We introduce an approach which allows a detailed structural and quantitative analysis of multipartite entanglement. The sets of states with different structures are convex and nested. Hence, they can be distinguished from each other using appropriate measurable witnesses. We derive equations for the construction of optimal witnesses and discuss general properties arising from our approach. As an example, we formulate witnesses for a 4-cluster state and perform a full quantitative analysis of the entanglement structure in the presence of noise and losses. The strength of the method in multimode continuous variable systems is also demonstrated by considering a dephased GHZ-type state.

This corrects the article DOI: 10.1103/PhysRevLett.118.110502.

Within the context of semiquantum nonlocal games, the trust can be removed from the measurement devices in an entanglement-detection procedure. Here, we show that a similar approach can be taken to quantify the amount of entanglement. To be specific, first, we show that in this context, a small subset of semiquantum nonlocal games is necessary and sufficient for entanglement detection in the local operations and classical communication paradigm. Second, we prove that the maximum payoff for these games is a universal measure of entanglement which is convex and continuous. Third, we show that for the quantification of negative-partial-transpose entanglement, this subset can be further reduced down to a single arbitrary element. Importantly, our measure is measurement device independent by construction and operationally accessible. Finally, our approach straightforwardly extends to quantify the entanglement within any partitioning of multipartite quantum states. DOI: 10.1103/PhysRevLett.118.150505 Introduction.-Entanglement is a valuable resource for practical as well as fundamental applications of quantum theory, ranging from quantum computation and communication to metrology [1-3]. There are two major challenges in understanding entanglement that stimulates this research. First, it is extremely difficult to specify all the nonentangled bipartite or multipartite quantum states. In fact, the problem is known to be NP-hard [4,5]. Second, not surprisingly, the characterization of entangled states, i.e., the quantification of entanglement within quantum states, is an equally difficult task. The answer to the second challenge is practically very important because it tells us how well our protocols will perform using a given state [6][7][8][9].Focusing on the second challenge above, a first level of hardness is that the quantification of entanglement using almost any entanglement measure, e.g., entanglement of formation [10], negativity [11,12], or random robustness [13], requires estimating a large number of density matrix elements, a task which is difficult to perform on bipartite and multipartite quantum states. While this difficulty can be partially circumvented by making use of entanglement witnesses (EWs) when lower bounds on the entanglement are desired [14][15][16][17][18][19], errors and misalignments of the measurement devices can still lead to incorrect estimations of the quantities and thus, erroneous conclusions. A measurementdevice-independent approach is therefore desirable.Recent work by Buscemi [20] has introduced a new way to think about entanglement detection [21][22][23][24]. The idea is to map the problem onto a modified class of nonlocal games, called semiquantum nonlocal games (SQNLGs). In any such game, two players (Alice and Bob) share a possibly entangled state. A referee (Charlie) starts by asking them

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Entanglement witnesses are invaluable for efficient quantum entanglement certification without the need for expensive quantum state tomography. Yet, standard entanglement witnessing requires multiple measurements and its bounds can be elusive as a result of experimental imperfections. Here, we introduce and demonstrate a novel procedure for entanglement detection which simply and seamlessly improves any standard witnessing procedure by using additional available information to tighten the witnessing bounds. Moreover, by relaxing the requirements on the witness operators, our method removes the general need for the difficult task of witness decomposition into local observables. We experimentally demonstrate entanglement detection with our approach using a separable test operator and a simple fixed measurement device for each agent. Finally, we show that the method can be generalized to higher-dimensional and multipartite cases with a complexity that scales linearly with the number of parties. DOI: 10.1103/PhysRevLett.118.110502 Quantum entanglement provides many advantages beyond classical limits, including quantum communication, computation, and information processing [1,2]. Yet, determining whether a given quantum state is entangled or not is a theoretically and experimentally challenging task [3,4]. In particular, the ideal approach of reconstructing the full quantum state via quantum tomography is practically infeasible for all but the smallest systems.An elegant solution to this problem, known as entanglement witnessing, relies on the geometry of the set of nonentangled (separable) quantum states [2,[5][6][7]. Since these states form a convex set, it is always possible to find a hyperplane such that a given entangled state lies on one side of the hyperplane, while all separable states are on the other side, see Fig. 1. This hyperplane is a so-called entanglement witness (EW) and corresponds to a joint observable that has a bounded expectation value over all separable quantum states. Any quantum state that produces a value beyond the bound must be entangled. This simplification, however, comes at a cost: first, different entangled states in general require different EWs to be detected; second, not every EW can be practically realized, i.e., can be decomposed into operators corresponding to available local measurement devices (See also Refs. [7][8][9] for examples of the reverse procedure: constructing EWs from local observables); third, when such a decomposition is possible, it might require multiple measurement devices (with multiple settings) to be implemented; and fourth, witnessing bounds can be elusive in the presence of experimental imperfections. Consequently, the goal is to construct EWs that have a simple decomposition and, at the same time, detect a large set of entangled states.There are three main techniques to improve EWs. First, adding nonlinear terms to the original witness operator [10]; second, using collective measurements of EWs on multiple copies of the quantum state [11]; and third, op...

Determination of the quantum nature of correlations between two spatially separated systems plays a crucial role in quantum information science. Of particular interest is the questions of if and how these correlations enable quantum information protocols to be more powerful. Here, we report on a distributed quantum computation protocol in which the input and output quantum states are considered to be classically correlated in quantum informatics. Nevertheless, we show that the correlations between the outcomes of the measurements on the output state cannot be efficiently simulated using classical algorithms. Crucially, at the same time, local measurement outcomes can be efficiently simulated on classical computers. We show that the only known classicality criterion violated by the input and output states in our protocol is the one used in quantum optics, namely, phase-space nonclassicality. As a result, we argue that the global phase-space nonclassicality inherent within the output state of our protocol represents true quantum correlations.Introduction.-Correlations play an undeniable role in our understanding of the physical world. In our macroscopic classical description of commonplace phenomena, classical physics and classical information theory are in perfect agreement in characterization and quantification of correlated events. At a microscopic level, where quantum theory is our best candidate for explaining phenomena, however, the situation is different. Our informational inspections of a quantum world, i.e., quantum information theory, is based on our intuition from classical information theory and classical probability theory, mostly the notion of quantum entropy [1, 2]. However, a discrepancy emerges when physical constraints are taken into account to distinguish between quantum physics and classical physics, hence splitting quantum information approach to correlations from that of quantum optics.In quantum optics it is common to study nonclassical features of bosonic systems in a quantum analogue of the classical phase space [3]. While in a classical statistical theory in phase-space the state of the system is represented by a probability distribution, the quantum phase-space distributions can have negative regions, and hence, fail to be legitimate probability distributions [4]. The negativities are thus considered as nonclassicality signatures. Within multipartite quantum states, the phase-space nonclassicality is tempted to be interpreted as quantum correlations, due to the fact that in a classical description of the joint system no such effects are present [5,6].The sharpest contrast between the definition of quantum correlations in quantum information science and that of quantum optics has been demonstrated very recently by Ferraro and Paris [7]. They showed that the two definitions from quantum information and quantum optics are maximally inequivalent, meaning that every quantum state which is classically correlated with respect to the quantum information definition of quantum correlations is nece...

We find a sufficient condition to imprint the single-mode bosonic phase-space nonclassicality onto a bipartite state as modal entanglement and vice versa using an arbitrary beam splitter. Surprisingly, the entanglement produced or detected in this way depends only on the nonclassicality of the marginal input or output states, regardless of their purity and separability. In this way, our result provides a sufficient condition for generating entangled states of arbitrary high temperature and arbitrary large number of particles. We also study the evolution of the entanglement within a lossy Mach-Zehnder interferometer and show that unless both modes are totally lost, the entanglement does not diminish.

The general class of Gaussian Schmidt-number witness operators for bipartite systems is studied. It is shown that any member of this class is reducible to a convex combination of two types of Gaussian operators using local operations and classical communications. This gives rise to a simple operational method, which is solely based on measurable covariance matrices of quantum states. Our method bridges the gap between theory and experiment of entanglement quantification. In particular, we certify lower bounds of the Schmidt number of squeezed thermal and phase-randomized squeezed vacuum states, as examples of Gaussian and non-Gaussian quantum states, respectively.