Lifting facets of the cut polytope

Simone, Caterina De

doi:10.1016/0167-6377(90)90029-5

Cited by 17 publications

(12 citation statements)

References 5 publications

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“…If a Bell inequality for (m, n) settings is tight, then it is also a tight Bell inequality for (m ′ , n ′ ), where 1 ≤ m ≤ m ′ and 1 ≤ n ≤ n ′ . This can be proved directly [27], or as a corollary [2] of the zero-lifting theorem for the cut polytope of a general graph [11]. Here we prove that the same holds for correlation inequalities.…”

Section: Zero-lifting Of Correlation Inequalitiessupporting

confidence: 53%

See 1 more Smart Citation

On the relationship between convex bodies related to correlation experiments with dichotomic observables

Avis¹,

Imai²,

Ito³

2006

J. Phys. A: Math. Gen.

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In this paper we explore further the connections between convex bodies related to quantum correlation experiments with dichotomic variables and related bodies studied in combinatorial optimization, especially cut polyhedra. Such a relationship was established in Avis, Imai, Ito and Sasaki (2005 J. Phys. A: Math. Gen. 38 10971-87) with respect to Bell inequalities. We show that several well known bodies related to cut polyhedra are equivalent to bodies such as those defined by Tsirelson (1993 Hadronic J. S. 8 329-45) to represent hidden deterministic behaviors, quantum behaviors, and no-signalling behaviors. Among other things, our results allow a unique representation of these bodies, give a necessary condition for vertices of the no-signalling polytope, and give a method for bounding the quantum violation of Bell inequalities by means of a body that contains the set of quantum behaviors. Optimization over this latter body may be performed efficiently by semidefinite programming. In the second part of the paper we apply these results to the study of classical correlation functions. We provide a complete list of tight inequalities for the two party case with (m, n) dichotomic observables when m = 4, n = 4 and when min{m, n} ≤ 3, and give a new general family of correlation inequalities. PACS classificationnumbers: 03.65.Ud, 02.40.Ft, 02.10.Ud Keywords: Bell inequalities, quantum behaviors, the cut polytope, the no-signaling polytope, semidefinite programming 1 The convex sets arising from the classical, quantum and nosignaling correlation experiments Our terminology for bodies related to quantum correlations follows closely that of Tsirelson [33] whilst for cut polyhedra we follow that of Deza and Laurent [12]. We consider the following (quantum) correlation experiment. Suppose that two parties, say Alice and Bob, share a quantum mixed state ρ, or a nonnegative Hermitian operator ρ on a Hilbert space H A ⊗ H B with tr ρ = 1. Here H A and H B are Hilbert spaces representing the subsystems owned by Alice and Bob, respectively. Alice has m ±1-valued observables A 1 , . . . , A m in the space H A , i.e. Hermitian operators A i on H A whose eigenvalues are within [−1, 1]. Similarly Bob has n ±1-valued observables B 1 , . . . , B n in H B . Alice

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Section: Zero-lifting Of Correlation Inequalitiessupporting

confidence: 53%

“…If the inequality involves more than one coordinate, then the theorem follows by applying the zero-lifting theorem for the cut polytope of a general graph [11] repeatedly, in the same way as in the case [2] of Bell inequalities.…”

Section: Zero-lifting Of Correlation Inequalitiesmentioning

confidence: 99%

On the relationship between convex bodies related to correlation experiments with dichotomic observables

Avis¹,

Imai²,

Ito³

2006

J. Phys. A: Math. Gen.

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“…Thus deg s (f ) = 20, and hence according to (8) we get deg s (u A u B u C u D ) = 20. This is the case if and only if, up to a relabeling, |A| = |B| = 1 and |C| = |D| = 2.…”

Section: Examples and Further Remarksmentioning

confidence: 91%

Retracts and algebraic properties of cut algebras

Römer

Madani

2018

European Journal of Combinatorics

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“…for any A ⊆ V and e ∈ E. The cut polytope of G has been intensively studied by many authors; see, e.g., [6,9,10,12,28,27,33,34].…”

Section: Cycle Polytopes Of Matroidsmentioning

confidence: 99%

“…A prominent example with many applications is the cut polytope Cut (G) of a graph G which has taken the attention of researchers from many fields. Cut polytopes are well studied, see for example [3,9,10,14]. Indeed, cut polytopes are closely related to some well-known problems, like the MAXCUT problem (see for example [11,12,13]) and the four color theorem in graph theory (see [24]).…”

Section: Introductionmentioning

confidence: 99%

Cycle algebras and polytopes of matroids

Römer,

Madani

2021

Preprint

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Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an algebraic and geometric investigation of these polytopes by studying their toric algebras, called cycle algebras, and their defining ideals. Several matroid operations are considered which determine faces of cycle polytopes that belong again to this class of polyhedral objects. As a key technique used in this paper, we study certain minors of given matroids which yield algebra retracts on the level of cycle algebras. In particular, that allows us to use a powerful algebraic machinery. As an application, we study highest possible degrees in minimal homogeneous systems of generators of defining ideals of cycle algebras as well as interesting cases of cut polytopes and Eulerian subgraph polytopes.

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Lifting facets of the cut polytope

Cited by 17 publications

References 5 publications

On the relationship between convex bodies related to correlation experiments with dichotomic observables

On the relationship between convex bodies related to correlation experiments with dichotomic observables

Retracts and algebraic properties of cut algebras

Cycle algebras and polytopes of matroids

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