In a quantum network, distant observers sharing physical resources emitted by independent sources can establish strong correlations, which defy any classical explanation in terms of local variables. We discuss the characterization of nonlocal correlations in such a situation, when compared to those that can be generated in networks distributing independent local variables. We present an iterative procedure for constructing Bell inequalities tailored for networks: starting from a given network, and a corresponding Bell inequality, our technique provides new Bell inequalities for a more complex network, involving one additional source and one additional observer. We illustrate the relevance of our method on a variety of networks, demonstrating significant quantum violations, which could not have been detected using standard Bell inequalities.
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 characterized efficiently, e.g., using a finite set of Bell-like inequalities-an observation that enables the systematic study of quantum nonlocality and related applications under limited measurement independence.
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....
The possibility to explain quantum correlations via (possibly) unknown causal influences propagating gradually and continuously at a finite speed v > c has attracted some attention recently. In particular, it could be shown that this assumption leads to correlations that can be exploited for superluminal communication. This was achieved studying the set of possible correlations that are allowed within such a model and comparing them to correlations produced by local measurements on a four-party entangled quantum state. Here, we report on a quantum state that allows for the same conclusion involving only three parties.
Photons of a laser beam driving the upper motional sideband of an optomechanical cavity can decay into photon-phonon pairs by means of an optomechanical parametric process. The phononic state can subsequently be mapped to a photonic state by exciting the lower sideband, hence creating photon-photon pairs out of an optomechanical system. Here we show that these pairs can violate a Bell inequality when they are measured with photon counting techniques preceded by small displacement operations in phase space. The consequence of such a violation as well as the experimental requirements are intensively discussed. DOI: 10.1103/PhysRevLett.116.070405 Introduction.-Cavity optomechanics, which describes a mechanical oscillator controlled by an electromagnetic cavity mode via a generalized radiation pressure force, is the subject of intense research [1][2][3]. Most recent progress includes the cooling of mechanical oscillators down to the ground state [4][5][6], the readout of the mechanical position with a readout imprecision below the standard quantum limit [7] as well as optomechanical squeezing [8,9] and entanglement [10]. Reciprocally, the mechanical degrees of freedom can be used to control the cavity light, e.g., for fast and slow light [11,12], frequency conversions [13,14], squeezing [15], and information storage in long-lived mechanical oscillations [10,16].Optomechanical systems are also envisioned as test benches for physical theories [17][18][19][20][21][22][23]. As a step in this direction, quantum correlations between light and mechanics have been observed recently [10]. In this experiment, quantum features have been detected through an entanglement witness where one assumes that the measurement devices are well characterized and where quantum theory is used to predict the results of these measurements on separable states. It is interesting to wonder whether the nonclassical behavior of optomechanical systems can be certified outside of the quantum formalism, i.e., from a Bell test [24]. This is particularly relevant to test postquantum theories including explicit collapse models [25][26][27][28], where the assumption that the system behaves quantum mechanically may be questionable [29].In this Letter, we show how to perform such a Bell test in the experimentally relevant weak-optomechanical coupling and sideband-resolved regime. Our proposal, which starts with a mechanical oscillator close to its ground state, consists of two steps. First, the optomechanical system is excited by a laser tuned to the upper motional sideband of the cavity to create photon-phonon pairs via optomechanical parametric conversion. Second, a laser resonant with
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