Abstract
Entanglement is an important resource for quantum technologies. There are many ways quantum systems can be entangled, ranging from the twoqubit 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 stateoftheart 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 deviceindependence and assumptionfree certification.

Current challenges include the extension of wellunderstood methods for two qubits to manybody and/or highdimensional 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 Rosen^{1.} 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 localrealistic theory^{2.} 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 experiments^{3,4,5,6.} 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 Belllike inequalities^{7} is an active field of research, and recent experimental tests closed all loopholes^{8,9,10}, 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 phenomenon^{11.} Whereas entanglement is needed to violate Bell inequalities, it is still not known whether (and in what sense) entanglement always allows for Bell violation^{12,13,14,15.} From a contemporary perspective, Bell inequalities are seen as deviceindependent certifications of entanglement. However, the question of whether all entangled states can be certified deviceindependently 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 subfield of quantum information. Previous reviews have captured various aspects of the research in this subfield, focusing, for example, on the nature of nonentangled states^{16} and on the quantification of entanglement as a resource^{17,18}, or providing detailed collections of works on entanglement theory^{19} and entanglement detection^{20}.
In quantum communication, certifiable entanglement forms the basis for the next generation of secure quantum devices^{21,22,23,24.} 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 devices^{25.} In a quantum simulation, a large amount of entanglement can serve as an indicator of the difficulty of classically simulating the corresponding quantum states^{26,27,28.} Nonetheless, the precise role of entanglement in quantum computation and simulation is less clearcut than in quantum communication. Finally, entanglement can be understood as a means of bringing about speedups^{29,30,31,} parallelization^{32} and even flexibility^{33} in quantum metrology^{34.} It is not a coincidence that these four areas also form the central pillars of the European flagship programme on quantum technologies^{35.}
With the development of the first largescale 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 finitedimensional systems, which are the focus of this Review; for continuousvariable entanglement and infinitedimensional systems, we refer the interested reader to existing reviews^{36,37,38.}
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 (finitedimensional) 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 states^{11.}
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 {p_{i}}_{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 nondeterministic polynomialtime (NP)hard problem^{41.} 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 NPhard^{42.}
Pure states, separable or entangled, admit a Schmidt decomposition into biorthogonal product vectors, that is, one can write them as \({ \psi \rangle }_{AB}\) = \({\sum }_{i=0}^{k1}{\lambda }_{i} ii\rangle \). The coefficients \({\lambda }_{i}\in {{\mathbb{R}}}^{+}\) are called the Schmidt coefficients. Their squares, which are equal to the eigenvalues of the marginals ρ_{A/B} := Tr_{B/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 nonzero 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 theorem^{39,43}: 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 \(\left\,{{\rm{\Phi }}}^{+}\right\rangle \) := \(\frac{1}{\sqrt{d}}{\sum }_{i=0}^{d1} ii\rangle \) 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(ρ_{A/B}) = \({\sum }_{i=0}^{k1}{\lambda }_{i}^{2}{{\rm{log}}}_{2}\left({\lambda }_{i}^{2}\right)\), and the Rényi zeroentropy 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 E_{oF} (refs^{44,45}). 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 cost^{44,46,} the asymptotic LOCC interconversion rate from m twoqubit Bell states \({\left\psi \right\rangle }^{\otimes m}\) = \(\frac{1}{\sqrt{2}}{\left( 00\rangle +\left11\right\rangle \right)}^{\otimes m}\) to n copies of ρ, or ρ^{⊗n}. Conversely, one may define distillable entanglement as the asymptotic LOCC conversion rate from nonmaximally entangled states to Bell states^{47,48.} If the E_{oF} were additive, it would coincide with the entanglement cost. However, as shown by Hastings^{49,} the entanglement of formation is only subadditive. 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 indepth review, we refer the interested reader to refs^{17,18}. Whereas these and many other measures have very instructive and operational interpretations, even deciding whether they are nonzero is, in general, an NPhard 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 smallscale demonstrations^{51,52.} 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) criterion^{53,54.} Partially transposing a separable state leads to a positive semidefinite 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 Λ_{P}[ρ] ≥ 0 lead to positive semidefinite matrices when applied to positive semidefinite matrices, such as quantum states. Completely positive maps (Λ_{CP} ⊗ 𝟙_{d})[ρ] ≥ 0 \(\forall d\in {{\mathbb{Z}}}^{+}\), on the other hand, lead to positive semidefinite 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 subsystem^{54.}
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 distillation^{47,48,} 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 × 2dimensional 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 nonpositive under partial transposition (NPT) are distillable is still an open problem^{56,} but it is known that for any finite number of copies, the answer is negative^{57.}
The PPT map is also commonly used to quantify entanglement through the logarithmic negativity^{58,} 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 nonnegative. The logarithmic negativity is a prominent example of an entanglement monotone^{59} (as is the negativity^{60,61}), that is, a quantity that is nonincreasing under LOCC like any entanglement measure but that does not need to be nonzero 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 witnesses^{54} even if only partial or imprecise information about the state is available. Suppose one is provided with a theoretical target state ρ_{T} that is not positive semidefinite under a positive (but not completely positive) map Λ_{P}, Λ_{P}[ρ_{T}] ≱ 0. Then there exist vectors (for example, preferably the eigenvector ψ^{−}〉 of Λ_{P}[ρ_{T}] 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 ρ_{T}. Some illustrative examples of linear and nonlinear (in ρ) entanglement witnesses (negative values detected), positive (but not completely positive) maps (resulting nonpositive 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 \(\left\,\psi \right\rangle \) = \(\frac{1}{\sqrt{d}}{\sum }_{i=0}^{d1}{ i\rangle }^{\otimes n}\) for (n, d) = (2, 2), (2, 3) and (3, 2), respectively.
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 noncommuting singleparty observables, say (A_{1}, A_{2}) for party A and (B_{1}, B_{2}) for party B. Because the A_{i} do not commute with each other, their uncertainties cannot both be zero simultaneously. The same is true for the B_{i}. However, in the joint system, the uncertainties of the collective observables M_{i} = A_{i} ⊗ 𝟙+ 𝟙 ⊗ B_{i} 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(A^{2}ρ) − Tr(Aρ)^{2}. The sum \({\left({\rm{\Delta }}{A}_{1}\right)}_{{\rho }_{A}}^{2}\) + \({\left({\rm{\Delta }}{A}_{2}\right)}_{{\rho }_{A}}^{2}\) ≥ U_{A} must have a nonzero lower bound U_{A} > 0 for all singleparty states ρ_{A} whenever the two observables do not commute. Similarly, \({\left({\rm{\Delta }}{B}_{1}\right)}_{{\rho }_{B}}^{2}\) + \({\left({\rm{\Delta }}{B}_{2}\right)}_{{\rho }_{B}}^{2}\) ≥ U_{B} for all ρ_{B}. Thus, by simple concavity arguments, one can prove that \({\left({\rm{\Delta }}{M}_{1}\right)}_{{\rho }_{AB}}^{2}\) + \({\left({\rm{\Delta }}{M}_{2}\right)}_{{\rho }_{AB}}^{2}\) ≥ U_{A} + U_{B} must hold for all separable states ρ_{AB} = \({\sum }_{k}{p}_{k}{({\rho }_{A}\otimes {\rho }_{B})}_{k}\) (refs^{62,63,64,65}). This method hence combines two conceptual features: first, the LURs themselves — representing a tradeoff between information about different complementary (noncommuting) 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 entanglement^{34.} 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 formulation^{66,67,68.}
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 criterion^{69,70} 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) \(\rho \mapsto {\rho }_{{\rm{FNF}}}\) := (F_{A} ⊗ F_{B}) ρ (F_{A} ⊗ F_{B})^{†} such that ρ_{FNF} =\(\frac{1}{4}(\mathbb{1}_{4}+{\sum }_{i,j=x,y,z}{t}_{ij}{\sigma }_{i}\otimes {\sigma }_{j})\) , where σ_{k} (k = x,y,z) are the Pauli matrices. For local dimensions larger than two, the covariance matrix criterion can, in principle, be evaluated using semidefinite 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 transforms^{71,72.} In this context, let us define such a transform as \(\widehat{{\rm{E}}}(W)\) := sup_{ρ}[Tr(Wρ) − 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
where λ is real and Tr(Wρ) 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 relations^{73,74,75,76.} Another option is a direct construction of witnesses that have a natural connection between their expectation value and a suitably chosen entanglement measure^{77,78,79.}
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 operatorvalued 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 semidefinite operators M_{i} ≥ 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 {v_{i}}_{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 AB 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) offdiagonal 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 ddimensional systems (qudits) and any number of parties in terms of a generalized Bloch decomposition^{80} by expanding a quantum state in a basis of suitable matrices g_{i}, 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
where {σ_{i}}_{i} is a basis of the SU(d) algebra. The Bloch coefficients themselves are obtained as expectation values of local observables, t_{ij} = 〈σ_{i} ⊗ σ_{j}〉_{ρ}, 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 d^{2} − 1 orthogonal generators of SU(d), requiring a large amount of observables to be measured for tomographic purposes (the g_{i} generally do not have full rank), most of them can be represented through dichotomic operators and are thus often easier to implement than multioutcome 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 ≈ 10^{3}−10^{12} 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 Nparticle collective operators such as \({J}_{k}\) = \({\sum }_{i=1}^{n}{j}_{k}^{(i)}\). Such quantities are, in turn, directly related to interparticle correlations, potentially providing information about entanglement.
Multioutcome 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 multioutcome 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 multioutcome measurements, each resulting in one detector event, the outcome i is obtained S_{i} times, such that \({\sum }_{i=1}^{d}{S}_{i}=N\), and the expected value of M_{i} is estimated to be Tr(M_{i} ρ) ≈ S_{i}/N. However, in singleoutcome 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 S_{i} 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 nonorthonormal bases, this approach can still be used with minor modifications^{81.} Crucially, the data corresponding to a doutcome measurement can also be obtained from d individual singleoutcome 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 × ddimensional Hilbert space can be obtained using d^{2} 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 multioutcome 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(M_{i} ρ) ≈ S_{i}/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 twoqubit data^{82,} but similar reasoning also applies to methods directly aimed at state estimation^{83,84.} 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 estimation^{86}). 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 downconverted 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.
Key challenges
Highdimensional entanglement
Entanglement dimensionality
Highdimensional 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 highdimensional 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 techniques^{88,89,} semidefinite programmes^{89,90,} uncertainty relations^{91} and mutually unbiased bases^{81,92,93} provide versatile tools for quantifying highdimensional 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 highdimensional 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 state^{94.} Whereas this witness faithfully certifies highdimensional 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 measurements^{81,87.}
Highdimensional entanglement can also be ascertained using suitable quantitative measures. For instance, certifying an entanglement of formation beyond log_{2}(k) also implies (k + 1)dimensional entanglement. Alternatively, highdimensional entanglement can also be quantified directly by the gconcurrence^{95,} the bounds for which can be obtained from nonlinear witness operators^{96.}
From a local Hilbert space perspective, multiple copies of entangled qubit pairs can be considered as equivalent to highdimensionally entangled systems. However, this equivalence breaks down for distributed quantum systems; that is, genuine highdimensionally entangled systems can feature correlations that are, in principle, unattainable by multiple copies of twoqubit entangled states^{97,} which has recently been used in a photonic experiment to verify genuine highdimensional entanglement^{98.}
Besides practical challenges, many open questions still remain concerning the mathematical structure of highdimensional entanglement. Whereas it is known to generically occur in highdimensional Hilbert spaces^{99,} few techniques are known for constructing witnesses detecting PPT entanglement in high dimensions (or, dual to that problem, nondecomposable kpositive maps^{54}). Even among PPT states, highdimensional entanglement is generic^{100} but, at the same time, not maximal^{101.}
Photonic highdimensional entanglement
Photonic systems, which are inherently multimode in the temporal and spatial degrees of freedom, naturally lend themselves to the creation and measurement of highdimensional entanglement. Here, we discuss some landmark experiments and accompanying theoretical techniques used for demonstrating the highdimensional entanglement of two photons in their orbital angular momentum (OAM), transverse spatial position–momentum, time–frequency and path degrees of freedom.
The highdimensional entanglement of two photons in the spatial or temporal degrees of freedom usually results from the conservation of energy and momentum in a secondorder nonlinear process, such as spontaneous parametric downconversion. 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 twophoton state depends on the spectral and spatial properties of the pump beam, as well as on the phasematching 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 OAMentangled state^{102.}
Some of the first demonstrations of highdimensional entanglement were performed with photons entangled in their OAM, which is a discrete quantum property resulting from a spatially varying amplitude and phase distribution^{103,104.} This type of entanglement was first demonstrated with Schmidt number d_{ent} = 3 in an experiment that measured a generalized Belltype inequality^{105} with singleoutcome, holographic projective filters that allowed the measurement of coherent superpositions of OAM at the singlephoton level^{106.} In recent years, the development of computerprogrammable wavefrontshaping devices, such as spatial light modulators, has allowed the measurement of OAMentangled states with everincreasing dimension. Examples of such experiments include the certification of d_{ent} = 100 spatialmode entanglement with a visibilitybased entanglement witness^{107} and d_{ent} = 11 OAMentanglement with a generalized Belltype test^{108,} both with certain assumptions on the state. More recently, an assumptionfree entanglement witness was implemented with spatial light modulators certifying d_{ent} = 9 OAM entanglement with only two measurement settings^{81.}
A natural second basis for observing highdimensional 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 singlephoton cameras. Pixel entanglement was first observed with arrays of three and six fibres^{109,} and entanglement was certified by violating the EPR–Reid criterion^{66} by setting a lower bound on the product of conditional variances in position and momentum: Δ^{2}(ρ_{1} − ρ_{2})Δ^{2}(p_{1} + p_{2}) ≥ \(\frac{{\hslash }^{2}}{4}\). More recently, electronmultiplying cameras that exhibit a high singlephoton detection efficiency have been used to violate the EPR–Reid criterion by very high values, albeit by subtracting a large, uncorrelated background^{110,111.} Other approaches that aim to reduce the number of measurements required to certify position–momentum entanglement have been developed, such as using compressedsensing techniques to measure such states in a sparse basis^{112} or employing periodic masks to increase photoncounting rates^{113.}
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 introduced^{114} to describe pure continuousvariable systems, in which the Schmidt rank of pure twomode squeezed states is infinite while any proper entanglement entropy is still finite (in particular, the inverse purity is the exponential of the Rényi2 entropy of entanglement).
The development of silicon integrated photonic circuits presents another versatile platform for highdimensional 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 qubits^{115,116,} 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 twoqutrit space^{117.} A more recent experiment certified up to d_{ent} = 14 through the use of nonlinear deviceindependent dimension witnesses in a largescale 16mode photonic integrated circuit and demonstrated violations of a generalized Belltype inequality^{105} and the recently developed Salavrakos–Augusiak–Tura–Wittek–Acín–Pironio (SATWAP) inequality^{118} in up to d_{ent} = 8 (ref.^{119}).
Alongside position–momentum encoding, the time–frequency domain presents yet another powerful platform available for the investigation of highdimensional entanglement. Early experiments in this direction demonstrated highdimensional entanglement in photonic time bins generated by spontaneous parametric downconversion with a modelocked, pulsed pump laser^{120,121.} A central challenge in certifying timebin 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 d_{ent} = 18 entanglement with 4.1 ebits of entanglement of formation^{89.} In parallel, experiments certifying highdimensional frequencymode entanglement have also been demonstrated, for example, by the manipulation of broadband spontaneous parametric downconversion through spatial light modulators^{122} or through electrooptic phase modulation of photons generated by spontaneous fourwave mixing in integrated microring resonators^{123.} 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 test^{124.} More recently, a hyperentangled state of polarization and energy–time was transmitted over 1.2 km of free space, and highdimensional entanglement in d_{ent} = 4 was certified through an entanglement witness relating visibility to state fidelity^{90.}
In addition to photonic systems, highdimensional quantum states have been realized in other systems, such as Caesium atoms^{125,} transmon superconducting qubits^{126} and nitrogen vacancy centres^{127.} Recent progress has also been made on entangling two micromechanical oscillators consisting of nanostructured silicon beams^{128.} Matterbased 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 communication^{129,130,131,132,} it is a key resource for beating the standard quantum limit in quantum metrology^{133,} it is important in quantum error correcting codes^{134,} and it appears as a generic ingredient in quantum algorithms^{135} and as the principal resource in measurementbased quantum computation^{136.} The latter two topics motivated the introduction of quantum states representable by graphs^{137} or hypergraphs^{138.} As these are locally equivalent to socalled stabilizer states, the two concepts are often used synonymously and a lot of effort has been invested in certifying entanglement for stabilizer states^{139,140.}
Furthermore, apart from practical, technologically oriented applications, (multipartite) entanglement is closely connected to important physical phenomena, from the physics of manybody systems to quantum thermodynamics and quantum gravity. In thermodynamics, the entanglement of manybody systems is a crucial ingredient for reaching thermodynamic equilibrium^{141,} whereas the growth of entanglement entropies with subsystem areas or volumes is of high importance in the field of condensedmatter physics^{26,28.} 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 noncritical 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 manybody states^{26,27,28.}
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 fewbody systems, such as photons and ion traps (Boxes 2 and 3, respectively), and manybody systems, such as atomic gases (Box 4), noting that other promising realizations of multipartite entanglement exist (for instance, using superconducting qubits^{143,144}), 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 manyparticle 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 kseparable pure states as \( {{\rm{\Psi }}}_{k{\rm{sep}}}\rangle \) := \({\otimes }_{i=1}^{k} {{\rm{\Phi }}}_{{\alpha }_{i}}\rangle \), where the α_{i} ⊆ {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 α_{i} ∩ α_{j} = ∅∀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 kseparability 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 kpartition. 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 2separable, 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 kproducible if it can be decomposed as a mixture of products of kparticle states, ρ_{k−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 kproducible has a depth of entanglement of at least k + 1 (refs^{73,145}). The two notions of kseparability and kproducibility are hence quite different but match in the extremal cases: a fully separable state is also 1producible, whereas a genuine Npartite entangled state also has an entanglement depth of N (that is, it is Nproducible 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 ensembles^{146.}
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 \(\left\,{{\rm{\Psi }}}_{N}\right\rangle \) = \({\sum }_{i}{\lambda }_{i}{\lefti\right\rangle }^{\otimes N}\). Nonetheless, there are two prominent ways to generalize the Schmidt rank for multipartite pure states. The first is the tensor rank r_{T}, which is defined as the minimum number of coefficients λ_{i}, 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, r_{T} = 1 implies full separability of the state. It is at least NPhard to determine the tensor rank even for pure states^{147.} Moreover, the tensor rank is not additive under tensor products^{148} and is known only for very few exemplary multipartite states with particular symmetries^{149.} 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 vector^{79} \({\left[{\overrightarrow{r}}_{{\rm{S}}}\right]}_{i}:={r}_{{\alpha }_{i}}\). Because there are 2^{N−1} − 1 possible bipartitions of the system, this vector has exponentially many components, and a state is fully separable if and only if \({\parallel {\overrightarrow{r}}_{{\rm{S}}}\parallel }^{2}\) = 2^{N−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 cone^{150.} A consistent generalization of multipartite entanglement dimensionality can then be given as d_{GME}(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 refs^{151,152}, 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 Wstate \(\left\,{\rm{W}}\right\rangle \) := \(\frac{1}{\sqrt{3}}\left(\left\,001\right\rangle +\left\,010\right\rangle +\left\,100\right\rangle \right)\). 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 general^{153,} many cases allow finding finite maximally entangled sets of resource states to reach every other state (except for some isolated ‘islands’) through LOCC^{154.} Another option is volumebased 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 LOCC^{155.} 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 subfield of entanglement characterization using stochastic LOCC, which was first solved for four qubits^{156} and later for all states that allow for a ‘normal form’, which can be filtered to localmaximally mixed states^{157,} comprising all states except for a measurezero subset. GME of photons and trappedion 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 correction^{134} and quantum secret sharing^{158} 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 ρ_{AB} such that E_{A:B}(ρ_{AB}) = log_{2}(min[d_{A}, d_{B}]). This precludes any further entanglement with a third party. This example, however, is strictly true if and only if d_{A} = d_{B}, in which case, maximal entanglement additionally implies purity of the state ρ_{AB} and thus a tensor product structure with respect to any third party. Quantitatively, monogamy relations are often written in the form
The first prominent example valid for three qubits is the Coffman–Kundu–Wootters relation^{160,} in which the respective entanglement measure is the squared concurrence^{45.} This was later generalized to n qubits^{161} but proved not to hold for qutrits or higher dimensional systems^{162.} Moreover, it has been shown that monogamy is a feature only for entanglement measures in a strict sense^{163} and that monogamy and ‘faithfulness’ (in a geometric sense) are mutually exclusive features of entanglement measures in general dimensions^{164.} Meanwhile, additive measures, such as squashed entanglement^{165,} 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 threetangle, which is nonzero 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 polynomials^{166,167.}
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 semidefinite programming. This follows from the simple observation that a state that is decomposable into biproduct states is, for instance, also decomposable into PPT states. This insight has led to the concept of PPT mixers^{168,} yielding effective numerical tools for low dimensions. At the same time, this connection can be used to effectively lift bipartite witnesses for multipartite usage^{169,170} and to obtain generalizations to maps that are positive on biseparable states^{171.}
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 witness^{172} of the form W_{GME} := \({{\rm{\max }}}_{{\alpha }_{i}}{\parallel {\rho }_{{\alpha }_{i}}\parallel }_{\infty }\mathbb{1} {\psi }_{{\rm{T}}}\rangle \langle {\psi }_{{\rm{T}}} \). Apart from this generically applicable method, most available GME witnesses are tailored towards detecting specific multipartite entangled states, such as graph states^{173} or stabilizer states^{139,140,} Dicke states^{174} or generally symmetric states^{175.}
Leaving the regime of linear operators and moving on to nonlinear functions of density matrix elements, more powerful certification techniques exist. In refs^{176,177}, nonlinear inequalities for detecting multipartite entanglement in GHZ and Wlike 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 GMEconcurrence^{77.} In fact, one can leverage positive semidefinite programming techniques to numerically evaluate multiple suitable convexroofbased entanglement measures^{178.} In a separate approach, separability eigenvalues were introduced as a means to construct multipartite entanglement witnesses^{179.}
Entanglement and spin squeezing
A wellunderstood manybody system is an ensemble of N (pseudo)spins manipulated (and measured) collectively in a trap (Box 4). To detect entanglement, spinsqueezing 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 spin1/2 reads
which also directly connects entanglement with enhanced sensitivity in Ramsey spectroscopy with totally polarized ensembles of atoms^{73,180,181,182.} Here, (ΔJ_{z})^{2} is the smallest variance in a direction orthogonal to the polarization, such as 〈J_{y}〉 ≈ N/2, and a spinsqueezed 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 spinsqueezing inequalities, which have the geometrical shape of a closed convex polytope and define a more general spinsqueezing quantifier, has been derived for spin1/2 ensembles^{183,184} and later generalized to all higher spinj ensembles and to su(d) observables different from angular momentum components^{185,186.} 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 deviceindependent certification of entanglement based on collective measurements^{187.}
Entanglement depth is typically used as a quantifier of entanglement in spinsqueezed states, which can also be witnessed with spinsqueezing parameters by making use of the Legendre transform method^{73,74,75.} The general picture is that one can find a hierarchy of bounds on some collective quantities that depend on the entanglement depth, such as (ΔJ_{z})^{2} ≥ \(Nj{F}_{J}(\frac{\langle {J}_{y}\rangle }{Nj})\), where F_{J} is a certain convex function that can be obtained through Legendre transforms. A state with the property that the variance on the lefthand side is smaller than the quantity on the righthand side for a certain F_{J} 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 states^{74,188} and planar quantum squeezed states^{75,189,} as well as based on other quantities, such as the quantum Fisher information^{133,190,191,192.}
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 spinsqueezing 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 flight^{193,194.} Furthermore, some collective quantities with thermodynamical significance, such as energy^{195,196} or susceptibilities^{197,198,199} (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 sections^{76,194,198,200,201.} Some of these methods have been used for a first experimental demonstration (and quantification) of entanglement in a bosonic optical lattice^{200,} whereas other recent experiments^{202,203,204} 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 wellcharacterized 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 deviceindependent 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 deviceindependent certification techniques, increasing the security in quantum communication and the trust in the correct functionality of quantum devices.
Finally, whereas the use of bipartite highdimensional entanglement is well established, the unfathomable complexity of multipartite quantum correlations has so far only found few applications in manyparty protocols, and for some applications, they may not be useful at all (for example, universal quantum computation^{205}). 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 manybody quantum states.
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Acknowledgements
The authors acknowledge support from the Austrian Science Fund (FWF) through the START project Y879N27 and from the joint Czech–Austrian project MultiQUEST (I3053N27 and GF1733780L). N.F. acknowledges support from the FWF through project P 31339N27. M.M. acknowledges support from the QuantERA ERANET cofund (FWF Project I3773N36) and from the UK Engineering and Physical Sciences Research Council (EPSRC) (EP/P024114/1). G.V. acknowledges support from the FWF through the LiseMeitner project M 2462N27.
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