In the last decades, great progress has been made in both quantum information and quantum technologies [1–4] even with focus on possible realizations and applications to be carried out in space [5]. A very important resource for applications in quantum information are nonlocal quantum correlations [6–9]. In this review, we summarize the state of the art of entanglement and Einstein–Podolsky–Rosen (EPR) steering.

Nonlocal quantum correlations are strong correlations which cannot be described by classical physics. In the seminal paper by Einstein et al [10], two spatially separated particles with perfect correlations were considered. Then, if a measurement is performed on either momentum or position in one of the particles, the other particle will have a defined momentum or position. This is what is nowadays known as the EPR paradox. In the EPR paradox, the premise of local realism, which implies that one particle cannot influence another spatially separated particle, is assumed for a quantum state, finally leading to the conclusion that quantum mechanics is incomplete. In a series of articles starting in the same year, Schrödinger [11–15], as a reply to the ongoing discussion, brought forward the concepts of steering and entanglement.

In 1964 Bell [16, 17] established the Bell inequality, which showed the incompatibility between local hidden variables, which are the proposal to maintain the local properties of classical physics, and quantum mechanics. Later, other Bell-type inequalities, some of them experimentally accessible, were proposed. For instance, two that are amply used in the present are the Clauser–Horne [18] and the Clauser–Horne–Shimony–Holt inequalities [19], which are experimentally accessible and thus have had a great relevance in the investigation of quantum correlations since they were proposed. In the decades of the 1970–1980's the first experiments of the Bell inequality were carried out [20–22], proving that nonlocal quantum correlations are present in nature and therefore showing that entanglement is an important resource for processes and tasks in quantum information [23].

Moreover, the EPR paradox shows the existence of the nonlocal quantum correlations [7, 8, 22], pioneering the three nonlocal quantum correlations, which are quantum entanglement, quantum steering and Bell nonlocality. Mathematically, an entangled state is a composite state that cannot be described in a separable or factorizable form, this means that one party is strongly correlated with the other, even if they are spatially separated [6, 11, 12]. Bell nonlocality and quantum steering are stronger forms of quantum correlations. Quantum steering is an asymmetric correlation, since for two spatially separated parties local measurements performed by one of the parties can steer the state of the other, and is the manifestation of the EPR paradox. This property is what had been long known as 'spooky action at a distance'. Steering can also be expressed in terms of violations of the so named local hidden state (LHS) models or defined in terms of a quantum task [8, 24, 25]. Due to the asymmetry property, steering has applications in one-sided device independent quantum key distribution (QKD), where only one of the measurement apparatuses of the parties is trusted, which in turn requires the demonstration of steering [26]. Steering is also important in quantifying the randomness that can be extracted in an scenario where one of the parties does not trust their device [27].

Bipartite entanglement refers to the type of entanglement that manifests/occurs between two systems or particles. This form of entanglement has been thoroughly investigated [2, 6, 28–36] from a quantitatively and qualitatively point of view, as well as both in a theoretical and in an experimental way. However, more recently there have been described and studied other forms of entanglement, which include high dimensional entanglement [37, 38], hybrid entanglement [39, 40] and multipartite entanglement [41–45], in which several systems are involved. In particular, hybrid entanglement is that between different types of subsystems, including at least one continuous variable (CV) subsystem and one discrete variable (DV) subsystem, in contrast with continuous entanglement which is entanglement between CVs only, and discrete entanglement, which is only between DV subsystems. The inclusion of the two different types of variables allows for novel applications of entanglement. For instance, this has been implemented in QKD protocols using the spin and orbital angular momentum [46], or polarization and orbital angular momentum [47]. In the sense of high dimensional entanglement, a higher dimensional Hilbert space might allow to increase the capacity of and generally improve quantum communication protocols [48–50], since more bits can be encoded in the system [51–53]. On the other hand, multipartite entanglement can be implemented in integrated optics [54], and can be used to enhance sensitivity [55] or for quantum metrology [56].

Besides the applications, it is important to investigate how quantum correlations can be generated. This can be done through nonlinear processes. In the bipartite case, entanglement and steering can be produced by parametric processes, as harmonic cascades with sum-frequency or by a direct third-harmonic generation [57]. In [58], a nonlinear optical apparatus where the steering can be controlled and even set as one-way steering is described, which shows the asymmetry of quantum steering. For the tripartite case, entanglement can be generated by an optical parametric oscillator [59]. Both genuine tripartite entanglement and steering have been investigated for a system of optical modes in a combined down conversion and sum of frequency process [60], while other proposals concerning only genuine tripartite entanglement are given by spontaneous parametric down conversion processes of spatial degrees of freedom of the photons [61]. There are also recent studies considering triple photon generation [62–65] or a nonlinear process of third order [66]. The generation of higher order multipartite steering and entanglement is a subject of current investigation [67, 68] which may require multiple nonlinear cascaded processes [69, 70].

In this review, we first give an overview of quantum correlations, in section 2, with focus on their definition and relationship. Next, the different measures to certify either entanglement and steering for both bipartite and multipartite entanglement are described in section 3. Section 4 shows the applications of multipartite quantum correlations. Concluding remarks are given in section 5, the last section of this review.

In quantum theory there exist quantum correlations in what is observed between two systems that do not correspond to classical causality. This important fact is known and was developed from the beginnings of quantum theory [71–76]. Currently, our understanding is that there exist three different types of quantum correlations and they represent an appropriately complete panorama.

2.1. Quantum correlations2.1.1. Bell nonlocality

In the EPR paper [10], local hidden variable theories (LHVTs) were proposed to complete the description that quantum mechanics gives of physical systems, in such a way that classical realism is recovered in the predictions we make of the measurement results. Bell identifies the local causality principle as the component of LHVTs that disagrees with the correlations predicted in quantum theory. The kind of quantum correlations that, by Bell's theorem, do not satisfy local causality are named nonlocality.

Additionally, nonlocality and Bell's inequalities experimental proofs have allowed us in the past decade to observe the existence of quantum correlations in quantum mechanical systems [20].

Nonlocality, Bell inequalities and their applications were reviewed more in depth by Brunner et al [22], while Bell inequalities for mesoscopic and macroscopic systems were reviewed by Teh et al [77]. As this is outside of the scope of this work, the interested reader is directed to these references.

2.1.2. Entanglement

The concept of entanglement surges from the idea that was early observed by Schrödinger [11, 12, 78–80], that quantum systems do not obey the independence of correlated physical systems, like classical systems do. In this way, entanglement is defined as the proper non-separability of quantum systems, in the sense that in order to describe a quantum system composed of two or more subsystems, it can not be described by only the description of the subsystems individually, but must be described in full.

The conceptual and mathematical sense of the phenomenon of entanglement will be revised in more detail in the following section discussing the detection of quantum correlations, section 3.1.

2.1.3. EPR steering

In discussing the situation presented by Einstein et al [10], Schrödinger [13–15] proposed the term steering to refer to the influence of a part of the system over the other, in search of a more general discussion than that of the conjugate variables position and momentum [72, 74]. In this way, steering can be described as the ability to affect a system that we do not have access to, through performing local measurements in a different system.

A formal definition of steering in terms of quantum operations and information theory was given in 2007 by Wiseman et al and Jones et al [24, 25], through the attentive revision of Schrödinger's works. To reflect the fact that the concept of steering carries the essence of the correlations in the case considered by EPR, it was named EPR steering. In the information operational sense, steering is defined as the impossibility of a description through a LHS model. In steering, the EPR effect is present, but the correlations are not strong enough to discard all hidden variable models. Steering is a quantum correlation that, in strength, is located between entanglement and Bell nonlocality [24, 25].

A remarkable work about steering's growth since its conception is given by Uola et al [7]. There, the theoretical basis and historical milestones are collected, giving a general steering framework as guidance for future research. A recent review of the state of the art of quantum steering was made by Xiang et al [81], where the main collaborations that had arisen through the years about the cited topic are recognized. Also a series of open questions and research paths to follow are presented. This review [81], gathers together theory, experiments and the scope of quantum steering for the years to come. It is important to comment that this work gives a wide outlook about steering detection with entropic criteria. In addition, different criteria are compared in different references, for instance [82–84].

2.1.3.1. Steering asymmetry

In a different manner as for the other quantum correlations, steering possesses a peculiar characteristic, that of asymmetry. Given that it is so rare in the context of quantum correlations, this asymmetry is an important property and promises interesting applications [85]. The asymmetry has been realized in different scenarios, in which there is also evidence of manipulation of this asymmetry [58, 86–93]. In this way, the asymmetry of steering provides important advantages for applications, such as in QKD, where now protocols in which we only require one party to be trustworthy are realizable [94], as well as permitting different characteristics that improve the security of quantum teleportation [95].

The steering asymmetry also implies that in some states it can be certified only in one direction between the two parties, this is known as one way steering [24, 96]. This means that for an entangled state, an observer Alice can steer Bob's system, but no the other way around.

Different measures to characterize this asymmetry have been proposed. For example by defining an steering radius it is possible to quantify the steerability of the system and also demonstrate the asymmetry of the system [97], other measures include the use of entropic measures [98] (see section 3.2.2.4). The asymmetry of steering for Gaussian measurements has been theoretically shown in an intracavity nonlinear Kerr coupler [99]. Gaussian one way steering was first experimentally observed by Händchen et al [100].

Later experiments demonstrate genuine one way steering, but with the added important properties of being for a general type of measurements and with no detection loopholes, which include the positive operator value measurements (POVMs) and the assumption of the quantum state [101]. A rigorous experimental detection loophole-free test for a two qubit system was performed in [102]. For two-qubit states, a theoretical and experimental certification of one-way steering with no assumptions on state fidelity or measurement settings have been performed [103]. Since this asymmetry can be detected in several systems, a proposal for generating asymmetric steering in macroscopic steering is given in [104].

Due to the asymmetry property of steering being important for potential applications, there has been research in ways to generate this asymmetry. For example, it was studied in microwave photons by using a superconducting circuit system [105], a quantum interference from the incoherent pumping of an atomic system [106], or through atomic coherent effects in a resonant four-level system [107].

Related to the direction of steering, is the investigation of the decoherence of steering when an entangled system is coupled to a reservoir. In this case, steering can change depending if system A is steered by B or otherwise [108]. For multipartite systems the asymmetry property has been experimentally performed for optical networks [109], and theoretically for Gaussian states by using a modulating scheme [89].

2.2. The relationship between different quantum correlations

From the work of [24, 25], we know that every nonlocal state possesses steering and any steerable state is entangled, however, the converse relationships are not true. Moreover, it was found that the concepts of steering and the uncertainty principle are the two elemental components of nonlocality [110].

Trust in the measuring devices is an important characteristic of the tasks that are carried out in quantum information, and corresponds to the characterization of the measuring processes. Here also we can see the hierarchy of the quantum correlations. In entanglement all of the measurements are well-characterized and we have a full description of the quantum system. Then, for Bell-nonlocality we do not need a characterization of the measurements, just to know the resulting probabilities. Steering is found between these two extremes: in the bipartite case only one system is completely characterized, conventionally the measurements of Bob. This allows for one-sided device-independent protocols [26, 111, 112]. Even more, in the steering case presented by [113], Bob's system is not completely characterized and only its dimension is known, with which the certification of steering can be seen as a separability problem.

To date there are a great variety of different criteria to certify bipartite entanglement, with important contributions in the cases of multipartite and continuous entanglement. As well, there exist diverse manners to certify steering in different kinds of systems. A recent review with focus on theoretical and experimental certification of entanglement is given in [114]. Other reviews of entanglement include [6, 33, 34, 36, 115, 116]. For steering, the recent reviews of [7, 8, 81, 117] discuss aspects of certification. Below we review the different measures or witnesses used to certify quantum entanglement and steering.

3.1. Entanglement detection measures3.1.1. Bipartite entanglement

An entangled state is a state of a quantum system for which its parties are strongly correlated; entanglement is a very important phenomenon on quantum mechanics. Nowadays, it is a fundamental question to know whether a state presents nonlocal correlations. These can be defined for bipartite and multipartite systems, being the bipartite systems the most studied. For these, entanglement can be both certified and quantified.

Let us consider a bipartite pure state , in the Hilbert space . The system is entangled if the state cannot be expressed as a factorized form of product of the states of each of the systems, which correspond to and , respectively, for system A and B [28, 33]

A separable state is a state that can be written as a factorized form of product of the states of each of the systems. In this case, the state is uncorrelated, and measurements made in system A are independent of measurements of B.

For mixed states, the states of each system are described by a density matrix, ρA for system A and ρB for B, respectively, which are defined in the corresponding Hilbert spaces and . A bipartite state ρAB is separable if it can be written as a product of the form

Depending on the system under consideration there are different criteria that one can consider in order to certify whether the system is entangled. The quantum systems that have an infinite-dimensional Hilbert space, and a continuous spectrum for the observables, are called CV systems, while the quantum systems with a finite-dimensional Hilbert space and discrete spectrum are DV systems. Examples of CV systems are harmonic oscillators and electromagnetic modes. They are characterized by the quadrature operators

Here and are bosonic operators, while c is a constant which values are usually 2 or . Gaussian states are examples of CV systems. These systems present entanglement and steering in their quadrature operators. Thus, as an example, entanglement witnesses are given in terms of variances of these operators.

3.1.2. Entanglement witnesses for bipartite systems3.1.2.1. Schmidt decomposition

The simplest case of entanglement is seen in a bipartite quantum state. In this occurrence, the Schmidt decomposition of the state seen as a vector in the product state space, allows to determine unequivocally the separability of the state from its Schmidt rank.

The Schmidt decomposition corresponds to the construction of coefficients that can be used to describe the compound state trough tensor products between the subsystem states [36]. For an state the Schmidt Theorem assures the existence of bases and of A and B, respectively, such that

The Schmidt rank is defined as the number of non-zero coefficients in equation (4). Then, an state is entangled iff its Schmidt rank is greater than one [36].

Several measures can also be written using the Schmidt coefficients, , for instance, the concurrence (section 3.1.3.8) and entanglement of formation (EoF) (section 3.1.3.3), further details can be found in [36]. In the case of multipartite systems, the Schmidt rank vector, constructed from the Schmidt rank of the possible bipartitions, can be used to characterize the entanglement [118, 119], see also [37].

3.1.2.2. Positivity criteria

The two main results of positivity criteria, that of Peres–Horodecki [28, 120] for DV and of Simon [121] for CV, are of great importance due to the simplicity of the determination of entanglement they represent, specially when compared to the most straightforward approach of resolving the non-separability of a quantum state, and have been amply studied as a result. Here we briefly present some of the results in this direction.

3.1.2.3. The positive partial transpose (PPT) criterion

The PPT criterion was developed by Peres [120] and Horodecki et al [28] and is an operational criterion to detect entanglement based upon the fact that a given state is separable if its matrix representation has a PPT. It is noteworthy that this condition is just necessary, and not sufficient, for the separability in discrete systems of general dimensions [28].

3.1.2.4. The Simon positivity criterion

There exists a generalization of the criterion for CV that was proposed in 2000 by Simon [121], such generalization utilizes second order momenta of the canonical operators.

3.1.2.5. Momentum matrices positivity criteria

The positivity criteria can be further extended in terms of the momentum matrices [122–124]. The criteria of [123, 124] also allows for the detection of nonseparability for DV systems of general dimensions. For the case of non-Gaussian continuous states, the Simon criterion [121], becomes a hierarchy of positivity conditions on the minor matrices obtained from the quadrature momentum matrices in the approach of [122].

Even further, there is a generalization of the work of [122], which puts constrictions on the CV systems in order to generalize the criterion for hybrid systems [125].

3.1.2.6. Entanglement witnesses

An important disadvantage of the PPT criterion and its extensions is that they just offer necessary conditions to detect entanglement in a given system in most cases. Fortunately, there are entanglement witnesses [6], which are another type of tests that in turn use quantum observables to detect quantum entanglement.

It is said that an observable is an entanglement witness if at least it is capable of distinguishing one entangled state from the set of all separable states. The detection provided by these witnesses is defined thanks to a general operational condition [126]. The entanglement witnesses are Hermitian operators W acting over the compounded Hilbert space of a system, and they always have positive mean values for all pure separable states , even though they are not positive definite, then pointing out the presence of entanglement with the existence of negative mean values. This is, if , then ρ is entangled.

Clearly, a downside of the witnesses is that they are not global, because one single operator is not capable of detecting entanglement in every general state. Conversely, for each entangled state, there exist multiple witnesses capable of distinguishing it from all separable states. Even more, there are entangled states that can not be detected utilizing any witnesses [36]. Regarding the mathematical formulation, entanglement witnesses are defined in a geometrical manner starting from the convexity of the set of separable states. For two qubits, when the two parties perform Pauli measurements σx and σz , there exist closed descriptions for every witness [127]. More introductory details on this important topic can be found in [6, 36].

3.1.2.7. Information theory

Entanglement, as all quantum correlations, is closely related to the transmission of information from one system to another. This can be seen both in the initial discussions of correlations and intuitive understandings, and the upsurging fields of quantum information and quantum communications, in which these correlations are fundamental concepts and building blocks.

Since it results that many of the concepts developed in this current of research provide closed expressions for the amount of entanglement in bipartite DV states, a more detailed basic review appears in section 3.1.3.1.

3.1.2.8. Entropic entanglement criteria

Entropic criteria and quantities have been able to be defined in order to detect the presence and give a measure of quantum correlations. Entanglement criteria from entropic uncertainty relations have been obtained in references such as [128–131]. In an interesting analysis, a general formalism to obtain criteria from local uncertainty relations, which bound the variances, is proposed in [132]. This is then upgraded to a formalism to obtain both entanglement and steering criteria from entropic uncertainty relations in [131], see section 3.2.2.4.

In order to certify entanglement, we require to know the measurements of both systems. For separable states, it follows from the existence of generalized entropic formulations of the uncertainty principle, that there is an entropic limit on the probabilities of both subsystems [132]

Therefore, we can find for the generalized conditional entropies the following bound for separable states

3.1.2.9. Entanglement criteria from local uncertainty relations

The previous limit based on the uncertainty principle can be also obtained for the regular uncertainty relations, where the uncertainty is given in terms of variances instead of entropies. In this case, equation (6) becomes a bound on the variances of the global measurements