Tsirelson bounds for generalized Clauser-Horne-Shimony-Holt inequalities

Quantum theory imposes a strict limit on the strength of non-local correlations. It only allows for a violation of the CHSH inequality up to the value 2 √ 2, known as Tsirelson’s bound. In this paper, we consider generalized CHSH inequalities based on many measurement settings with two possible measurement outcomes each. We demonstrate how to prove Tsirelson bounds for any such generalized CHSH inequality using semidefinite programming. As an example, we show that for any shared entangled state and observables X1, . . . , Xn and Y1, . . . , Yn with eigenvalues ±1 we have |〈X1Y1〉+ 〈X2Y1〉+ 〈X2Y2〉+ 〈X3Y2〉+ . . .+ 〈XnYn〉− 〈X1Yn〉| ≤ 2n cos (π/(2n)) . It is well known that there exist observables such that equality can be achieved. However, we show that these are indeed optimal. Our approach can easily be generalized to other inequalities for such observables. Non-local correlations arise as the result of measurements performed on a quantum system shared between two spatially separated parties. Imagine two parties, Alice and Bob, who are given access to a shared quantum state |Ψ〉, but cannot communicate. In the simplest case, each of them is able to perform one of two possible measurements. Every measurement has two possible outcomes labeled ±1. Alice and Bob now measure |Ψ〉 using an independently chosen measurement setting and record their outcomes. In order to obtain an accurate estimate for the correlation between their measurement settings and the measurement outcomes, they perform this experiment many times using an identically prepared state |Ψ〉 in each round. Both classical and quantum theories impose limits on the strength of such non-local correlations. In particular, both do not violate the non-signaling condition of special relativity. That is, the local choice of measurement setting does not allow Alice and Bob to transmit information. Limits on the strength of correlations which are possible in the framework of any classical theory, i.e. a framework based on local hidden variables, are known as Bell inequalities [1]. The best known Bell inequality is the Clauser, Horne, Shimony and Holt (CHSH) inequality [5] |〈X1Y1〉+ 〈X1Y2〉+ 〈X2Y1〉 − 〈X2Y2〉| ≤ 2, where {X1,X2} and {Y1, Y2} are the observables representing the measurement settings of Alice and Bob respectively. 〈XiYj〉 = 〈Ψ|Xi ⊗ Yj |Ψ〉 denotes the mean value of Xi and Yj . Quantum mechanics allows for a violation of the CHSH inequality, but curiously still limits the strength of nonlocal correlations. Tsirelson’s bound [17] says that for quantum mechanics |〈X1Y1〉+ 〈X1Y2〉+ 〈X2Y1〉 − 〈X2Y2〉| ≤ 2 √ 2. 1 Peres demonstrated how to derive Bell inequalities [12] even for more than two settings. As Froissart and Tsirelson [16] have shown, these inequalities correspond to the faces of a polytope. Computing the boundary of the space of correlations that can be attained using a classical theory therefore corresponds to determining the faces of this polytope. However, determining bounds on the correlations that quantum theory allows remains an even more difficult problem [4]. All Tsirelson’s bounds are known for CHSH-type inequalities (also known as correlation inequalities) with two measurement settings and two outcomes for both Alice and Bob [16]. Filipp and Svozil [7] have considered the case of three measurement settings analytically and conducted numerical studies for a larger number of settings. Finally, Buhrman and Massar have shown a bound for a generalized CHSH inequality using three measurement settings with three outcomes each [4]. In this paper, we investigate the case where Alice and Bob can choose from n measurement settings with two outcomes each. We use a completely different approach based on semidefinite programming in combination with Tsirelson’s seminal results [17, 15, 16]. This method is similar to methods used in computer science for the two-way partitioning problem [2] and the approximation algorithm for MAXCUT by Goemans and Williamson [9]. Cleve et al. [6] have also remarked that Tsirelson’s constructions leads to an approach by semidefinite programming in the context of multiple interactive proof systems with entanglement. Semidefinite programming allows for an efficient way to approximate Tsirelson’s bounds for any CHSH-type inequalities numerically. However, it can also be used to prove Tsirelson type bounds analytically. As an illustration, we first give an alternative proof of Tsirelson’s original bound using semidefinite programming. We then prove a new Tsirelson’s bound for the following generalized CHSH inequality [11, 3]. Classically, it can be shown that | n ∑