horne-Zeilinger (GHZ) case and for two particles with spin state measurements along three directions.
The use of Bell's theorem in any application or experiment relies on the assumption of free choice or, more precisely, measurement independence, meaning that the measurements can be chosen freely. Here, we prove that even in the simplest Bell test - one involving 2 parties each performing 2 binary-outcome measurements - an arbitrarily small amount of measurement independence is sufficient to manifest quantum nonlocality. To this end, we introduce the notion of measurement dependent locality and show that the corresponding correlations form a convex polytope. These correlations can thus be char…
We introduce the measures, Bell discord (BD) and Mermin discord (MD), to characterize bipartite quantum correlations in the context of nonsignaling (NS) polytopes. These measures divide the full NS polytope into four regions depending on whether BD and/or MD is zero. This division of the NS polytope allows us to obtain a 3-decomposition that any bipartite box with two binary inputs and two binary outputs can be decomposed into Popescu-Rohrlich (PR) box, a maximally local box, and a local box with BD and MD equal to zero. BD and MD quantify two types of nonclassicality of correlations arising f…
We give a multidimensional generalisation of the complete set of Bell-correlation inequalities given by Werner and Wolf, and by Zukowski and Brukner, for the two-dimensional case. Our construction applies for the n parties, two-observables case, where each observable is d-valued. The d^{d^n} inequalities obtained involve homogeneous polynomials. They define the facets of a polytope in a complex vector space of dimension d^n. We also show that these inequalities are violated by Quantum Mechanics. We exhibit examples in the three-dimensional case.
In this work, we study a recently proposed operational measure of nonlocality by Fonseca and Parisio~[Phys. Rev. A 92, 030101(R) (2015)] which describes the probability of violation of local realism under randomly sampled observables, and the strength of such violation as described by resistance to white noise admixture. While our knowledge concerning these quantities is well established from a theoretical point of view, the experimental counterpart is a considerably harder task and very little has been done in this field. It is caused by the lack of complete knowledge about the facets of the …
The demonstration and use of Bell-nonlocality, a concept that is fundamentally striking and is at the core of applications in device independent quantum information processing, relies heavily on the assumption of measurement independence, also called the assumption of free choice. The latter cannot be verified or guaranteed. In this paper, we consider a relaxation of the measurement independence assumption. We briefly review the results of Phys. Rev. Lett. 113, 190402 (2014), which show that with our relaxation, the set of so-called measurement dependent local (MDL) correlations is a polytope,…
The class of d-setting, d-outcome Bell inequalities proposed by Ji and collaborators [Phys. Rev. A 78, 052103] are reexamined. For every positive integer d > 2, we show that the corresponding non-trivial Bell inequality for probabilities provides the maximum classical winning probability of the Clauser-Horne-Shimony-Holt-like game with d inputs and d outputs. We also demonstrate that the general classical upper bounds given by Ji et al. are underestimated, which invalidates many of the corresponding correlation inequalities presented thereof. We remedy this problem, partially, by providing the…
Tests of local realism and macrorealism have historically been discussed in very similar terms: Leggett-Garg inequalities follow Bell inequalities as necessary conditions for classical behavior. Here, we compare the probability polytopes spanned by all measurable probability distributions for both scenarios and show that their structure differs strongly between spatially and temporally separated measurements. We arrive at the conclusion that, in contrast to tests of local realism where Bell inequalities form a necessary and sufficient set of conditions, no set of inequalities can ever be neces…
When conducting a Bell test, it is normal to assume that the preparation of the quantum state is independent of the measurements performed on it. Remarkably, the violation of local realism by entangled quantum systems can be certified even if this assumption is partially relaxed. Here, we allow such measurement dependence to correlate multiple runs of the experiment, going beyond previous studies that considered independent and identically distributed (i.i.d.) runs. To do so, we study the polytope that defines block-i.i.d. measurement-dependent local models. We prove that non-i.i.d. models are…
