Technical Review | Published:

Entanglement certification from theory to experiment

Abstract

Entanglement is an important resource for quantum technologies. There are many ways quantum systems can be entangled, ranging from the two-qubit case to entanglement in high dimensions or between many parties. Consequently, many entanglement quantifiers and classifiers exist, corresponding to different operational paradigms and mathematical techniques. However, for most quantum systems, exactly quantifying the amount of entanglement is extremely demanding, if at all possible. Furthermore, it is difficult to experimentally control and measure complex quantum states. Therefore, there are various approaches to experimentally detect and certify entanglement when exact quantification is not an option. The applicability and performance of these methods strongly depend on the assumptions regarding the involved quantum states and measurements, in short, on the available prior information about the quantum system. In this Review, we discuss the most commonly used quantifiers of entanglement and survey the state-of-the-art detection and certification methods, including their respective underlying assumptions, from both a theoretical and an experimental point of view.

Key points

  • Entanglement detection and certification are of high significance for ensuring the security of quantum communication, improving the sensitivity of sensing devices, and benchmarking devices for quantum computation and simulation.

  • Recent years have seen continuous progress in the development of tools for entanglement certification and an increase in control over a wide variety of experimental setups suitable for entanglement creation.

  • Goals for the development of entanglement detection techniques are device-independence and assumption-free certification.

  • Current challenges include the extension of well-understood methods for two qubits to many-body and/or high-dimensional quantum systems and their application in entanglement experiments with ions, atoms and photons.

  • An important focus of recent research is the reduction in the number of measurements required for entanglement certification to cope with increasing system dimensions.

Introduction

Quantum entanglement rose to prominence as the central feature of the famous thought experiment by Einstein, Podolsky and Rosen1. Initially disregarded as a mathematical artefact showcasing the supposed incompleteness of quantum theory, the properties of entanglement were largely ignored until 1964, when John Bell proposed an experimentally testable inequality able to distinguish between the predictions of quantum mechanics and those of any local-realistic theory2. With the advent of the first experimental tests emerged the realization that entanglement constitutes a resource for information processing and communication tasks, confirmed in a series of experiments3,4,5,6. With the development of quantum information theory, the understanding of entanglement has advanced and diversified, and many links have been established with other disciplines. Today, the study of Bell-like inequalities7 is an active field of research, and recent experimental tests closed all loopholes8,9,10, proving that entanglement is an indispensable ingredient for the description of nature and that quantum technologies can produce, manipulate and certify it.

However, in the early days of quantum information, Werner had already realized that entanglement and the violation of Bell inequalities are not necessarily the same phenomenon11. Whereas entanglement is needed to violate Bell inequalities, it is still not known whether (and in what sense) entanglement always allows for Bell violation12,13,14,15. From a contemporary perspective, Bell inequalities are seen as device-independent certifications of entanglement. However, the question of whether all entangled states can be certified device-independently is still an open problem. In the same paper, Werner also gave the first formal mathematical definition of entanglement. Since then, entanglement theory as a means to characterize and quantify entanglement has developed into an entire sub-field of quantum information. Previous reviews have captured various aspects of the research in this sub-field, focusing, for example, on the nature of non-entangled states16 and on the quantification of entanglement as a resource17,18, or providing detailed collections of works on entanglement theory19 and entanglement detection20.

In quantum communication, certifiable entanglement forms the basis for the next generation of secure quantum devices21,22,23,24. However, it is important to note that entanglement certification goes beyond entanglement estimation, in the sense that the latter may rely on reasonable assumptions about the system state or measurement setup, whereas the requirements for certification are stricter. In quantum computation, the certified presence of entanglement points towards the use of genuine quantum resources, which is crucial for trusting the correct functionality of devices25. In a quantum simulation, a large amount of entanglement can serve as an indicator of the difficulty of classically simulating the corresponding quantum states26,27,28. Nonetheless, the precise role of entanglement in quantum computation and simulation is less clear-cut than in quantum communication. Finally, entanglement can be understood as a means of bringing about speed-ups29,30,31, parallelization32 and even flexibility33 in quantum metrology34. It is not a coincidence that these four areas also form the central pillars of the European flagship programme on quantum technologies35.

With the development of the first large-scale quantum devices and more complex quantum technologies come the challenges of experimentally certifying and quantifying entanglement in quantum systems too complex for conventional tomography. These problems arise in finite-dimensional systems, which are the focus of this Review; for continuous-variable entanglement and infinite-dimensional systems, we refer the interested reader to existing reviews36,37,38.

Entanglement detection and quantification

Entanglement and separability

Entanglement is conventionally defined through a contrapositive: separability. A pure quantum state is called separable with respect to a tensor factorization \({{\mathscr{H}}}_{A}\)\({{\mathscr{H}}}_{B}\) of its (finite-dimensional) Hilbert space if and only if it can be written as a product state \({\left|\psi \right\rangle }_{AB}\) := \({\left|\phi \right\rangle }_{A}\)\({\left|\chi \right\rangle }_{B}\). A general (mixed) quantum state ρ is called separable if it can be written as a probabilistic mixture of separable pure states11.

$${\rho }_{{\rm{sep}}}:=\sum _{i}{p}_{i}\left|{\phi }_{i}\right\rangle {\left\langle {\phi }_{i}\right|}_{A}\otimes \left|{\chi }_{i}\right\rangle {\left\langle {\chi }_{i}\right|}_{B}$$
(1)

All the infinitely many pure state decompositions of a density matrix can be interpreted as a concrete instruction for preparing the quantum state via mixing the states \({| {\phi }_{i}\rangle }_{A}\) \({| {\chi }_{i}\rangle }_{B}\) drawn from a classical probability distribution {pi}i. Because each of these pure states is separable, mixed separable states can easily be prepared by coordinated local operations, that is, local operations and classical communication (LOCC)39,40. Conversely, any state that is not separable is called entangled and cannot be created by LOCC. The fact that there are infinitely many ways to decompose a density matrix into pure states is at the root of the central challenge in entanglement theory: to conclude that a state is indeed entangled, one needs to rule out that there is any decomposition into product states. Answering this question for general density matrices is a non-deterministic polynomial-time (NP)-hard problem41. To be precise, even the relaxed problem allowing for a margin of error that is inversely polynomial (in contrast to inversely exponential, as in the original proof by Gurvits) in the system dimension remains NP-hard42.

Pure states, separable or entangled, admit a Schmidt decomposition into bi-orthogonal product vectors, that is, one can write them as \({| \psi \rangle }_{AB}\) = \({\sum }_{i=0}^{k-1}{\lambda }_{i}| ii\rangle \). The coefficients \({\lambda }_{i}\in {{\mathbb{R}}}^{+}\) are called the Schmidt coefficients. Their squares, which are equal to the eigenvalues of the marginals ρA/B := TrB/A\(| \psi \rangle {\langle \psi | }_{AB}\), are usually arranged in decreasing order and collected in a vector \(\vec{\lambda }\) with components \({[\vec{\lambda }]}_{i}:={\lambda }_{i}^{2}\). The number k of non-zero Schmidt coefficients is called the Schmidt rank or sometimes the dimensionality of entanglement, as it represents the minimum local Hilbert space dimension required to faithfully represent the correlations of the quantum state. One of the fundamental pillars of state manipulation under LOCC is Nielsen’s majorization theorem39,43: a quantum state with (squared) Schmidt coefficients \(\vec{\lambda }\) can be transformed to another state with Schmidt coefficients \({\vec{\lambda }}^{^{\prime} }\) by a LOCC transformation if and only if \(\vec{\lambda }\prec {\vec{\lambda }}^{^{\prime} }\), that is, the vector of squared Schmidt coefficients of the output state majorizes the corresponding vector of the input state. This also conveniently captures two extremal cases. On the one hand, a separable state has a corresponding vector of (1, 0, …, 0), majorizing every other vector, and thus cannot be transformed into any entangled state by LOCC. On the other hand, in dimensions d, the vector \((\frac{1}{d},\frac{1}{d},\ldots ,\frac{1}{d})\) is majorized by every other vector. The corresponding state \(\left|\,{{\rm{\Phi }}}^{+}\right\rangle \) := \(\frac{1}{\sqrt{d}}{\sum }_{i=0}^{d-1}| ii\rangle \) can thus be transformed into any other quantum state and is therefore referred to as a maximally entangled state.

Entanglement quantification

Any meaningful entanglement quantifier for pure states is hence a function of the Schmidt coefficients. The two most prominent representatives are the entropy of entanglement, that is, the von Neumann entropy of the marginals, or equivalently the Shannon entropy of the squared Schmidt coefficients \(E({| \psi \rangle }_{AB})\) := S(ρA/B) = \(-{\sum }_{i=0}^{k-1}{\lambda }_{i}^{2}{{\rm{log}}}_{2}\left({\lambda }_{i}^{2}\right)\), and the Rényi zero-entropy or the logarithm of the marginal rank. For mixed states, the fact that there exist infinitely many pure state decompositions complicates the quantification of entanglement. How is one to unambiguously quantify the entanglement of a state that admits different decompositions into states with various degrees of entanglement? A straightforward answer presents itself in the form of an average over the entanglement \(E(| {\psi }_{i}\rangle )\) within a given decomposition, minimized over all decompositions \({\mathscr{D}}(\rho )\), that is, \(E(\rho )\) := \({{\rm{\inf }}}_{{\mathscr{D}}(\rho )}{\sum }_{i}{p}_{i}E(| {\psi }_{i}\rangle )\). When the entropy of entanglement is the measure of choice, this convex roof construction leads to the entanglement of formation EoF (refs44,45). Its regularization \({{\rm{lim}}}_{n\to \infty }\frac{1}{n}{E}_{{\rm{oF}}}\left({\rho }^{\otimes n}\right)\) has a convenient operational interpretation as the entanglement cost44,46, the asymptotic LOCC interconversion rate from m two-qubit Bell states \({\left|\psi \right\rangle }^{\otimes m}\) = \(\frac{1}{\sqrt{2}}{\left(| 00\rangle +\left|11\right\rangle \right)}^{\otimes m}\) to n copies of ρ, or ρn. Conversely, one may define distillable entanglement as the asymptotic LOCC conversion rate from non-maximally entangled states to Bell states47,48. If the EoF were additive, it would coincide with the entanglement cost. However, as shown by Hastings49, the entanglement of formation is only sub-additive. For other measures, such as the Schmidt rank, a more appropriate generalization is to maximize (instead of averaging) over all states within a given decomposition. In this way, the Schmidt number of mixed quantum states, defined as \({d}_{{\rm{ent}}}\) := \({{\rm{\inf }}}_{{\mathscr{D}}(\rho )}{{\rm{\max }}}_{| {\psi }_{i}\rangle \in {\mathscr{D}}(\rho )}\) \({\rm{rank}}({{\rm{Tr}}}_{A}(| {\psi }_{i}\rangle \langle {\psi }_{i}| ))\)50, directly inherits the operational interpretation of the Schmidt rank for pure states. These are just two examples of generally inequivalent entanglement measures and monotones. For an in-depth review, we refer the interested reader to refs17,18. Whereas these and many other measures have very instructive and operational interpretations, even deciding whether they are non-zero is, in general, an NP-hard problem, even if the density matrix is known to infinite precision. However, not only will uncertainties be associated to the different matrix elements obtained in actual experiments but the sheer amount of information that needs to be collected renders full state tomography too cumbersome to be practical beyond small-scale demonstrations51,52. This is exacerbated in the multipartite case, in which the system dimension grows exponentially with the number of parties.

An implication of this observation is that the amount of actual entanglement in a quantum system not only depends on the measure used (and hence the context or task for which it is applied) but also is impossible to ascertain exactly. However, it is possible to certify the presence of and even to provide a lower bound on the amount of entanglement for various useful quantifiers through few experimentally realizable measurements, which is the main focus of this Review.

Partial transposition and entanglement distillation

A recurring feature among entanglement tests is overcoming the hardness of the separability problem by detecting only a subset of entangled states. An example (that nonetheless requires knowledge of the entire density matrix) is the positive partial transpose (PPT) criterion53,54. Partially transposing a separable state leads to a positive semi-definite density matrix. However, this need not be the case for entangled states because the partial transposition is an instance of a positive, but not completely positive, map. By contrast, positive maps ΛP[ρ] ≥ 0 lead to positive semi-definite matrices when applied to positive semi-definite matrices, such as quantum states. Completely positive maps (ΛCP 𝟙d)[ρ] ≥ 0 \(\forall d\in {{\mathbb{Z}}}^{+}\), on the other hand, lead to positive semi-definite operators even when applied to marginals. In fact, it was proved that a state is separable if and only if it remains positive under all positive maps applied to a subsystem54.

In addition to serving as an easily implementable entanglement test (provided the density matrix is known), the partial transposition provides a simple sufficient criterion for distillation. As shown in ref.55, the process of entanglement distillation47,48, that is, the simultaneous local processing of multiple copies of pairwise distributed quantum states to concentrate the entanglement in one pair, is possible only if there exists at least a 2 × 2-dimensional subspace of the multicopy state space that is not PPT. Because any tensor products of PPT states are also PPT, this directly implies that even though many PPT states are entangled, none of them are distillable. Conversely, whether all states that are non-positive under partial transposition (NPT) are distillable is still an open problem56, but it is known that for any finite number of copies, the answer is negative57.

The PPT map is also commonly used to quantify entanglement through the logarithmic negativity58, defined as the logarithm of the trace norm of the partially transposed density matrix \({\mathscr{N}}(\rho )\) := \({{\rm{log}}}_{2}({\left\Vert {{\rm{\Lambda }}}_{{\rm{P}}}[\rho ]\right\Vert }_{1})\). Loosely speaking, it captures how much the partial transpose fails to be non-negative. The logarithmic negativity is a prominent example of an entanglement monotone59 (as is the negativity60,61), that is, a quantity that is non-increasing under LOCC like any entanglement measure but that does not need to be non-zero for all entangled states.

Whereas calculating the result of applying a positive map requires knowledge of the entire density matrix, it is still possible to harness positive maps to construct powerful entanglement witnesses54 even if only partial or imprecise information about the state is available. Suppose one is provided with a theoretical target state ρT that is not positive semi-definite under a positive (but not completely positive) map ΛP, ΛP[ρT]  0. Then there exist vectors (for example, preferably the eigenvector |ψ〉 of ΛP[ρT] corresponding to the smallest eigenvalue) for which \(\left\langle {\psi }^{-}\right|{{\rm{\Lambda }}}_{{\rm{P}}}\left[{\rho }_{{\rm{T}}}\right]| {\psi }^{-}\rangle \) = \({\rm{Tr}}({{\rm{\Lambda }}}_{{\rm{P}}}[{\rho }_{{\rm{T}}}]| {\psi }^{-}\rangle \langle {\psi }^{-}| )\) < 0. Through the dual map \({{\rm{\Lambda }}}_{{\rm{P}}}^{* }\), this is equivalent to the statement \({\rm{Tr}}({\rho }_{{\rm{T}}}{{\rm{\Lambda }}}_{{\rm{P}}}^{* }[| {\psi }^{-}\rangle \langle {\psi }^{-}| ])\) < 0, whereas \({\rm{Tr}}\left(\sigma {{\rm{\Lambda }}}_{{\rm{P}}}^{* }[| {\psi }^{-}\rangle \langle {\psi }^{-}| ]\right)\) ≥ 0 for all separable states σ. The Hermitian operator \({{\rm{\Lambda }}}_{{\rm{P}}}^{* }[| {\psi }^{-}\rangle \langle {\psi }^{-}| ]\) is thus an example of an entanglement witness (Box 1), an observable that can, in principle, be measured to detect entangled states, at least in the vicinity of ρT. Some illustrative examples of linear and nonlinear (in ρ) entanglement witnesses (negative values detected), positive (but not completely positive) maps (resulting non-positive operators detected) detecting bipartite entanglement for two qubits, maximal entanglement dimensionality (Schmidt number 3) for two qutrits and genuine multipartite entanglement (GME) for three qubits are shown in Table 1. The exemplary techniques detect the entanglement, Schmidt number or GME for the generalized state \(\left|\,\psi \right\rangle \) = \(\frac{1}{\sqrt{d}}{\sum }_{i=0}^{d-1}{| i\rangle }^{\otimes n}\) for (n, d) = (2, 2), (2, 3) and (3, 2), respectively.

Table 1 Examples of entanglement detection methods

Beyond linear witnesses

To improve over linear witnesses, a very useful experimentally applicable method makes use of local uncertainty relations (LURs). The idea to derive entanglement criteria by means of LURs has some analogies with the original Einstein–Podolsky–Rosen (EPR)–Bell approach in the sense that it considers pairs of non-commuting single-party observables, say (A1, A2) for party A and (B1, B2) for party B. Because the Ai do not commute with each other, their uncertainties cannot both be zero simultaneously. The same is true for the Bi. However, in the joint system, the uncertainties of the collective observables Mi = Ai 𝟙+ 𝟙 Bi can both vanish at the same time, provided that the state is entangled.

A powerful and instructive example is given in terms of the variance \({\left({\rm{\Delta }}A\right)}_{\rho }^{2}\) = Tr(A2ρ) − Tr()2. The sum \({\left({\rm{\Delta }}{A}_{1}\right)}_{{\rho }_{A}}^{2}\) + \({\left({\rm{\Delta }}{A}_{2}\right)}_{{\rho }_{A}}^{2}\) ≥ UA must have a non-zero lower bound UA > 0 for all single-party states ρA whenever the two observables do not commute. Similarly, \({\left({\rm{\Delta }}{B}_{1}\right)}_{{\rho }_{B}}^{2}\) + \({\left({\rm{\Delta }}{B}_{2}\right)}_{{\rho }_{B}}^{2}\) ≥ UB for all ρB. Thus, by simple concavity arguments, one can prove that \({\left({\rm{\Delta }}{M}_{1}\right)}_{{\rho }_{AB}}^{2}\) + \({\left({\rm{\Delta }}{M}_{2}\right)}_{{\rho }_{AB}}^{2}\) ≥ UA + UB must hold for all separable states ρAB = \({\sum }_{k}{p}_{k}{({\rho }_{A}\otimes {\rho }_{B})}_{k}\) (refs62,63,64,65). This method hence combines two conceptual features: first, the LURs themselves — representing a trade-off between information about different complementary (non-commuting) observable quantities — and second, the fact that those (nonlinear) quantities are either concave or convex. Thus, analogous reasoning can be applied to other quantifiers of uncertainty, for instance, the quantum Fisher information, introduced in the context of quantum metrology and proved to be related to metrological applications of entanglement34. In addition, LURs in the form of a product of uncertainties (variances) can be used (although requiring a somewhat more complicated mathematical treatment) to derive entanglement criteria resembling Heisenberg uncertainty relations in their original formulation66,67,68.

It is also worth mentioning that all nonlinear entanglement witnesses arising from sums of variances can be cast in a compact form in terms of the covariance matrix \({{\rm{\Gamma }}}_{ij}(\rho )\) = \(\frac{1}{2}{\langle {g}_{i}{g}_{j}+{g}_{j}{g}_{i}\rangle }_{\rho }\) − \({\langle {g}_{i}\rangle }_{\rho }{\langle {g}_{j}\rangle }_{\rho }\) of a local basis of observables. The resulting covariance matrix criterion69,70 was proved to be necessary and sufficient for the special case of two qubits, provided that one makes use of local filterings that map the state to its filtered normal form (FNF) \(\rho \mapsto {\rho }_{{\rm{FNF}}}\) := (FAFB) ρ (FAFB) such that ρFNF =\(\frac{1}{4}(\mathbb{1}_{4}+{\sum }_{i,j=x,y,z}{t}_{ij}{\sigma }_{i}\otimes {\sigma }_{j})\) , where σk (k = x,y,z) are the Pauli matrices. For local dimensions larger than two, the covariance matrix criterion can, in principle, be evaluated using semi-definite programmes, but in its general form, this is still a difficult task, even for bipartite systems.

Bounding witnessed entanglement

When using an approach based on witnesses, one is also interested in quantitative statements about the detected entanglement based on the data of the (preferably) few measurements required for the witness itself. A simple yet general method to compute lower bounds on convex functions of quantum states E(ρ) (such as entanglement measures) using only a few expectation values is based on Legendre transforms71,72. In this context, let us define such a transform as \(\widehat{{\rm{E}}}(W)\) := supρ[Tr() − E(ρ)], where the supremum is taken over quantum states ρ. Note that for a given convex function E(ρ), the quantity \(\widehat{{\rm{E}}}(W)\) depends only on the chosen witness W. Then, a tight lower bound on E(ρ) for the underlying (unknown) system state ρ is obtained through another Legendre transformation, which leads to

$$E(\rho )\ge \mathop{{\rm{\sup }}}\limits_{\lambda }\left[\lambda {\rm{Tr}}(W\rho )-\widehat{{\rm{E}}}(\lambda W)\right]$$
(2)

where λ is real and Tr() is obtained from measurements. The applicability of this technique largely depends on whether \(\widehat{{\rm{E}}}(W)\) (and hence E(ρ) for a given ρ) can be efficiently computed, but this technique has turned out to be a powerful tool to quantify multipartite entanglement based on uncertainty relations73,74,75,76. Another option is a direct construction of witnesses that have a natural connection between their expectation value and a suitably chosen entanglement measure77,78,79.

Measurement strategies and restrictions

Identifying measurement strategies

The previous discussion of bipartite entanglement showcases one of the central challenges for experimental verification: entanglement quantification and detection methods are available in abundance but are often defined in a formal way. Some allude to observable quantities, some to maps on density matrices and others to positive operator-valued measures (POVMs). Identifying the most suitable and efficient practical method for a specific experimental setup is hence not straightforward. For instance, the types of measurements that can be most easily (or at all) implemented depend on the experimental platform, and their identification and comparison may be obfuscated by varying terminologies. A consistent challenge across all platforms and paradigms is the exponential number of potential measurements that could be required for the desired task. Moreover, this number is often specified in terms of different quantifiers, such as the number of global settings, the number of local settings, the number of observables or the number of density matrix elements. To provide a comparative overview of the complexity of different detection methods, we give more precise definitions, briefly review some practical methods of data acquisition and identify which tests work well with what type of data.

Formally, all measurements can be described by POVMs, that is, sets of positive semi-definite operators Mi ≥ 0 with the property \({\sum }_{i=1}^{m}\,{M}_{i}{={\mathbb{1}}}_{d}\), where m is the number of distinguishable outcomes labelled by i. A special case is the projective measurement, where \({M}_{i}=\left|\,{v}_{i}\right\rangle \left\langle {v}_{i}\,\right|\) for all i and m = d. Each POVM can be thought of as a projective measurement on a larger system, and most experimental implementations indeed work directly with projective measurements. Repeated projective measurements allow estimation of the expectation values \({\rm{Tr}}(\rho {M}_{i})=\langle {v}_{i}| \rho | {v}_{i}\rangle \), that is, a complete set of diagonal density matrix elements with respect to a specific basis {vi}i, and, in turn, the expected values of all observables of the form \(O={\sum }_{i}{\lambda }_{i}| {v}_{i}\rangle \langle {v}_{i}| \).

Local versus global

It is useful to distinguish between different types of projective measurements. Most importantly, one differentiates between local and global measurement bases (or observables) depending on whether the basis vectors \(| {v}_{i}\rangle \) are product states \({| {v}_{i}\rangle }_{AB}\) = \({| {u}_{i}\rangle }_{A}\)\({| {w}_{i}\rangle }_{B}\) with respect to the chosen bipartition A|B or not. Here, the choice of basis \({\{{| {v}_{i}\rangle }_{AB}\}}_{i}\) is referred to as a global setting, whereas bases \({\{{| {u}_{i}\rangle }_{A}\}}_{i}\) and \({\{{| {w}_{j}\rangle }_{B}}_{j}\) are called local settings. In the standard scenario for quantum communication, whenever the constituents of the quantum system are spatially separated, local (product basis) measurements are the only possible measurements. In this case, detection, certification or quantification of entanglement requires the measurement of (at least some) off-diagonal density matrix elements. These can be obtained by measurements of diagonal matrix elements of specific (product) bases conjugate with respect to the original basis. Alternatively, it is often useful to work directly with a local operator basis. That is, the Bloch picture can be extended to d-dimensional systems (qudits) and any number of parties in terms of a generalized Bloch decomposition80 by expanding a quantum state in a basis of suitable matrices gi, ρ = \({\sum }_{{i}_{1},{i}_{2},\ldots ,{i}_{n}=0}^{{d}^{2}-1}{\rho }_{{i}_{1}{i}_{2}\ldots {i}_{n}}{g}_{{i}_{1}}\otimes {g}_{{i}_{2}}\otimes \ldots \otimes {g}_{{i}_{n}}\). For instance, for two qudits and an operator basis that includes the identity, one has

$$\rho =\frac{1}{{d}^{2}}({1}_{{d}^{2}}+{\overrightarrow{v}}_{A}\overrightarrow{\sigma }\otimes {1}_{d}+{\overrightarrow{v}}_{A}{1}_{d}\otimes \overrightarrow{\sigma }\,+\sum _{i,j}{t}_{ij}{\sigma }_{i}\otimes {\sigma }_{j}),$$
(3)

where {σi}i is a basis of the SU(d) algebra. The Bloch coefficients themselves are obtained as expectation values of local observables, tij = 〈σiσjρ, making the Bloch basis a convenient expression of quantum states only in terms of results of local measurements instead of abstract density matrix elements. Whereas, in general, there exist d2 − 1 orthogonal generators of SU(d), requiring a large amount of observables to be measured for tomographic purposes (the gi generally do not have full rank), most of them can be represented through dichotomic operators and are thus often easier to implement than multi-outcome measurements. In contrast to any local measurements, probes interacting with multiple constituents of the system simultaneously or global observables whose eigenstates do not factorize (such as the magnetization) can give rise to entangling measurements. These measurements are inherently global, and the individual detector events can be used directly to estimate the correlators necessary for measuring entanglement witnesses. This is particularly relevant experimentally when the number of involved parties becomes very large, n ≈ 103−1012 or larger, in which case a reconstruction of the full density matrix is prevented by the extremely large number of required measurements. At the same time, it is typically possible to measure level populations and consequently infer moments of N-particle collective operators such as \({J}_{k}\) = \({\sum }_{i=1}^{n}{j}_{k}^{(i)}\). Such quantities are, in turn, directly related to inter-particle correlations, potentially providing information about entanglement.

Multi-outcome versus single outcome

Measurements in any basis may be classified by the method by which the relative frequencies of different measurement outcomes are recorded. In multi-outcome measurements, the interaction of a measurement device with a single copy of the measured system described by ρ provides one of several (ideally one of d) different outcomes i associated with the projection into \(| {v}_{i}\rangle \). That is, the detector event may fall into one of d categories that can be distinguished by the experimenter. After N such rounds of multi-outcome measurements, each resulting in one detector event, the outcome i is obtained Si times, such that \({\sum }_{i=1}^{d}{S}_{i}=N\), and the expected value of Mi is estimated to be Tr(Mi ρ) ≈ Si/N. However, in single-outcome measurements, filters are used to select only one particular outcome i, for which the detector (such as a photodetector placed behind a polarization filter) responds with a ‘click’. In principle, one may think of a ‘no click’ event as a second outcome, but this works only if the imminent event is heralded. A much simpler alternative is usually to collect the number Si of clicks in the filter setting i during some fixed integration period and again associate \(\left\langle {v}_{i}\,\,\right|\rho \left|\,{v}_{i}\right\rangle \approx {S}_{i}\,{\rm{/}}\,N\) with \(N={\sum }_{i=1}^{d}{S}_{i}\) for the chosen orthonormal basis \({\left\{| {v}_{i}\rangle \right\}}_{i}\). For non-orthonormal bases, this approach can still be used with minor modifications81. Crucially, the data corresponding to a d-outcome measurement can also be obtained from d individual single-outcome measurements. In principle, this also applies to local measurements. For instance, diagonal density matrix elements with respect to the product basis \({\{{\left|{u}_{i}\right\rangle }_{A}\otimes {| {w}_{j}\rangle }_{B}\}}_{i,j=1}^{d}\) in a d × d-dimensional Hilbert space can be obtained using d2 pairs of local filter settings, provided that local detection events for filter settings i and j fall within a sufficiently close time interval to be combined to ‘coincidences’ \({C}_{{i}_{A}{j}_{B}}\). More generally, for n parties, temporal coincidence allows association of the localized single events at n detectors into coincidences \({C}_{{i}_{1}{i}_{2}\ldots {i}_{n}}\) and global density matrix elements \(\left\langle {i}_{1}{i}_{2}\ldots {i}_{n}\right|\,\rho \,\left|{i}_{1}{i}_{2}\ldots {i}_{n}\right\rangle \) = \({C}_{{i}_{1}{i}_{2}\ldots {i}_{n}}{\rm{/}}{\sum }_{{i}_{1},{i}_{2},\ldots ,{i}_{n}}{C}_{{i}_{1}{i}_{2}\ldots {i}_{n}}\).

Statistical error and finite data

The discussion above illustrates that the number of measurement settings required for entanglement tests depends not only on the chosen theoretical method but also on what is counted, such as local or global bases and operators, filter settings (single outcome), dichotomic observables (two outcomes, such as for Bloch decompositions) or multi-outcome measurements. However, regardless of the method used, each single measurement setting still requires a number of repetitions of individual measurements to ensure the desired statistical confidence in the result. That is, the association Tr(Mi ρ) ≈ Si/N is exact only in the limit of infinitely many repetitions, and any real experiment using a finite number of measurements may estimate only probabilities or expected values from frequencies of occurrence of certain measurement outcomes. The confidence in these estimates is then guaranteed by a sufficiently large sample size (number of repetitions). How many samples can be taken with reasonable effort and time largely depends on the specific experimental setup. For instance, whereas many thousands of coincidences can be recorded every second in photonic setups used in communications and the resulting statistical error can be easily computed and does not heavily influence the conclusions drawn, state preparation in other systems is often tedious and not straightforwardly repeatable. In such scenarios, statistical errors and sufficiently narrow confidence intervals become prominent challenges that have to be addressed. Certifying entanglement with finite data was first addressed with simulated two-qubit data82, but similar reasoning also applies to methods directly aimed at state estimation83,84. In this context, ref.85 also provides a cautionary tale against density matrix reconstruction techniques, as neglecting errors can lead to a systematic overestimation of entanglement and underestimation of fidelity (maximum likelihood reconstructions have thus recently been deemed inappropriate for fidelity estimation86). In general, different measurement techniques come at different experimental cost for entanglement estimation or state tomography. This cost can be quantified in the number of states needed for achieving statistical certainty (see, for instance, ref.87 for optimal strategies in the bipartite case). Nonetheless, if enough repetitions for meaningful statistics are possible (for example, for down-converted photons), the number of different measurement bases and settings remains the principal measure of efficiency. An overview of this figure of merit for the most common measurement strategies is shown in Table 2.

Table 2 Minimal number of measurement settings

Key challenges

High-dimensional entanglement

Entanglement dimensionality

High-dimensional Hilbert spaces enable an encoding of more bits per photon and thus promise increased communication capacities over quantum channels. However, if the security of these channels is to be ensured by entanglement, a major challenge is the certification of high-dimensional entanglement because it should be done with as few measurements as possible and without introducing unwarranted assumptions that may lead to exploitable loopholes in the certification. In this context, matrix completion techniques88,89, semi-definite programmes89,90, uncertainty relations91 and mutually unbiased bases81,92,93 provide versatile tools for quantifying high-dimensional entanglement in different contexts.

The canonical witnesses for known target states \(| {\psi }_{{\rm{T}}}\rangle \) shown in Box 1 can readily be generalized to detect high-dimensional entanglement in the same way. One defines \({W}_{k}\) := \({\sum }_{i=1}^{k}\,{\lambda }_{i}^{2}{\mathbb{1}}-| {\psi }_{{\rm{T}}}\rangle \langle {\psi }_{{\rm{T}}}| \), where \({\sum }_{i=1}^{k}{\lambda }_{i}^{2}\) denotes the sum over the k largest squared Schmidt coefficients of the target state94. Whereas this witness faithfully certifies high-dimensional entanglement of any pure target state, it is decomposable (for instance, it detects only NPT states) and features a weak resistance to noise. However, it requires only an estimate of the target state fidelity, which can be efficiently obtained with few measurements81,87.

High-dimensional entanglement can also be ascertained using suitable quantitative measures. For instance, certifying an entanglement of formation beyond log2(k) also implies (k + 1)-dimensional entanglement. Alternatively, high-dimensional entanglement can also be quantified directly by the g-concurrence95, the bounds for which can be obtained from nonlinear witness operators96.

From a local Hilbert space perspective, multiple copies of entangled qubit pairs can be considered as equivalent to high-dimensionally entangled systems. However, this equivalence breaks down for distributed quantum systems; that is, genuine high-dimensionally entangled systems can feature correlations that are, in principle, unattainable by multiple copies of two-qubit entangled states97, which has recently been used in a photonic experiment to verify genuine high-dimensional entanglement98.

Besides practical challenges, many open questions still remain concerning the mathematical structure of high-dimensional entanglement. Whereas it is known to generically occur in high-dimensional Hilbert spaces99, few techniques are known for constructing witnesses detecting PPT entanglement in high dimensions (or, dual to that problem, non-decomposable k-positive maps54). Even among PPT states, high-dimensional entanglement is generic100 but, at the same time, not maximal101.

Photonic high-dimensional entanglement

Photonic systems, which are inherently multimode in the temporal and spatial degrees of freedom, naturally lend themselves to the creation and measurement of high-dimensional entanglement. Here, we discuss some landmark experiments and accompanying theoretical techniques used for demonstrating the high-dimensional entanglement of two photons in their orbital angular momentum (OAM), transverse spatial position–momentum, time–frequency and path degrees of freedom.

The high-dimensional entanglement of two photons in the spatial or temporal degrees of freedom usually results from the conservation of energy and momentum in a second-order nonlinear process, such as spontaneous parametric down-conversion. This process entails the annihilation of one pump photon with energy ħω and zero OAM in a nonlinear crystal, resulting in the creation of two daughter photons with energy \(\frac{1}{2}\hslash \omega \). Whereas formally the dimension of the Hilbert space relating to modal properties is infinite, only a finite number of modes will be populated significantly. Thus, the effective dimensionality of the resulting two-photon state depends on the spectral and spatial properties of the pump beam, as well as on the phase-matching function governing the nonlinear process. For example, the width of the pump beam and the length of the nonlinear crystal determine the dimensionality of an OAM-entangled state102.

Some of the first demonstrations of high-dimensional entanglement were performed with photons entangled in their OAM, which is a discrete quantum property resulting from a spatially varying amplitude and phase distribution103,104. This type of entanglement was first demonstrated with Schmidt number dent = 3 in an experiment that measured a generalized Bell-type inequality105 with single-outcome, holographic projective filters that allowed the measurement of coherent superpositions of OAM at the single-photon level106. In recent years, the development of computer-programmable wavefront-shaping devices, such as spatial light modulators, has allowed the measurement of OAM-entangled states with ever-increasing dimension. Examples of such experiments include the certification of dent = 100 spatial-mode entanglement with a visibility-based entanglement witness107 and dent = 11 OAM-entanglement with a generalized Bell-type test108, both with certain assumptions on the state. More recently, an assumption-free entanglement witness was implemented with spatial light modulators certifying dent = 9 OAM entanglement with only two measurement settings81.

A natural second basis for observing high-dimensional entanglement is found in the transverse photonic position–momentum degrees of freedom. A discretized version of the transverse position can be thought of as a ‘pixel’ basis, which is particularly relevant today with the development of sensitive single-photon cameras. Pixel entanglement was first observed with arrays of three and six fibres109, and entanglement was certified by violating the EPR–Reid criterion66 by setting a lower bound on the product of conditional variances in position and momentum: Δ2(ρ1 − ρ22(p1 + p2) ≥ \(\frac{{\hslash }^{2}}{4}\). More recently, electron-multiplying cameras that exhibit a high single-photon detection efficiency have been used to violate the EPR–Reid criterion by very high values, albeit by subtracting a large, uncorrelated background110,111. Other approaches that aim to reduce the number of measurements required to certify position–momentum entanglement have been developed, such as using compressed-sensing techniques to measure such states in a sparse basis112 or employing periodic masks to increase photon-counting rates113.

It is important to note here that in several experimental works, the term Schmidt number is used to define a different concept than the canonical one mentioned in the introduction. This surrogate quantity refers to the inverse purity, which for pure states is related to the Schmidt coefficients via \({\rm{PR}}(| \psi \rangle )\) = \({({\sum }_{i}{\lambda }_{i}^{4})}^{-1}\) and is supposed to roughly quantify the number of local dimensions that relevantly contribute to the observed coincidences. This approach was introduced114 to describe pure continuous-variable systems, in which the Schmidt rank of pure two-mode squeezed states is infinite while any proper entanglement entropy is still finite (in particular, the inverse purity is the exponential of the Rényi-2 entropy of entanglement).

The development of silicon integrated photonic circuits presents another versatile platform for high-dimensional entanglement, in which quantum states are simply encoded in different optical paths of a circuit. Whereas such circuits have been used extensively for quantum information processing with qubits115,116, their first implementation for qutrit entanglement was demonstrated only recently, with integrated multiport devices enabling the realization of any desired local unitary transformation in a two-qutrit space117. A more recent experiment certified up to dent = 14 through the use of nonlinear device-independent dimension witnesses in a large-scale 16-mode photonic integrated circuit and demonstrated violations of a generalized Bell-type inequality105 and the recently developed Salavrakos–Augusiak–Tura–Wittek–Acín–Pironio (SATWAP) inequality118 in up to dent = 8 (ref.119).

Alongside position–momentum encoding, the time–frequency domain presents yet another powerful platform available for the investigation of high-dimensional entanglement. Early experiments in this direction demonstrated high-dimensional entanglement in photonic time bins generated by spontaneous parametric down-conversion with a mode-locked, pulsed pump laser120,121. A central challenge in certifying time-bin entanglement is measuring coherent superpositions of multiple time bins. Usually performed with unbalanced interferometers, this method can measure only a single 2D subspace at a time and faces problems of scalability and stability. A recent experiment overcame these problems through the use of matrix completion methods that required only coherent superpositions of adjacent time bins in order to certify dent = 18 entanglement with 4.1 ebits of entanglement of formation89. In parallel, experiments certifying high-dimensional frequency-mode entanglement have also been demonstrated, for example, by the manipulation of broadband spontaneous parametric down-conversion through spatial light modulators122 or through electro-optic phase modulation of photons generated by spontaneous four-wave mixing in integrated micro-ring resonators123. Finally, multiple photonic degrees of freedom can be combined to produce what is referred to as hyperentanglement. This was first demonstrated with photonic OAM, time–frequency and polarization, in which entanglement was certified in each degree of freedom through a Bell Clauser–Horne–Shimony–Holt test124. More recently, a hyperentangled state of polarization and energy–time was transmitted over 1.2 km of free space, and high-dimensional entanglement in dent = 4 was certified through an entanglement witness relating visibility to state fidelity90.

In addition to photonic systems, high-dimensional quantum states have been realized in other systems, such as Caesium atoms125, transmon superconducting qubits126 and nitrogen vacancy centres127. Recent progress has also been made on entangling two micromechanical oscillators consisting of nanostructured silicon beams128. Matter-based systems such as these may provide yet another playground for exploring the types of complex entanglement achieved thus far only with photonic systems.

Multipartite entanglement

Applications of multipartite entanglement

The controlled generation and manipulation of multipartite entangled states are big challenges in current experiments. Multipartite entangled states feature across different disciplines, and thus our Review cannot do justice to the complexity of this topic. To name just a few, multipartite entanglement forms the basis for quantum networking proposals in quantum communication129,130,131,132, it is a key resource for beating the standard quantum limit in quantum metrology133, it is important in quantum error correcting codes134, and it appears as a generic ingredient in quantum algorithms135 and as the principal resource in measurement-based quantum computation136. The latter two topics motivated the introduction of quantum states representable by graphs137 or hypergraphs138. As these are locally equivalent to so-called stabilizer states, the two concepts are often used synonymously and a lot of effort has been invested in certifying entanglement for stabilizer states139,140.

Furthermore, apart from practical, technologically oriented applications, (multipartite) entanglement is closely connected to important physical phenomena, from the physics of many-body systems to quantum thermodynamics and quantum gravity. In thermodynamics, the entanglement of many-body systems is a crucial ingredient for reaching thermodynamic equilibrium141, whereas the growth of entanglement entropies with subsystem areas or volumes is of high importance in the field of condensed-matter physics26,28. The entanglement of thermal states is drastically influenced by quantum criticality: a high degree of entanglement appears in ground states across a quantum phase transition, with a scaling law that depends on the universality class of the transition. Thus far, theoretical studies in this framework have been devoted to entanglement across bipartitions, especially in ground states (quantified by the concurrence of two sites crossing the partition or the von Neumann entropy of a block), as well as to multipartite entanglement in thermal states (with criteria arising from collective quantities)20,142. In the former case, the area law of entanglement for non-critical systems emerged as a major result together with the corresponding classification of entangled states as tensor networks, a notion closely connected to classical simulability of many-body states26,27,28.

Given the different types of entanglement that can exist between multiple constituents and the different physical platforms, approaches to entanglement certification vary. First, we overview a selection of theoretical techniques and then provide examples of their application in few-body systems, such as photons and ion traps (Boxes 2 and 3, respectively), and many-body systems, such as atomic gases (Box 4), noting that other promising realizations of multipartite entanglement exist (for instance, using superconducting qubits143,144), but their detailed description goes beyond the scope of this Review.

Genuine multipartite entanglement

The definitions for entanglement across bipartitions of the systems straightforwardly carry over to the many-particle case, but there is a much deeper structure underlying the potential ways in which multipartite systems can be entangled. To unravel this structure, we revisit the definition of separability. There exist states of multipartite systems that can be factored into tensor products of multiple parts. This leads to the definition of k-separable pure states as \(| {{\rm{\Psi }}}_{k-{\rm{sep}}}\rangle \) := \({\otimes }_{i=1}^{k}| {{\rm{\Phi }}}_{{\alpha }_{i}}\rangle \), where the αi {1, 2, , N} refer to specific subsets of systems in the collection of N parties, that is, \({\cup }_{i=1}^{k}{\alpha }_{i}\) = {1, 2, , N} and αiαj = i ≠ j. States for which k = N are called fully separable, as there is no entanglement in the system. In the other extreme of k = 1, states are called multipartite entangled, because for all possible partitions of the system, one finds entanglement. Considering general (mixed) quantum states adds another layer of complexity to this notion, as k-separability has to be defined as \({\rho }_{k-{\rm{sep}}}:={\sum }_{i}{p}_{i}| {\Psi }_{k-{\rm{sep}}}^{i}\rangle {\langle {\Psi }_{k-{\rm{sep}}}^{i}| }_{}^{}\), where each of the \({| {{\rm{\Psi }}}_{k-{\rm{sep}}}^{i}\rangle }_{}^{}\) can be separable with respect to a different k-partition. Whereas states with k = N are still fully separable and can be prepared purely by LOCC, the case of k = 1 is referred to as genuine multipartite entanglement. Here, the word genuine emphasizes the fact that the state indeed cannot be prepared by LOCC without the use of multipartite entangled pure states. Hence, in contrast to the pure state case, there exist density matrices that are entangled across every partition and yet do not require multipartite entanglement for their creation.

Entanglement depth

Whereas the above definition reveals one aspect of entanglement in multipartite systems, it is far from a complete characterization. Consider the two states \(| {\psi }_{1}\rangle \otimes | {\psi }_{234}\rangle \) and \(| {\psi }_{12}\rangle \otimes | {\psi }_{34}\rangle \). Both are 2-separable, yet one describes a tripartite entangled system decoupled from a fourth party, whereas the other represents a pair of independent bipartite entangled states. The concept of entanglement depth attempts to capture this distinction, quantifying the number of entangled subsystems in a multipartite state. In the above example, the entanglement depths would be three and two, respectively. Analogous to GME, the generalization to mixed states makes use of a contrapositive: a state is called k-producible if it can be decomposed as a mixture of products of k-particle states, ρk−prod = \({\sum }_{i}{p}_{i}{({\rho }_{{\beta }_{1}}\otimes \ldots \otimes {\rho }_{{\beta }_{M}})}_{i}\), where the \({\rho }_{{\beta }_{m}}\) are states of at most k parties. By contrast, a state that is not k-producible has a depth of entanglement of at least k + 1 (refs73,145). The two notions of k-separability and k-producibility are hence quite different but match in the extremal cases: a fully separable state is also 1-producible, whereas a genuine N-partite entangled state also has an entanglement depth of N (that is, it is N-producible but not (N − 1)-producible). The concept of entanglement depth is particularly useful for systems with a large number of particles, that is, approaching the thermodynamic limit, because the resulting hierarchy is (somewhat) independent of the total number of particles N. Entanglement depth is therefore often used in experiments with atomic ensembles146.

Tensor rank and Schmidt rank vectors

In contrast to the bipartite case, for multipartite systems, there is no such thing as a Schmidt decomposition (at least not in the same sense). That is, not every multipartite state can be written as \(\left|\,{{\rm{\Psi }}}_{N}\right\rangle \) = \({\sum }_{i}{\lambda }_{i}{\left|i\right\rangle }^{\otimes N}\). Nonetheless, there are two prominent ways to generalize the Schmidt rank for multipartite pure states. The first is the tensor rank rT, which is defined as the minimum number of coefficients λi, such that the state can be written as \(\left|\,{{\rm{\Psi }}}_{N}\right\rangle \) = \({\sum }_{i=1}^{{r}_{{\rm{T}}}}{\lambda }_{i}{\otimes }_{x=1}^{N}\left|\,{v}_{i}^{x}\right\rangle \), so \({\otimes }_{x=1}^{N}\langle {v}_{i}^{x}| {v}_{j}^{x}\rangle \) = \({\delta }_{ij}\). Similar to the Schmidt rank, rT = 1 implies full separability of the state. It is at least NP-hard to determine the tensor rank even for pure states147. Moreover, the tensor rank is not additive under tensor products148 and is known only for very few exemplary multipartite states with particular symmetries149. One can, however, bound the tensor rank from below by considering the Schmidt ranks with respect to all possible partitions \({\alpha }_{i}| {\bar{\alpha }}_{i}\), which we denote by \({r}_{{\alpha }_{i}}\) because it is also the rank of the corresponding reduced density matrix, \({r}_{{\alpha }_{i}}\) = \({\rm{rank}}({{\rm{Tr}}}_{{\bar{\alpha }}_{i}}\left|\,{{\rm{\Psi }}}_{N}\right\rangle \left\langle {\Psi }_{N}\,\right|)\). Using this definition, it is easy to see that \({r}_{{\rm{T}}}\ge {{\rm{\max }}}_{i}{r}_{{\alpha }_{i}}\). The second generalization used as an alternative to the tensor rank is the collection of the marginal ranks in the Schmidt rank vector79 \({\left[{\overrightarrow{r}}_{{\rm{S}}}\right]}_{i}:={r}_{{\alpha }_{i}}\). Because there are 2N−1 − 1 possible bipartitions of the system, this vector has exponentially many components, and a state is fully separable if and only if \({\parallel {\overrightarrow{r}}_{{\rm{S}}}\parallel }^{2}\) = 2N−1 − 1, that is, if every marginal rank is equal to one. Although this vector admits different ranks across different partitions, strict inequalities exist that limit the possible vectors to a nontrivial cone150. A consistent generalization of multipartite entanglement dimensionality can then be given as dGME(ρ) := \({{\rm{\inf }}}_{{\mathscr{D}}(\rho )}\,{{\rm{\max }}}_{| {\psi }_{i}\rangle \in {\mathscr{D}}(\rho )}\) \({{\rm{\min }}}_{{\alpha }_{i}}{r}_{{\alpha }_{i}}(| {\psi }_{i}\rangle )\).

GME classes

The tensor rank and Schmidt rank vector give further insight into multipartite entanglement structures beyond qubits, but there is still a more complex structure hidden beneath. This was first realized in refs151,152, proving that even genuinely multipartite states of three qubits can be inequivalent under LOCC with the famous examples of the Greenberger–Horne–Zeilinger (GHZ) state \(\left|\,{\rm{GHZ}}\right\rangle \) := \(\frac{1}{\sqrt{2}}\left(\left|\,000\right\rangle +\left|\,111\right\rangle \right)\) and the W-state \(\left|\,{\rm{W}}\right\rangle \) := \(\frac{1}{\sqrt{3}}\left(\left|\,001\right\rangle +\left|\,010\right\rangle +\left|\,100\right\rangle \right)\). This already excludes easy operational measures of entanglement that could be interpreted as asymptotic resource conversions, such as in the bipartite case. In other words, there cannot be a single universal multipartite entangled reference state from which every other state can be created by LOCC (such as the maximally entangled state for bipartite systems). Whereas infinitely many states are needed for such a source set in general153, many cases allow finding finite maximally entangled sets of resource states to reach every other state (except for some isolated ‘islands’) through LOCC154. Another option is volume-based approaches, such as the volume of all states reachable by LOCC and the volume of all states from which a state can be reached by LOCC155. States for which the source volume is zero are extremal resources, whereas the target volume gives a good insight into the general utility of resource states for state transformations. Beyond deterministic transformations, one can also ask when a transformation from a state to another is possible probabilistically. This forms the basis for work in the sub-field of entanglement characterization using stochastic LOCC, which was first solved for four qubits156 and later for all states that allow for a ‘normal form’, which can be filtered to local-maximally mixed states157, comprising all states except for a measure-zero subset. GME of photons and trapped-ion qubits is discussed in Boxes 2 and 3.

Maximal entanglement

Whereas the previous examples show that a universal notion of maximal entanglement cannot exist in the context of LOCC resource theories, one can, in principle, define states to contain the maximum amount of entanglement if they are maximally entangled across every bipartition. Such states are used in quantum error correction134 and quantum secret sharing158 and are called absolutely maximally entangled (AME) states. It can be shown that for every number n of parties, there is a local dimension d admitting an AME state. However, for n qubits, AME states only exist for n = 2, 3, 5 and 6 (ref.159).

Monogamy of entanglement

Another signature of entanglement in multipartite systems is the phenomenon commonly referred to as monogamy of entanglement. The name alludes to the fact that entanglement is not arbitrarily sharable among many parties. To illustrate this point, an often invoked example is that of two parties, Alice and Bob, sharing a maximally entangled state ρAB such that EA:B(ρAB) = log2(min[dA, dB]). This precludes any further entanglement with a third party. This example, however, is strictly true if and only if dA = dB, in which case, maximal entanglement additionally implies purity of the state ρAB and thus a tensor product structure with respect to any third party. Quantitatively, monogamy relations are often written in the form

$${E}_{A:BC}\left({\rho }_{ABC}\right)\le {E}_{A:B}\left({\rho }_{AB}\right)+{E}_{A:C}\left({\rho }_{AC}\right)$$
(4)

The first prominent example valid for three qubits is the Coffman–Kundu–Wootters relation160, in which the respective entanglement measure is the squared concurrence45. This was later generalized to n qubits161 but proved not to hold for qutrits or higher dimensional systems162. Moreover, it has been shown that monogamy is a feature only for entanglement measures in a strict sense163 and that monogamy and ‘faithfulness’ (in a geometric sense) are mutually exclusive features of entanglement measures in general dimensions164. Meanwhile, additive measures, such as squashed entanglement165, are monogamous for general dimensions. The inequivalence of the two sides of the inequality (Eq. 4) can, in fact, be used to quantify and classify multipartite entanglement. For the squared concurrence of three qubits, their difference yields the three-tangle, which is non-zero only for GHZ states and can thus be used to distinguish it from biseparable or W states. A prominent property of the tangle is its invariance not only under local unitaries (SU(d)) but also under the complexification of SU(d) to SL(d) to encompass stochastic local operations. This led to the general research line of classifying multipartite entanglement in terms of stochastic LOCC using SL(d)-invariant polynomials166,167.

PPT mixers

Analogous to the bipartite case, the convex structure of (partial) separability permits the construction of multipartite entanglement witnesses. However, the additional challenge of the potentially different partitions of density matrix decomposition elements prevents the applicability of many techniques for bipartite witnesses in multipartite systems. In particular, positive maps and their resulting witnesses are inherently connected to bipartite structures. Nonetheless, they can be harnessed as constraints for positive semi-definite programming. This follows from the simple observation that a state that is decomposable into bi-product states is, for instance, also decomposable into PPT states. This insight has led to the concept of PPT mixers168, yielding effective numerical tools for low dimensions. At the same time, this connection can be used to effectively lift bipartite witnesses for multipartite usage169,170 and to obtain generalizations to maps that are positive on biseparable states171.

GME witnesses

A canonical form of GME witnesses can be obtained by harnessing the different Schmidt decompositions across bipartitions. For instance, for a pure target state \(| {\psi }_{{\rm{T}}}\rangle \), computing all marginal eigenvalues allows defining a witness172 of the form WGME := \({{\rm{\max }}}_{{\alpha }_{i}}{\parallel {\rho }_{{\alpha }_{i}}\parallel }_{\infty }\mathbb{1}-| {\psi }_{{\rm{T}}}\rangle \langle {\psi }_{{\rm{T}}}| \). Apart from this generically applicable method, most available GME witnesses are tailored towards detecting specific multipartite entangled states, such as graph states173 or stabilizer states139,140, Dicke states174 or generally symmetric states175.

Leaving the regime of linear operators and moving on to nonlinear functions of density matrix elements, more powerful certification techniques exist. In refs176,177, nonlinear inequalities for detecting multipartite entanglement in GHZ and W-like states were introduced, which were proved to be strictly more powerful than the canonical form introduced above. Moreover, these nonlinear inequalities were later shown to provide lower bounds on a particular measure of genuine multipartite entanglement, the GME-concurrence77. In fact, one can leverage positive semi-definite programming techniques to numerically evaluate multiple suitable convex-roof-based entanglement measures178. In a separate approach, separability eigenvalues were introduced as a means to construct multipartite entanglement witnesses179.

Entanglement and spin squeezing

A well-understood many-body system is an ensemble of N (pseudo)spins manipulated (and measured) collectively in a trap (Box 4). To detect entanglement, spin-squeezing criteria for entanglement have been derived. These are based on an analogy with bosonic quadratures and are connected with uncertainty relations of collective spin components. A necessary condition for all fully separable states of N particles with spin-1/2 reads

$${\xi }_{{\rm{S}}}^{2}:=N\frac{{\left({\rm{\Delta }}{J}_{z}\right)}^{2}}{{\left\langle {J}_{x}\right\rangle }^{2}+{| {J}_{y}\rangle }^{2}}\ge 1,$$

which also directly connects entanglement with enhanced sensitivity in Ramsey spectroscopy with totally polarized ensembles of atoms73,180,181,182. Here, (ΔJz)2 is the smallest variance in a direction orthogonal to the polarization, such as |〈Jy〉| ≈ N/2, and a spin-squeezed state is obtained when \({\xi }_{{\rm{S}}}^{2} < 1\), where the boundary value defines the coherent spin states.

As a generalization, a full set of spin-squeezing inequalities, which have the geometrical shape of a closed convex polytope and define a more general spin-squeezing quantifier, has been derived for spin-1/2 ensembles183,184 and later generalized to all higher spin-j ensembles and to su(d) observables different from angular momentum components185,186. Thus, witnessing entanglement through the squeezing of the collective spin of an ensemble is convenient because this notion is captured by a simple polytope in the space of collective spin variances. A similar simple structure remains even for device-independent certification of entanglement based on collective measurements187.

Entanglement depth is typically used as a quantifier of entanglement in spin-squeezed states, which can also be witnessed with spin-squeezing parameters by making use of the Legendre transform method73,74,75. The general picture is that one can find a hierarchy of bounds on some collective quantities that depend on the entanglement depth, such as (ΔJz)2 ≥ \(Nj{F}_{J}(\frac{\langle {J}_{y}\rangle }{Nj})\), where FJ is a certain convex function that can be obtained through Legendre transforms. A state with the property that the variance on the left-hand side is smaller than the quantity on the right-hand side for a certain FJ is detected with a depth of entanglement of at least k = J/j, where j is the spin quantum number of the individual particles. Similar entanglement depth criteria have also been derived for different target states, like Dicke states74,188 and planar quantum squeezed states75,189, as well as based on other quantities, such as the quantum Fisher information133,190,191,192.

Entanglement in optical lattices

A current challenge is to demonstrate and exploit multipartite entanglement in spatially extended systems, such as optical lattices. Here, as for localized traps, the most common measurement consists of releasing the gas from the trap (the lattice potential) and imaging the expanding gas, inferring the momentum distribution of the original system of particles. Besides spin-squeezing methods that could also be used in these systems, criteria to detect entanglement in optical lattices have been proposed based on quantities obtained from density measurements after a certain time of flight193,194. Furthermore, some collective quantities with thermodynamical significance, such as energy195,196 or susceptibilities197,198,199 (for instance, to external magnetic fields), could be used for entanglement detection in such extended systems. These quantities can be extracted from the structure factors coming from neutron scattering cross sections76,194,198,200,201. Some of these methods have been used for a first experimental demonstration (and quantification) of entanglement in a bosonic optical lattice200, whereas other recent experiments202,203,204 demonstrated entanglement between two spins in a lattice or a superlattice.

Outlook

Entanglement certification cannot be exhaustively covered in a single review. For the sake of brevity, we mainly discussed the case of well-characterized measurement devices and system Hamiltonians. It is indeed possible to transcend this paradigm and obtain robust entanglement certification techniques that do not require a detailed physical understanding of the measurement procedure or the investigated system. These device-independent certification techniques currently require more resources and suffer from poor robustness to experimental noise. As quantum technologies evolve, the logical next step is to move towards more device-independent certification techniques, increasing the security in quantum communication and the trust in the correct functionality of quantum devices.

Finally, whereas the use of bipartite high-dimensional entanglement is well established, the unfathomable complexity of multipartite quantum correlations has so far only found few applications in many-party protocols, and for some applications, they may not be useful at all (for example, universal quantum computation205). Finding further compelling quantum information protocols would motivate a deeper investigation of the structure of multipartite entanglement and guide theoretical and experimental efforts towards the preparation, manipulation and certification of novel many-body quantum states.

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Acknowledgements

The authors acknowledge support from the Austrian Science Fund (FWF) through the START project Y879-N27 and from the joint Czech–Austrian project MultiQUEST (I3053-N27 and GF17-33780L). N.F. acknowledges support from the FWF through project P 31339-N27. M.M. acknowledges support from the QuantERA ERA-NET co-fund (FWF Project I3773-N36) and from the UK Engineering and Physical Sciences Research Council (EPSRC) (EP/P024114/1). G.V. acknowledges support from the FWF through the Lise-Meitner project M 2462-N27.

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The authors declare no competing interests.

Correspondence to Marcus Huber.

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