Accessible quantification of multiparticle entanglement

Entanglement is a key ingredient for quantum technologies and a fundamental signature of quantumness in a broad range of phenomena encompassing many-body physics, thermodynamics, cosmology, and life sciences. For arbitrary multiparticle systems, entanglement quantification typically involves nontrivial optimisation problems, and may require demanding tomographical techniques. Here we develop an experimentally feasible approach to the evaluation of geometric measures of multiparticle entanglement. Our approach provides analytical results for particular classes of mixed states of N qubits, and computable lower bounds to global, partial, or genuine multiparticle entanglement of any general state. For global and partial entanglement, useful bounds are obtained with minimum effort, requiring local measurements in just three settings for any N. For genuine entanglement, a number of measurements scaling linearly with N is required. We demonstrate the power of our approach to estimate and quantify different types of multiparticle entanglement in a variety of N-qubit states useful for quantum information processing and recently engineered in laboratories with quantum optics and trapped ion setups.


I. INTRODUCTION
The fascination with quantum entanglement has evolved over the last eight decades, from the realm of philosophical debate [1] to a very concrete recognition of its resource role in a range of applied sciences [2,3]. While considerable progress has been achieved in the detection of entanglement [4][5][6][7][8][9][10][11][12], its experimentally accessible quantification remains an open problem for any real implementation of an entangled system [13][14][15][16][17][18][19][20][21][22][23]. Quantifying entanglement is yet necessary to gauge precisely the quantum enhancement in information processing and computation [2,3,24], and to pin down exactly how much a physical or biological system under observation departs from an essentially classical behaviour [25]. This is especially relevant in the case of complex, multiparticle systems, for which only quite recently have notable advances been reported on the control of entanglement [26][27][28][29].
An intuitive framework for quantifying the degree of multiparticle entanglement relies on a geometric perspective [30][31][32]. Within this approach, one first identifies a hierarchy of non-entangled multiparticle states, also referred to as Mseparable states for 2 ≤ M ≤ N, where N is the number of particles composing the quantum system of interest; see Figure 1. Introducing then a distance functional D respecting natural properties of contractivity under quantum operations and joint convexity (see Methods) [33], the quantity E D M defined as is a valid geometric measure of (M-inseparable) multiparticle entanglement in the state . Some special cases are prominent in this hierarchy. For M = N, the distance from N-separable (also known as fully * cianciaruso.marco@gmail.com † thomas.r.bromley@gmail.com ‡ gerardo.adesso@nottingham.ac.uk separable) states defines the global multiparticle entanglement E D N , which accounts for any form of entanglement distributed among two or more of the N particles. Geometric measures of global entanglement have been successfully employed to characterise quantum phase transitions in many-body systems [34] and directly assess the usefulness of initial states for Grover's search algoritms [35]. On the other extreme of the hierarchy, for M = 2, the distance from 2-separable (also known as biseparable) states defines instead the genuine multiparticle entanglement E D 2 , which quantifies the entanglement shared by all the N particles, that is the highest degree of inseparability. Genuine multiparticle entanglement is an essential ingredient for quantum technologies including multiuser quantum cryptography [36], quantum metrology [37], and measurement-based quantum computation [38]. Finally, for any intermediate M, we can refer to E D M as partial multiparticle entanglement. The presence of partial entanglement is relevant in quantum informational tasks such as quantum secret sharing [10] and may play a relevant role in biological phenomena [25,39]. Probing and quantifying different types of entanglement can shed light on which nonclassical features of a mixed multiparticle state are necessary for quantumenhanced performance in specific tasks [7] and can guide the understanding of the emergence of classicality in multiparticle quantum systems of increasing complexity [40].
The quantitative amount of multiparticle entanglement, be it global or genuine (or any intermediate type), has an intuitive operational meaning when adopting the geometric approach. Namely, E D M measures how distinguishable a given state is from the closest M-separable state. Given some widely adopted metrics, such a distinguishability is directly connected to the usefulness of for quantum information protocols relying on multiparticle entanglement. For instance, the trace distance of entanglement is operationally related to the minimum probability of error in the discrimination between and any M-separable state with a single measurement [33]. Furthermore, the geometric entanglement with respect to relative entropy or Bures distance sets quantitative bounds on the number of orthogonal states that can be discriminated by local operations and classical communication (LOCC) [41]. The geometric entanglement based on infidelity [31] (monotonically related to Bures distance) has also a dual interpretation based on the convex roof construction [42], that is, it quantifies the minimum price (in units of pure-state entanglement) that has to be spent on average to create a given density matrix as a statistical mixture of pure states. It is therefore clear that finding the minimum in Eq. (1), and hence evaluating geometric measures of multiparticle entanglement defined by meaningful distances, is a central challenge to benchmark quantum technologies. However, obtaining such a solution for general multiparticle states is in principle a formidable problem. Even if possible, there would remain major challenges for experimental evaluation, which would in general require a complete reconstruction of the state through full tomography. For multiparticle states of any reasonable number of qubits, full state tomography places significant demands on experimental resources, and it is thus highly desirable to provide quantitative guarantees on the geometric multiparticle entanglement present in a state, via non-trivial lower bounds, in an experimentally accessible way [13][14][15][16][17][18][19][20][21][22][23].
Here we provide substantial advances towards addressing this problem in a general fashion. We identify a general framework for the provision of experimentally friendly quantitative guarantees on the geometric multiparticle entangle-ment present in a state. This approach consists of: (1) Choosing a set of reference states: Find a restricted family of N-qubit states with the property that any state may be mapped into this family through a fixed procedure of single-qubit LOCC. This reference family should be simple to characterise, and can be chosen from experimental or theoretical considerations.
(2) Identifying M-separable reference states: Apply the fixed LOCC procedure to the general set of Mseparable states, hence identifying the subset of M-separable states within the reference family.
(3) Calculating E D M for the reference states: Solve the optimisation problem for the geometric entanglement of reference states. This is dramatically simplified by using the properties of contractivity and joint convexity, that hold for any distance functional D defining a valid entanglement measure, and imply in particular that one of the closest M-separable states to any reference state is to be found itself within the reference family.
(4) Deriving optimised lower bounds for any state: Exploit the freedom to apply single-qubit unitaries to any N-qubit state in order to find the corresponding reference state with the highest geometric entanglement, providing an optimised lower bound to E D M ( ). This process presents a versatile and comprehensive approach to obtain lower bounds on geometric multiparticle entanglement measures according to any valid distance. While building on some previously utilised methods for steps (1) [13][14][15][16] and (4) [13,15,16], it introduces novel techniques in steps (2) and most importantly (3), which are crucial for completing the framework and making it effective in practice (see e.g. Appendix C, D, E, and F of the Supplementary Material).
To illustrate the power of our approach, we focus initially on a reference family of mixed states of N qubits, that we label M 3 N states, which form a subset of the class of states having all maximally mixed marginals. This family includes maximally entangled Bell states of two qubits and their mixtures, as well as multiparticle bound entangled states [43][44][45][46]. For any N, these states are completely specified by three easily measurable quantities, given by the correlation functions c j = σ ⊗N j , where {σ j } j=1,2,3 are the Pauli matrices. In the following we show how every entanglement monotone E D M can be evaluated exactly for any even N on these states, by revealing an intuitive geometric picture common to all valid distances D. For odd N, the results are distance-dependent; we show nonetheless that E D M can still be evaluated exactly if D denotes the trace distance. The results are nontrivial for all M > N/2 in the hierarchy of Figure 1. A central observation, in line with the general framework, is that an arbitrary state of N qubits can be transformed into an M 3 N state by a LOCC procedure, which cannot increase entanglement by definition. This implies that our exact formulae readily provide practical lower bounds to the degree of global and partial multiparticle entanglement in completely general states. Importantly, the bounds are obtained by measuring only the three correlation functions {c j } for any number of qubits, and can be further improved by adjusting the local measurement basis (see Figure 2 for an illustration).
Furthermore, we discuss how our results can be extended to allow for the quantitative estimation of genuine multiparticle entanglement as well, at the cost of performing extra measurements. Since M 3 N states are always biseparable, we must consider a different reference family. We focus on the class of N-qubit states obtained as mixtures of Greenberger-Horne-Zeilinger (GHZ) states [5,47], the latter being central resources for quantum communication and estimation; this class of states depends on 2 N − 1 real parameters. We calculate exactly distance-based measures of genuine multiparticle entanglement E D 2 for these states, for every valid D. Once more, these analytical results provide lower bounds to geometric measures of genuine entanglement for any general state of N qubits, obtainable experimentally in this case by performing at least N + 1 local measurements [48].

A. Global and partial multiparticle entanglement
We begin by choosing as our reference family the set of N-qubit M 3 N states. An M 3 N state is defined as = where I is the 2 × 2 identity matrix. These states are invariant under permutations of any pair of qubits and enjoy a nice geometrical representation in the space of the three correlation functions c j , corresponding to a tetrahedron for even N and to the unit ball for odd N, as depicted in Figure 2. We can then characterise the subset of M-separable M 3 N states for any N. We find that, if M > N/2 , then the M-separable M 3 N states fill a subset corresponding to an octahedron in the space of the correlation functions (see Figure 2). When M ≤ N/2 , all M 3 N states are instead M-separable. The proofs are deferred to the Supplementary Material.
We can now tackle the quantification of global and partial multiparticle entanglement in these states; for the latter, we will always focus on the nontrivial case M > N/2 throughout this section. First, we observe that the closest M-separable state ς to an M 3 N state , which solves the optimisation in Eq. (1), can always be found within the subset of M-separable M 3 N states, yielding a considerable simplification of the general problem. To find the exact form of ς , and consequently of E D M ( ), we approach the cases of even and odd N sepa-rately. For even N, we prove that there exists a unique solution to the minimisation in Eq. (1), independent of the specific choice of contractive and jointly convex distance D. Namely, the closest M-separable state ς is on the face of the octahedron bounding the corner of the tetrahedron in which is located, and is identified by the intersection of such octahedron face with the line connecting to the vertex of the tetrahedron corner, as depicted in Figure 2(d). It follows that, for any nontrivial M, valid D, and even N, the multiparticle entanglement E D M ( {c j } ) of an M 3 N state {c j } with correlation functions {c j } is only a monotonically increasing function of the Euclidean distance between the point of coordinates {c j } and the closest octahedron face, which is in turn proportional to h = 1 2 ( 3 j=1 |c j | − 1) (notice that h equals the bipartite measure known as concurrence for N = 2 [24,53]). We have then a closed formula for any valid geometric measure of global and partial multiparticle entanglement on an arbitrary M 3 N state {c j } with even N, given by where f D denotes a monotonically increasing function whose explicit form is specific to each distance D. In Table II we present the expression of f D for relevant distances in quantum information theory. For odd N, the closest M-separable state ς to any M 3 N state is still independent of (any nontrivial) M. However, different choices of D in Eq.
(1) are minimised by different states ς . We focus on the important but notoriously hard-toevaluate case of the trace distance D Tr ( , ς) (see Table II). In the representation of Figure 2(c), the trace distance amounts to half the Euclidean distance on the unit ball. It follows that the closest M-separable state ς to is the Euclidean orthogonal projection onto the boundary of the octahedron, see Figure 2(e). We can then get a closed formula for the trace distance measure of global and partial multiparticle entanglement E D Tr M ( {c j } ) of an arbitrary M 3 N state {c j } with odd N as well, given by The usefulness of the just derived analytical results for multiparticle entanglement is not limited to the M 3 N states. In accordance with our general framework, a crucial observation is that the M 3 N states are extremal among all quantum states with given correlation functions {c j }. Specifically, any general state of N qubits can be transformed into an M 3 N state with the same {c j } by means of a procedure that we name M 3 N -fication, involving only LOCC (see Methods). This immediately implies that, for any M > N/2 , the multiparticle entanglement   [22,23] Variable Optimisation required 2 ≤ M ≤ N Any convex and continuous measure [21] O(2 N ) Optimisation required 2 Genuine multiparticle concurrence [19,20] O(N) Closed formula 2 Robustness of entanglement [18] O(N) Closed formula 2 Genuine multiparticle concurrence [17] O(N) Closed formula 2 Genuine multiparticle concurrence [15,16] O(N) Closed formula 2 Polynomial invariant (three-tangle) [14] O(N) Closed formula 2 Genuine multiparticle negativity [13] O(N)  From a practical point of view, one needs only to measure the three correlation functions {c j }, as routinely done in optical, atomic, and spin systems [4,44,49,50], to obtain an estimate of the global and partial multiparticle entanglement content of an unknown state with no need for a full state reconstruction.
Furthermore, the lower bound can be improved if a partial knowledge of the state is assumed, as is usually the case for experiments aiming to produce specific families of states for applications in quantum information processing [44,45,49]. In those realisations, one typically aims to detect entanglement by constructing optimised entanglement witnesses tailored on the target states [4]. By exploiting similar ideas, we can optimise the quantitative lower bound in Eq. (4) over all possible single-qubit local unitaries applied to the state before the M 3 N -fication, where Tr U ⊗ U † ⊗ σ ⊗N j =c j and U ⊗ = N α=1 U (α) denotes a single-qubit local unitary operation. Experimentally, the optimised bound can then be still accessed by measuring a triple of correlations functions {c j } given by the expectation val-ues of correspondingly rotated Pauli operators on each qubit, c j = U † ⊗ σ ⊗N j U ⊗ , as illustrated in Figure 2(a). Optimality in Eq. (5) can be achieved by the choice of U ⊗ such that h = 1 2 ( 3 j=1 |c j | − 1) is maximum. The optimisation procedure can be significantly simplified when considering a state which is invariant under permutations of any pair of qubits. In such a case, one may need to optimise only over three angles {θ, ψ, φ} parameterising a generic unitary applied to each single qubit; the optimisation can be equivalently performed over an orthogonal matrix acting on the Bloch vector of each qubit (see Methods).
We can now investigate how useful our results are on concrete examples. Table III presents a compendium of optimised analytical lower bounds on the global and partial multiparticle entanglement of several relevant families of N-qubit states [46,47,51,52,[54][55][56][57][58][62][63][64], up to N = 8. All the bounds are experimentally accessible by measuring the three correlation functions {c j }, corresponding to optimally rotated Pauli operators (see also Figure 2).
Let us comment on some cases where our analysis is particularly effective. For GHZ states, cluster states, and halfexcited Dicke states, which constitute primary resources for quantum computation and metrology [37,38], we get the maximum h = 1 for any even N. This means that our bounds remain robust to estimate global and partial entanglement in noisy versions of these states (i.e. when one considers mixtures of any of these states with probability q and the maximally mixed state with probability 1 − q) for all q > 1/3. Notably, for values of q sufficiently close to 1, our bounds to global entanglement can be tighter than the (more experimentally demanding) ones derived very recently in Ref. [13], as shown in Figure 3(a). Focusing on noisy GHZ states, we observe however that our scale-invariant threshold q > 1/3, obtained by measuring the three canonical Pauli operators for each qubit, is weaker than the well-established inseparability threshold q > 1/(1 + 2 N−1 ) [9]. Nevertheless, we note that our simple quantitative bound given by Eq. (4) becomes tight in the paradigmatic limit of pure GHZ states (q = 1) of any even number N of qubits, thus returning the exact value of their global multiparticle entanglement via Eq. (2), despite the fact that such states are not (and are very different from) M 3 N states. Eq. (4) also provides a useful nonvanishing lower bound to the global (and partial) N-particle entanglement of Wei states in the interval x ∈ 1 2 , 1 , for any even N. A comparison between such a bound (with D denoting the relative entropy), which requires only three local measurements, and the true value of the relative entropy of global N-particle entanglement for these states [56], whose experimental evaluation would conventionally require a complete state tomography, is presented in Figure 3(b).

B. Genuine multiparticle entanglement
We now show how general analytical results for geometric measures of genuine multiparticle entanglement can be obtained as well within our approach. The results from the previous section, while quite versatile, cannot provide useful bounds for the complete hierarchy of multiparticle entanglement, because M 3 N states are M-separable for all M ≤ N/2 , and thus in particular biseparable for any number N of qubits. Therefore, to investigate genuine entanglement we consider a different reference set of states, specifically formed by mixtures of GHZ states, hence incarnating archetypical represen- TABLE III. Applications of our framework to construct accessible lower bounds on global and partial (M-inseparable) multiparticle entanglement (which are nonzero for any M > N/2 when j |c j | > 1), for the families of Nqubit states listed as follows. (i) N-qubit GHZ states [47] |GHZ (N) = 1 √ 2 (|00 · · · 00 + |11 · · · 11 ) with N ≥ 3. (ii) N-qubit W states [54] |W (N) = 1 √ N (|00 · · · 01 + |00 · · · 10 + · · · + |10 · · · 00 ) with N ≥ 3. (iii) N-qubit Wei states [55,56] and P k is the projector onto the binary N-qubit representation of , which are superpositions of all states with k qubits in the excited state |1 and N − k qubits in the ground state |0 , with the symbol  [43,46,61] hence their entanglement quantification is exact. The asterisk * indicates non-permutationally invariant states for which the optimisation of the bounds requires different angles for each qubit (not reported here). Notice that in the table we listed mostly pure states. In general, if the triple {c j } is optimal for a pure N-qubit state |Φ (N) , then for the mixed state (N) (q) = q|Φ (N) Φ (N) | + 1−q 2 N I ⊗N , obtained by mixing |Φ (N) with white noise, one still gets nonzero lower bounds to global and partial entanglement for all q > 1/ 3 j=1 |c j |, as shown in Figure 3 for some representative examples.
tatives of full inseparability [5,47]. Any such state ξ, which will be referred to as a GHZ-diagonal (in short, G N ) state, can be written as i=0 denotes the binary ordered N-qubit computational basis). The G N states have been studied in recent years as testbeds for multiparticle entanglement detection [5,48], and specific algebraic measures of genuine multiparticle entanglement such as the N-particle concurrence [18,21] and negativity [14] have been computed for these states. Here, we calculate exactly the whole class of geometric measures of genuine multiparticle entanglement E D 2 defined by Eq.
(1), with respect to any contractive and jointly convex distance D, for G N states of an arbitrary number N of qubits.
By applying our general framework, we can prove that, for every valid D, the closest biseparable state to any G N state can be found within the subset of biseparable G N states (see Supplementary Material for detailed derivations). The latter subset is well characterised [5], and is formed by all, and only, the G N states with eigenvalues such that p max ≡ max i,± p ± i ≤ 1/2. We can then show that the closest biseparable G N state to an arbitrary G N state has maximum eigenvalue equal to 1/2, which allows us to solve the optimisation in the definition of E D 2 , with respect to every valid D. We have then a closed formula for the geometric multiparticle entanglement of any G N state ξ with maximum eigenvalue p max , given by where g D denotes a monotonically increasing function whose explicit form is specific to each distance D, as reported in Table IV for typical instances. Let us comment on some particular results. The genuine multiparticle trace distance of entanglement E D Tr 2 is found to coincide with the genuine multiparticle negativity [14] and with half the genuine multiparticle concurrence [18] for all G N states, thus providing the latter entanglement measures with an insightful geometrical interpretation on this important set of states. Examples of G N states include several resources for quantum information processing, such as the noisy GHZ states and Wei states introduced in the previous section. In particular, for noisy GHZ states (described by a pure-state  Table III, as a function of the probability x of obtaining a GHZ state. The dashed red line E D RE N ( (N) Wei ) = x denotes the exact value of the global relative entropy of entanglement as computed in [56]. The solid blue line denotes our accessible lower bound, obtained by combining Eqs. (2) and (5) with the expressions in Tables II and III, and given explicitly by The bound becomes tight for x = 1, thus quantifying exactly the global multiparticle entanglement of pure GHZ states. We further show that our lower bound to global entanglement coincides with the exact genuine multiparticle entanglement of Wei states, , that is computed in the next section of this paper. The results are scale-invariant and hold for any even N. probability q as detailed in Table III), we recover that every geometric measure of genuine multiparticle entanglement is nonzero if and only if q [5] and monotonically increasing with q, as expected; for q = 1 (pure GHZ states), genuine and global entanglement coincide, i.e. the hierarchy of Figure 1 collapses, meaning that all the entanglement of N-qubit GHZ states is genuinely shared among all the N particles [32]. On the other hand, the relative entropy of genuine multiparticle entanglement of Wei states [55,56] can be calculated exactly via Eq. (6); interestingly, for even N it is found to coincide with the lower bound to their global entanglement that we had obtained by M 3 Nfication, plotted as a solid line in Figure 3(b). This means that for these states also the genuine multiparticle entanglement can be quantified entirely by measuring the three canonical correlation functions {c j }, for any N. More generally, for arbitrary G N states, all the genuine entanglement measures given by Eq. (6) can be obtained by measuring the maximum GHZ overlap p max , which requires N + 1 local measurement settings given explicitly in Ref. [48]. This is remarkable, since with the same experimental effort needed to detect full inseparability [5] we have now a complete quantitative picture of genuine entanglement in these states based on any geometric measure, agreeing with and extending the findings of [14,18]. Furthermore, as evident from Eq. (6), all the geometric measures are monotonic functions of each other: our analysis thus reveals that there is a unique ordering of genuinely entangled G N states within the distance-based approach of Fig. 1.
In the same spirit as the previous section, and in compliance with our general framework, we note that the exact results obtained for the particular reference family of G N states provide quantitative lower bounds to the genuine entanglement of general N-qubit states. This follows from the observation that any N-qubit state can be transformed into a G N state with eigen- by a LOCC procedure that we may call GHZ-diagonalisation [14]. Therefore, given a completely general state , one only needs to measure its overlap with a suitable reference GHZ state; if this overlap is found larger than 1/2, then by using Eq. (6) with p max equal to the measured overlap one obtains analytical lower bounds to the genuine multiparticle entanglement E D 2 of with respect to any desired distance D. As before, the bounds can be optimised in situations of partial prior knowledge, e.g. by applying local unitaries on each qubit before the GHZ-diagonalisation, which has the effect of maximising the overlap with a chosen particular GHZ vector in the basis {|β ± i }. The bounds then remain accessible for any state by N + 1 local measurements [48], with exactly the same demand as for just witnessing entanglement [5].
For instance, for the singlet state |Ψ (4) [60], which is a relevant resource in a number of quantum protocols including multiuser secret sharing [65][66][67], one has p max = β + 3 |Ψ (4) Ψ (4) |β + 3 = 2/3 > 1/2, obtainable by measuring the overlap with the GHZ basis state |β + 3 = (|0011 +|1100 )/ √ 2. Optimised bounds to the genuine multiparticle entanglement of half-excited Dicke states |D (N) N/2 (for even N ≥ 4), defined in Table III [58,59], can be found as well based on GHZ-diagonalisation, and are expressed by p (N) max = N N/2 2 1−N , meaning that they become looser with increasing N and stay nonzero only up to N = 8. In this respect, we note that alternative methods to detect full inseparability of Dicke states for any N are available [4,51,52], but quantitative results are lacking in general. Nevertheless, applying our general approach to an alternative reference family more tailored to the Dicke states could yield tighter lower bounds that do not vanish beyond N = 8.
Finally, notice that a lower bound to a distance-based measure of genuine multiparticle entanglement, as derived in this section, is automatically also a lower bound to corresponding measures of global and any form of partial entanglement, as evident by looking at the geometric picture in Figure 1. However, for states which are entangled yet not genuinely entangled, the simple bound from the previous section remains instrumental to assess their inseparability with minimum effort. M 3 N states are themselves instances of such states (in fact, for even N, M 3 N states are also G N states, but with p max ≤ 1/2 for N > 2).

C. Applications to experimental states
In this section, we benchmark the applicability of our results to real data from recent experiments [44,[49][50][51][52]68].
In Refs. [44,45], the authors used quantum optical setups to prepare an instance of a bound entangled four-qubit state, known as Smolin state [61]. Such a state cannot be written as a convex mixture of product states of the four qubits, yet no entanglement can be distilled out of it, thus incarnating the irreversibility in entanglement manipulation while still representing a useful resource for information locking and quantum secret sharing [3,46]. It turns out that noisy Smolin states are particular types of M 3 N states (for any even N) [43,46], that in the representation of Figure 2(b) are located along the segment connecting the tetrahedron vertex {(−1) N/2 , (−1) N/2 , (−1) N/2 } with the origin. Therefore, this work provides exact analytical formulae for all the nontrivial hierarchy of their global and partial entanglement, as mentioned in Table III. In the specific experimental implementation of Ref. [44] for N = 4, the global entanglement was detected (but not quantified) via a witness constructed by measuring precisely the three correlation functions {c j }. Based on the existing data alone (and without assuming that the produced state is within the M 3 N family), we can then provide a quantitative estimate to the multiparticle entanglement of this experimental bound entangled state in terms of any geometric measure E D M , by using Table II. The results are reported in Table V(a) for the illustrative case of the trace distance.
Remaining within the domain of quantum optics, recently two laboratories reported the creation of six-photon Dicke states |D (6) 3 [51,52]. Dicke states [58] are valuable resources for quantum metrology, computation, and networked communication, and emerge naturally in many-body systems as ground states of the isotropic Lipkin-Meshkov-Glick model [59]. Based on the values of the three correlation functions {c j }, which were measured in Refs. [51,52] to construct some entanglement witnesses, we can provide quantitative bounds to their global and partial geometric entanglement E D M (for 4 ≤ M ≤ 6) from Eq. (4); see Table V(a).
A series of experiments at Innsbruck [40,49,50,68] resulted in the generation of a variety of relevant multi-qubit states with trapped ion setups, for explorations of fundamental science and for the implementation of quantum protocols. In those realisations, data acquisition and processing for the purpose of entanglement verification was often a more demanding task than running the experiment itself [49]. Focusing first on global and partial entanglement, we obtained full datasets for experimental density matrices corresponding to particularly noisy GHZ and W states of up to four qubits, produced during laboratory test runs [68]. Despite the relatively low fidelity with their ideal target states, we still obtain meaningful quantitative bounds from Eq. (5). The results are compactly presented in Table V(a).
Regarding now genuine multiparticle entanglement, the authors of Ref. [50] reported the creation of (noisy) GHZ states of up to N = 14 trapped ions. In each of these states, full inseparability was witnessed by measuring precisely the maximum overlap p max with a reference pure GHZ state, without the need for complete state tomography. Thanks to Eq. (6), we can now use the same data to obtain a full quantification of the genuine N-particle entanglement of these realistic states, according to any measure E D 2 , at no extra cost in terms of experimental or computational resources. The results are in Table V(b), for all the representative choices of distances enumerated in Table IV. Notice that we do not need to assume that the experimentally produced states are in the G N set: the obtained results can be still safely regarded as lower bounds.

III. DISCUSSION
We have introduced a general framework for estimating and quantifying geometric entanglement monotones. This enabled us to achieve a compendium of exact results on the quantification of general distance-based measures of (global, partial, and genuine) multiparticle entanglement in some pivotal reference families of N-qubit mixed states. In turn, these results allowed us to establish faithful lower bounds to various forms of multiparticle entanglement for arbitrary states, accessible by  Table IV, obtained by Eq. (6) with p max given in each case by the measured fidelity with the pure reference GHZ state. All the reported entanglement estimates are obtained from the same data needed to witness full inseparability, which for general N-qubit states can be accessed by N + 1 local measurements without the need for a full tomography. few local measurements and effective on prominent resource states for quantum information processing.
Our results can be regarded as realising simple yet particularly convenient instances of quantitative entanglement witnesses [22,23], with the crucial advance that our lower bounds are analytical (in contrast to conventional numerical approaches requiring semidefinite programming) and hold for all valid geometric measures of entanglement, which are endowed with meaningful operational interpretations yet have been traditionally hard to evaluate [13,69].
A key aspect of our analysis lies in fact in the generality of the adopted techniques, which rely on natural informationtheoretic requirements of contractivity and joint convexity of any valid distance D entering Eq. (1). We can expect our general framework to be applicable to other reference families of states (for example, states diagonal in a basis of cluster states [14,69], or more general states with X-shaped density matrices [18]), thereby leading to alternative entanglement bounds for arbitrary states, which might be more tailored to different classes, or to specific measurement settings in laboratory.
Furthermore, our framework lends itself to numerous other applications. These include the obtention of accessible analytical results for the geometric quantification of other useful forms of multiparticle quantum correlations, such as Einstein-Podolsky-Rosen steering [70,71], and Bell nonlocality in many-body systems [59]. This can eventually lead to a uni-fying characterisation, resting on the structure of information geometry, of the whole spectrum of genuine signatures of quantumness in cooperative phenomena. We plan to extend our approach in this sense in subsequent works.
Another key feature of our results is the experimental accessibility. Having tested our entanglement bounds on a selection of very different families of theoretical and experimentally produced states with high levels of noise, we can certify their usefulness in realistic scenarios. We recall that, for instance, three canonical local measurements suffice to quantify exactly the global entanglement of GHZ states of any even number N of qubits, while N + 1 local measurements provide their exact genuine entanglement, according to every geometric measure for any N, when such states are realistically mixed with white noise. Compared to other complementary studies of accessible quantification of multiparticle entanglement [13][14][15][16][17][18][19][20][21][22][23], our study retains not only a comparably low resource demand but also crucial aspects such as efficiency and versatility, as shown in Table I. This can lead to a considerable simplification of quantitative resource assessment in future experiments based on large-scale entangled registers, involving e.g. two quantum bytes (16 qubits) and beyond [50,68].
Theorem. Any N-qubit state can be transformed into a corresponding M 3 N state through a fixed transformation, Θ, consisting of single-qubit LOCC, such that Here we sketch the form of the M 3 N -fication channel Θ. We begin by setting 2(N − 1) single-qubit local unitaries {U j } 2(N−1) is still a sequence of single-qubit local unitaries. Since Θ is a convex mixture of such local unitaries, it belongs to the class of single-qubit LOCC, mapping any M-separable set into itself. In the Supplementary Material, we show that Θ( ) = , concluding the proof.

AUTHOR CONTRIBUTIONS
M. C. and T. R. B. contributed equally to this work. All the authors conceived the idea, derived the technical results, discussed all stages of the project, and prepared the manuscript and figures.

COMPETING FINANCIAL INTERESTS
The authors declare that they have no competing financial interests.

CORRESPONDING AUTHOR
Correspondence to: Gerardo Adesso (gerardo.adesso@nottingham.ac.uk) When considering a multiparticle quantum system, there exist two different approaches to entanglement, one referring to a particular partition of the composite system under consideration (partition-dependent setting), and another which considers indiscriminately all the partitions with a set number of parties (partition-independent setting).
In order to characterise the possible partitions of an N-qubit system, we will employ the following notation [32] : • the positive integer M, 1 < M ≤ N, representing the number of subsystems; • the sequence of positive integers {K α } M α=1 := {K 1 , K 2 , · · · , K M }, where a given K α represents the number of qubits belonging to the α-th subsystem; • the sequence of sequences of positive integers , · · · , N} and Q α ∩ Q α = ∅ for α α , where a given sequence Q α represents precisely the qubits belonging to the α-th subsystem.
In the following we will say that {Q α } M α=1 identifies a generic M-partition of an N-qubit system.
The set of N-qubit separable states S {Q α } M α=1 with respect to the M-partition {Q α } M α=1 contains all, and only, states ς of the form where {p i } forms a probability distribution and τ (α) i are arbitrary states of the α-th subsystem. In other words, any {Q α } M α=1 -separable state can be written as a convex combination of product states that are all factorised with respect to the same partition {Q α } M α=1 . On the other hand, the set of N-qubit M-separable states S M contains all, and only, states that can be written as convex combinations of product states, each of which is factorised with respect to an M-partition that need not be the same. One can easily see that the set of M-separable states is the convex hull of the union of all the sets of {Q α } M α=1separable states obtained by considering all the possible Mpartitions {Q α } M α=1 . Any valid measure of multiparticle entanglement must be zero on the relevant set of separable states and monotonically non-increasing under LOCC. In the partition-dependent setting, a LOCC with respect to a particular {Q α } M α=1 -partition amounts to allowing each of the M parties to perform local operations on their qubits, and communicate with any other party via a classical channel [3]. Conversely, in the partitionindependent setting, one considers operations that are LOCC with respect to all of the M-partitions, which can be shown to be all and only the single-particle LOCC. A convex combination of single-particle local unitaries acting on a state , given is a particular type of single-particle LOCC (requiring only one-way communication). It can be physically achieved by allowing one of the subsystems α to randomly select a local unitary U (α) i by using the probability distribution {p i } and then to communicate the result to all the other subsystems.
One can also impose that a measure of N-particle entanglement is convex under convex combinations of quantum states, i.e.
for some probability q and quantum states and , which ensures that classical mixing of quantum states cannot lead to an increasing of entanglement. Any measure obeying these properties is referred to as a convex entanglement monotone.
In this work we adopt geometric measures of multiparticle entanglement, defined in terms of the distance to the relevant set of separable states. For a given distance D, generic distancebased measures of the multiparticle entanglement of an Nqubit state , quantifying how much is not {Q α } M α=1 -separable (resp., M-separable), are given by, respectively, where Eq. (A4) refers to the partition-dependent setting, and Eq. (A5) to the partition-independent one. It is sufficient for the distance D to obey contractivity and joint convexity (see Methods in the main text) for E D and E D M to be convex entanglement monotones [24]. In this appendix we show some relevant properties of the subclass of N-qubit states with all maximally mixed marginals that we refer to as M 3 N states. Their matrix representation in the computational basis is the following: where I is the 2 × 2 identity matrix, σ i is the i-th Pauli matrix, c i = Tr σ ⊗N i ∈ [−1, 1] and N > 1. These states are denoted by the triple {c 1 , c 2 , c 3 }.
The characterisation of the M 3 N states is manifestly different between the even and odd N case. For even N, the eigenvectors and eigenvalues are given by, respectively, and where i ∈ {1, · · · , 2 N−1 }, {|i } 2 N i=1 is the binary ordered N-qubit computational basis and finally p is the parity of |β ± i with respect to the parity operator along the z-axis Π 3 = σ ⊗N 3 , i.e.
The following Theorem is crucial for providing a lower bound to any multiparticle entanglement monotone of any state and for analytically computing the multiparticle geometric entanglement of any M 3 N state . Theorem C.1. Any N-qubit state can be transformed into a corresponding M 3 N state M 3 N through a fixed operation, Θ, that is a single-qubit LOCC and such that where c i = Tr( σ ⊗N i ). Proof The first part of the proof was sketched in the Methods section and is repeated here for completeness.
Herein, we will refer to M 3 N = Θ( ) as the M 3 N -fication of the state . Theorem C.1 has two major implications. The first implication applies to any multiparticle entanglement monotone, be it partition-dependent or independent. We have that where in the first equality we use M 3 N = Θ( ) and in the inequality we use the monotonicity under single-qubit LOCC of any measure of multiparticle entanglement and the fact that Θ is a single-qubit LOCC. In other words, the multiparticle entanglement of the M 3 N -fication M 3 N of any state provides us with a lower bound of the multiparticle entanglement of .
The second implication applies specifically to distancebased measures of multiparticle entanglement, although regardless of whether such a measure is partition-dependent or independent. We have that, for any M 3 N state and any separable state ς, where in the first equality we use the invariance of any M 3 N state through Θ and that Θ(ς) ≡ ς M 3 N is the M 3 -fication of ς, and in the inequality we use the contractivity of the distance through any completely positive trace-preserving channel. Moreover, the M 3 N -fication ς M 3 N of any separable state ς, regardless of whether ς is {Q α } M α=1 -separable or M-separable, is a separable M 3 N state of the same kind as ς, since Θ is a single-qubit LOCC and thus leaves any set of separable states invariant. Therefore, both the sets S i.e. that one of the closest {Q α } M α=1 -separable (resp., Mseparable) states ς to an M 3 N state is itself an M 3 N state. We now formalise these two results as corollaries.
Corollary C.1. For any N-qubit state , the multiparticle entanglement of the corresponding M 3 N -fied state M 3 N is always less than or equal to the multiparticle entanglement of , i.e.
for any {Q α } M α=1 -partition of the N-qubit system and any 2 ≤ M ≤ N.
Corollary C.2. For any contractive distance D and any M 3 N state , one of the closest {Q α } M α=1 -separable (resp., Mseparable) states ς to is itself an M 3 N state, i.e.
for any {Q α } M α=1 -partition of the N-qubit system and any 2 ≤ M ≤ N. partition such that K α is odd for more than one value of α.

Proof
In order to characterise the set of , we simply need to identify its representation in we know that such a representation is the set of Pauli correlation vectors corresponding to all the elements of S {Q α } M α=1 . Due to Eq. (A1), the Pauli correlation vector of any ς ∈ S {Q α } M α=1 is given by where in the final equality we denote c (α) i, j = Tr τ (α) i σ ⊗K α j as the j-th component of the Pauli correlation vector c (α) i,3 } corresponding to the arbitrary state τ (α) i of subsystem α. Eq. (D1) can be simplified further by introducing the Hadamard product as the componentwise multiplication of vectors, i.e. for u = {u 1 , u 2 , Using the Hadamard product gives Eq. (D1) as i.e., that the Pauli correlation vector of any {Q α } M α=1 -separable state is a convex combination of Hadamard products of Pauli correlation vectors corresponding to subsystem states. Due to Corollary C.3, we know that c (α) i ∈ B 1 when K α is odd and c (α) i ∈ T (−1) Kα/2 when K α is even, and so S is represented by the following set with where we define the Hadamard product between any two sets A and B as A • B = { a • b | a ∈ A , b ∈ B} and the convex hull conv(A) is the set of all possible convex combinations of elements in A. The commutativity and associativity of the Hadamard product allow us to rearrange the ordering in Eq. (D3) in the following way . By writing any vector in T ±1 as a convex combination of the vertices of T ±1 , one can easily show that so that µ:Kµeven where M − is the number of K µ with odd K µ /2. Similarly, one can see that Finally, we have that where we define n = {|b 1 |, |b 2 |, |b 3 |} and n = {|b 1 |, |b 2 |, |b 3 |}, respectively, as the vectors corresponding to b and b in the positive octant of the unit ball, and θ as the angle between these vectors. Now, due to Eqs. (D5), (D7), (D8) and (D9), and the fact that conv(A) = A for any convex set A, we identify four cases: 1. if K α is even for any α then where M − is the number of K µ with odd K µ /2; 2. if K α is odd for just one value of α then For any even N-qubit system, only a {K α } M α=1 partitioning within cases 1, 3 and 4 may be realised. In case 1, i.e. when K α is even for any α, we have S For any odd N-qubit system, only a {K α } M α=1 partitioning within cases 2, 3 and 4 may be realised. In case 2, i.e. when K α is odd for only one α, we have S  2. K α is odd for more than one value of α and |c 1 | + |c 2 | + |c 3 | ≤ 1 for c i = Tr( σ ⊗N i ). Now we are ready to characterise also the set of M- Within the partition-dependent setting we will restrict to any nontrivial partition {K α } M α=1 , i.e. such that K α is odd for at least two values of α, whereas within the partition-independent setting we will restrict to any non trivial number of parties M , i.e. such that M > N/2 . According to Appendix C and D, in both cases we simply need to find the minimal distance from to the set of M 3 N states inside the unit octahedron O 1 . In the even N case, the closest state is the same for any convex and contractive distance (note that every jointly convex distance is convex), while in the odd N case this is not true. states. In the following we will focus only on the corner containing the vertex {−1, (−1) N/2 , −1}, since all the M 3 N states belonging to the other three corners can be obtained from this by simply applying a single-qubit local unitary σ i ⊗ I ⊗N−1 , i ∈ {1, 2, 3}, under which any sort of multiparticle entanglement is invariant.
In order to characterise all the M 3 N states with even N belonging to the {−1, (−1) N/2 , −1}-corner, it will be convenient to move from the coordinate system {c 1 , c 2 , c 3 } to a new coordinate system (p, q, h), where we assign the coordinates to any other point in the corner. In order to avoid confusion between the above two coordinate systems, we will denote an M 3 N state with curly brackets when representing it in the {c 1 , c 2 , c 3 } coordinate system, whereas we will denote with round brackets when representing it in the (p, q, h) coordinate system. Specifically, the M 3 N states represented by the triples (p, q, h), with a fixed value of h ∈ [0, 1[, correspond in the {c 1 , c 2 , c 3 }-space to all, and only, the M 3 N states belonging to the triangle with the following vertices: in such a way that These triangles corresponding to constant values of h will play a crucial role, as they represent the sets of M 3 N states with constant {K α } M α=1 -and M -inseparable multiparticle entanglement for even N. In particular, for h = 0 we get one of the faces of the octahedron of {K α } M α=1 -and M -separable states, whereas with increasing h, we will prove that both the {K α } M α=1 -and M -inseparable multiparticle entanglement of the M 3 N states belonging to the corresponding triangle will increase monotonically. We will now show that the {K α } M α=1 -separable (resp., M -separable) state represented by the triple (p, q, 0) is one of the closest {K α } M α=1 -separable (resp., M -separable) states to the M 3 N state (p, q, h). Lemma E.1. For every even N, according to any convex and contractive distance, one of the closest {K α } M α=1 -separable (resp., M -separable) states ς to any M 3 N state belonging to the {−1, (−1) N/2 , −1}-corner is always an M 3 N state of the form (p , q , 0) for some p , q ∈ [0, 1], p + q ≤ 1.
In order to prove Eqs. (E27) and (E29), we just remark that exactly the same proof holds when substituting {K α } M α=1separability with M -separability.
We are now ready to apply the above general results to calculate the geometric multiparticle entanglement E D In this appendix we will adopt our approach to evaluate exactly the geometric genuine multiparticle entanglement E D 2 of any N-qubit GHZ-diagonal state ξ with respect to any contractive and jointly convex distance D.
Recall that any N-qubit state can be transformed via a single-qubit LOCC Γ into a GHZ-diagonal state GHZ ≡ Γ( ) with eigenvalues given by p ± i = β ± i | |β ± i [14], a procedure referred to as GHZ-diagonalisation of in the main text. The entanglement quantification is then based on the following two arguments.
First, we have that one of the closest 2-separable states to a GHZ-diagonal state is itself GHZ-diagonal. Indeed, for any GHZ-diagonal state ξ and any 2-separable state ς, we have that D(ξ, ς GHZ ) = D(Γ(ξ), Γ(ς)) ≤ D(ξ, ς), where in the first equality we use the invariance of any GHZdiagonal state through Γ and that Γ(ς) ≡ ς GHZ is the GHZdiagonalisation of ς, and in the inequality we use the contractivity of the distance through any completely positive tracepreserving channel. Moreover, the GHZ-diagonalisation ς GHZ of any 2-separable state ς is a 2-separable GHZ-diagonal state since Γ is a single-qubit LOCC. Therefore, the set S G 2 of 2-separable GHZ-diagonal states turns out to be the relevant one in order to compute exactly any distance-based measure of genuine multiparticle entanglement of a GHZ-diagonal state ξ, thus dramatically simplifying the ensuing optimisation as follows: Now, let us consider an arbitrary GHZ-diagonal state ξ and rearrange its GHZ eigenstates {|β i } 2 N i=1 in such a way that the corresponding eigenvalues {p ξ i } 2 N i=1 are in non-increasing order. It is well known that ξ is 2-separable if, and only if, p ξ 1 ≤ 1/2 [5]. For p ξ 1 > 1/2, we will show that one of the closest 2-separable GHZ-diagonal states ς ξ has eigenvalues {p corresponding again to the ordering of GHZ eigenstates {|β i } 2 N i=1 set by ξ. This result further simplifies the optimisation in Eq. F2.
Consider any 2-separable GHZ-diagonal state ς, it holds that there will always be a 2-separable GHZ-diagonal state ς with eigenvalues {p ς i } 2 N i=1 and p ς 1 = 1/2 such that ς = λξ + (1 − λ)ς for some λ ∈ [0, 1]. Now, for any convex distance, the following holds This inequality immediately implies that one of the closest 2-separable GHZ-diagonal states ς ξ to a 2-inseparable GHZdiagonal state ξ is of the form ς , which is formalised as a corollary below.
Corollary F.1. For any convex and contractive distance D and any fully inseparable GHZ-diagonal state ξ, whose GHZ eigenstates {|β i } 2 N i=1 are arranged such that the corresponding eigenvalues {p ξ i } 2 N i=1 are in non-increasing order, one of the closest 2-separable states ς ξ to ξ is itself a GHZ-diagonal state with eigenvalues {p ς ξ i } 2 N i=1 such that p ς ξ 1 = 1/2, with {p ς ξ i } 2 N i=1 corresponding again to the ordering of GHZ eigenstates {|β i } 2 N i=1 set by ξ.
We can now apply this Corollary to calculate the geometric genuine multiparticle entanglement E D 2 (ξ) of any GHZdiagonal state ξ for particular instances of D. Since the closest 2-separable state ς ξ to a GHZ-diagonal state ξ is also a GHZdiagonal state, they are diagonal in the same basis and their distance reduces to the corresponding classical distance between the probability distributions formed by their eigenvalues, denoted by P ξ and P ς ξ respectively. By using the expressions given earlier in Appendix E of the classical relative entropy, trace, infidelity, squared Bures, and squared Hellinger distance between two probability distributions P ξ and P ς ξ , and minimising it over all probability distributions P ς ξ such that p ς ξ 1 = 1/2, one easily obtains the desired expressions for E D 2 (ξ) expressed in the main text.