New concise upper bounds on quantum violation of general multipartite Bell inequalities

On general Bell inequalities, see Section II.

This follows from the definition of Grothendieck’s constant $KG(ℝ)$ and Theorem 2.1 in Ref. 3.

That is, Bell inequalities of an arbitrary type, either for correlation functions or for joint probabilities or of a more complicated form, see in Section II.

Throughout the article, the term “precise bound” means that, in this bound, not any universal constant is involved—in contrast to some bounds in Refs. 8, 10, 11, and 14 defined up to universal constants.

For this notion, see Section II.

See Proposition 3 in Ref. 19.

On the general framework for the probabilistic description of multipartite correlation scenarios, see Ref. 24.

For the main statements on the LHV modelling of a general multipartite correlation scenario, see Section 4 in Ref. 24.

Here, the term a tight LHV constraint means that, in the LHV frame, the bounds established by this constraint cannot be improved. On the difference between the terms a tight linear LHV constraint and an extreme linear LHV constraint in the case of, for example, the LHV constraints on a linear combination of correlation functions, see the end of Section 2.1 in Ref. 22.

See Proposition 1 in Ref. 9 for an N-partite case and Proposition 1 in Ref. 28 for a bipartite case.

See definition 2 in Section II of Ref. 9.

See Eq. (8) in Ref. 31.

To our knowledge, for the maximal quantum violation $ΥS1×⋯×SN(ρd,N)$ by a state $ρd,N$ of general $S1×⋯×SN$-setting Bell inequalities, the precise upper bounds are presented by Eq. (62) in Ref. 9 and by Eq. (19) of Ref. 13. The bipartite upper bound $ΥS×S(ρd,2)≤Cmin{d,S}$ presented by Eq. (01) in Ref. 15 (also, in Refs. 8 and 14) is defined up to a universal constant. Note also that the upper bounds in Refs. 10 and 14 on the maximal quantum violation of general Bell inequalities used in nonlocal games do not need to hold for all general Bell inequalities.