Family of Bell-like Inequalities as Device-Independent Witnesses for Entanglement Depth

Abstract

We present a simple family of Bell inequalities applicable to a scenario involving arbitrarily many parties, each of which performs two binary-outcome measurements. We show that these inequalities are members of the complete set of full-correlation Bell inequalities discovered by Werner-WolfZukowski-Brukner. For scenarios involving a small number of parties, we further verify that these inequalities are facet-defining for the convex set of Bell-local correlations. Moreover, we show that the amount of quantum violation of these inequalities naturally manifests the extent to which the underlying system is genuinely many-body entangled. In other words, our Bell inequalities, when supplemented with the appropriate quantum bounds, naturally serve as device-independent witnesses for entanglement depth, allowing one to certify genuine k-partite entanglement in an arbitrary n ≥ k-partite scenario without relying on any assumption about the measurements being performed, nor the dimension of the underlying physical system. A brief comparison is made between our witnesses and those based on some other Bell inequalities, as well as the quantum Fisher information. A family of witnesses for genuine k-partite nonlocality applicable to an arbitrary n ≥ k-partite scenario based on our Bell inequalities is also presented. One of the most important no-go theorems in physics concerns the impossibility to reproduce all quantum mechanical predictions using any locally-causal theory [1] -a fact commonly referred to as Bell's theorem [2]. An important observation leading to this celebrated result is that measurement statistics allowed by such theories must satisfy constraints in the form of an inequality, a Bell inequality. Since these inequalities only involve experimentally accessible quantities, their violation -a manifestation of Bell-nonlocality [3] -can be, and has been (modulo some arguably implausible loopholes [4]) empirically demonstrated (see, e.g., [3][4][5] and references therein).Clearly, Bell inequalities played an instrumental role in the aforementioned discovery. Remarkably, they also find applications in numerous quantum information and communication tasks, e.g., in quantum key distribution involving untrusted devices [6][7][8], in the reduction of communication complexity [9], in the expansion of trusted random numbers [10,11], in certifying the Hilbert space dimension of physical systems [12,13], in self-testing [14-18] of quantum devices, in witnessing [19][20][21] and quantifying [22-25] (multipartite) quantum entanglement using untrusted devices etc. For a recent review on these and other applications, see [3].Identifying interesting or useful Bell inequalities is nonetheless by no means obvious. For instance, the approach of solving for the complete set of optimal, i.e., facet-defining Bell inequalities for a given experimental scenario -though potentially useful for the identification of non-Bell-local (hereafter nonlocal) correlationstypically produces a large number of inequalities with no * yliang@phys.ethz....

Nonlocality ofWand Dicke states subject to losses

“…This is partly due to the complexity of multipartite entanglement [4,5] and to the lack of tools adapted to the study of the multipartite nonlocality (see, however, Ref. [6,7]). …”

Quantifying multipartite nonlocality via the size of the resource

“…In the case of even n, the result recently established in Ref. [33] (based on earlier work of Ref. [38]) allows one to conclude that the above-mentioned GHZ correlations have an MGS of at least n − 1 (for R = Q).…”

“…In this regard, we note that a family of such witnesses for a quantum resource (as well as a general nonsignaling resource) has recently been identified [33]. Similar results for other resources, especially one that is either optimal (in the sense of being facet defining) for the respective convex polytope, or one that involves a small number of terms to be measured experimentally, would certainly be desirable.…”

Bipartite Bell inequalities with three ternary-outcome measurements—from theory to experiments

“…To date, however, the bulk of such studies have focussed on the simplest Clauser-Horne-Shimony-Holt (CHSH) [26] Bell scenario, namely, one involving only two parties, each performing two binary-outcome measurements. While more complicated Bell scenarios, such as those involving more parties [25,27,28], or more measurement settings [29][30][31][32][33] or more measurement outcomes [34][35][36] have also been individually considered, scenarios involving a combination of these have so far received relatively little attention (see, however, [21,[36][37][38][39]). …”

“…Similarly, quantum nonlocality is also an essential ingredient for the self-testing [7][8][9][10][11] of quantum apparatus directly from measurement statistics. More recently, the paradigm of device-independent quantum information [3,12]-where the analysis of quantum information is based solely on the observed correlations-has also been applied in the context of randomness expansion [13,14], randomness extraction [15], dimension-witnessing [16][17][18], as well as robust certification [19,21,20], classification [22] and quantification [23][24][25] of (multipartite) entanglement etc.…”