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Detecting the dimensionality of genuine multiparticle entanglement
Detecting the dimensionality of genuine multiparticle entanglement
https://orcid.org/0009-0006-9090-8806
Cobucci Gabriele Conceptualization Data curation Formal analysis Investigation Methodology Project administration Resources Software Validation Visualization Writing - original draft Writing - review & editing
https://orcid.org/0000-0001-9136-7411
Tavakoli Armin Conceptualization Formal analysis Funding acquisition Methodology Project administration Software Supervision Validation Writing - original draft Writing - review & editing *
Physics Department and NanoLund, Lund University, Box 118, 22100 Lund, Sweden.
* Corresponding author. Email: armin.tavakoli@teorfys.lu.se
20 9 2024
20 9 2024
10 38 eadq446714 5 2024
15 8 2024
Copyright © 2024 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).
2024
The Authors
https://creativecommons.org/licenses/by-nc/4.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

Complex forms of quantum entanglement can arise in two qualitatively different ways: either between many qubits or between two particles with higher-than-qubit dimension. While both the many-qubit frontier and the high-dimension frontier are well established, state-of-the-art quantum technology is becoming increasingly able to create and manipulate entangled states that simultaneously feature many particles and high dimension. Here, we investigate generic states that can be considered both genuinely high-dimensional and genuine multiparticle entangled. We consider a natural quantity that characterizes this key property. To detect it, we develop three different classes of criteria. These enable us both to probe the ultimate noise tolerance of this form of entanglement and to make detection schemes using sparse or even minimal measurement resources. The approach provides a simple way of benchmarking entanglement dimensionality in the multiparticle regime and general, platform-independent, detection methods that readily apply to experimental use.

Simple and efficient methods are introduced to detect the dimensionality of entanglement between many quantum particles.

http://dx.doi.org/10.13039/100001388 Wenner-Gren Foundation Wallenberg Center for Quantum Technology
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pmcINTRODUCTION

Entanglement is a cornerstone of quantum theory and a paradigmatic resource for modern quantum information processing. It has been a subject of intense research for decades, both in its own interest (1, 2) and for its applications in, e.g., quantum cryptography (3–5), quantum-enhanced metrology (6–8), quantum computation (9, 10), and fundamental tests of quantum theory (11, 12). Therefore, detecting, quantifying, and characterizing interesting forms of entanglement is a central question for quantum theory in general and quantum information science in particular.

A central goal is to generate fully controlled entangled states featuring increasingly many particles. This is a key requirement for quantum computing advantages. In its basic form, a quantum state comprised of n particles is said to be entangled if it cannot be created via coordinated local operations on each of the particles. However, many times, the most appropriate form of entanglement goes beyond this elementary notion. For example, knowing that n qubits are entangled does not reveal whether it is a feature of all the n qubits, or whether just a pair of them are entangled (13). To ensure that the entanglement is truly n-partite, it is standard to consider a stronger notion, called genuine multipartite entanglement (GME), which ensures that the state cannot be generated by entangling only some of the qubits. Much theoretical (2) and experimental (14–22) research has been focused on detecting and realizing GME. GME states of up to 14, 32, and 51 qubits have been achieved with photonics (23), trapped ions (24), and superconducting circuits (25), respectively.

A second frontier for entanglement is systems of two particles with more than two internal levels. Such high-dimensional entanglement leads to improved rates and enhanced noise and loss properties in quantum communication (26–28); it makes possible teleportation of multiple degrees of freedom (29) and it leads to stronger quantum correlation phenomena (30–32). In analogy with the multiparticle case, knowing that two high-dimensional particles are entangled does not reveal whether it is truly a feature of all the levels. Verifying the entanglement dimension, called the Schmidt number, means to certify that the state cannot be generated using fewer levels (33). High-dimensional entanglement has received much attention (34, 35). A variety of theoretical criteria are known (36–42) and experiments can now reach far into double-digit Schmidt numbers (43–49).

A major challenge is to combine the two above frontiers and generate entangled states with many particles and high dimension. This is typically approached by concepts that are based on bipartite entanglement and Schmidt numbers (50). This is interesting not only because entanglement generation and distribution is a fundamental primitive for quantum information science, but also because it allows combining the quantum advantages associated with both regimes. Such states are also interesting for high-dimensional quantum computing (51, 52), quantum nonlocality (53, 54), and various other tasks (19, 55, 56). Only recently have experiments reported on realizations of controlled high-dimensional multiparticle states in free space and fiber photonics (57–60), integrated optics (61, 62), and superconducting systems (63). It currently remains hard to generate such states at low noise.

In view of this progress, it is increasingly important to characterize relevant notions of high-dimensional multiparticle entanglement and to develop methods for benchmarking these properties in the laboratory. For this purpose, both two-particle entanglement dimensionality and GME are unsatisfactory concepts. The former cannot be generalized to multiparticle states (64) and the latter does not reveal knowledge of the dimensionality, i.e., a GME state of n-particles with dimension d is not necessarily a true d-dimensional phenomenon. It is possible that the entanglement, albeit spread over all n particles, can be simulated by using entangled states with fewer than d levels.

As an illustrative example, consider a source that with probability p emits a three-qutrit Greenberger-Horne-Zeilinger (GHZ) state 13(∣000⟩+∣111⟩+∣222⟩)and with probability 1−p emits the state 13(∣000⟩⟨000∣+∣111⟩⟨111∣+∣222⟩⟨222∣) (see Fig. 1A). The former state is genuinely high-dimensional and genuinely three-particle entangled but the latter is entirely classical; it is just the state of flipping three correlated three-sided coins. Using the methods of Bourennane et al. (14), one can show that the mixture of these two states, which we denote τp, is GME whenever p > 0. However, we shall now construct a simulation that uses only entanglement between qubits to reproduce τp. Specifically, let us randomly prepare one of the three different qubit GHZ states: ∣ψ1⟩=12(∣000⟩+∣111⟩), ∣ψ2⟩=12(∣000⟩+∣222⟩), and ∣ψ3⟩=12(∣111⟩+∣222⟩), as in Fig. 1B. One can verify that the average state, namely, 13(ψ1+ψ2+ψ3) in fact is identical to τp when p=12. Hence, for 0<p≤12, the original source has physical dimension three and is GME, but its genuine dimension is actually two.

Fig. 1. Simulation example.

The source in (A) flips a biased coin ({p,1−p}) and outputs either the three-qutrit GHZ state or a classically correlated state. The mixed state is always genuinely three-particle entangled, but when p ≤ 1/2, it can nevertheless be simulated with just two-dimensional entanglement using the alternative source in (B).

Here, we study the dimensionality of multiparticle entangled states based on ideas that naturally extend the standard concepts of GME and Schmidt number. The approach is based on a quantity that we call the GME-dimension [see also (65)]. It admits a simple interpretation in terms of the smallest possible Schmidt number needed over all possible ways of bisecting the n particles to simulate the state. In its limiting cases, namely, just having two particles or just two dimensions, this picture reduces to the Schmidt number and GME respectively (see Fig. 2). The states associated with GME-dimensions ranging from 1 up to the physical dimension, d, therefore constitute a hierarchy of increasingly high-dimensional forms of GME.

Fig. 2. Entanglement concepts.

The Schmidt number determines the genuine dimensionality of two-particle entanglement. GME determines whether all n subsystems are entangled. The GME-dimension generalizes both these concepts to systems with arbitrary dimensionality and number of subsystems.

The key question is how to construct versatile criteria for detecting the GME-dimension of arbitrary, initially uncharacterized, states. We develop three different classes of detection criteria. First, we consider measuring the fidelity with a given target state and show how this implies a bound on the GME-dimension. This generalizes the standard fidelity witness method for GME (14), which is commonly used to benchmark many-qubit experiments [see, e.g., (15–17, 19, 23)]. This method has the advantage of being versatile because it applies to arbitrary pure target states and admits a simple characterization. Second, we develop resource-minimalistic detection criteria, tailored for experimental limitations. This is motivated by the fact that high-dimensional multiparticle entanglement sources are typically complex, leading to limited count rates. In addition, measuring the fidelity can require a lot of measurement settings, especially as d increases. Together, this can quickly become a substantial obstacle for experiments. Therefore, our method allows the fidelity to be estimated using only the minimal number of measurements, namely, two. We explicitly construct such minimal criteria for the two most broadly considered high-dimensional multiparticle states, namely, GHZ and cluster states. Third, we use convex programming methods to determine conditions for a state’s GME-dimension. We show that this leads to considerably stronger GME-dimension criteria than what is possible with fidelity witnesses. We also show that for experiments involving a few particles and a few levels, this method can offer particularly noise-robust detection criteria using only a small number of measurement settings.

Dimensionality of GME

The entanglement structure of quantum systems with n > 2 subsystems is richer than in the two-particle (bipartite) case. A pure state, ∣ψ⟩, is called biseparable if it is possible to partition the n particles into two nonoverlapping sets {S,S¯}, such that the state factors over this bipartition. That is, there exist states such that ∣ψ⟩1…n=∣ϕ⟩S⊗∣φ⟩S¯. This is straightforwardly extended to mixed states. A mixed state is biseparable if it can be generated by classically mixing pure biseparable states, i.e., ρ=∑S∣S¯‍∑j‍qS∣S¯(j)∣ϕj⟩⟨ϕj∣S⊗∣φj⟩⟨φj∣S¯, where qS∣S¯(j)≥0 and ∑S∣S¯,j‍qS∣S¯(j)=1. The index S∣S¯ runs over all choices of {S,S¯}, the number of which is 2n−1−1. If a state is not biseparable, it is called genuine multipartite entangled. Thus, GME states admit a simple interpretation: they are the states impossible to generate using classical randomness and arbitrary n-partite states, each separable with respect to some bipartition of the particle set.

To further develop the notion of GME so that it also addresses the dimensionality of the entanglement, we must suitably replace the factorizability condition of each pure state in the ensemble decomposition. A more appropriate constraint to capture the high-dimensional nature of the entanglement is to instead impose that the bipartite entanglement dimension, namely, the Schmidt rank, across {S,S¯}, is bounded from above. The Schmidt rank is the rank of the reduced density matrix, which is necessarily equal to one for a product state but larger for pure entangled states [it generalizes to the Schmidt number for mixed states (33)]. That is, for a generic n-partite and d-dimensional density matrix ρ, we consider decompositions of the formρ=∑S∣S¯‍qS∣S¯σS∣S¯(1)

for an arbitrary probability distribution {qS∣S¯} and states σS∣S¯ with local dimensions d∣S∣ and dS¯, whose Schmidt number is no more than r. The latter limitation means that for each {S,S¯}, there exists a pure-state decomposition σS∣S¯=∑i‍pi,S∣S¯∣ψi,S∣S¯⟩⟨ψi,S∣S¯∣ where the largest Schmidt rank over all pure states {∣ψi,S∣S¯⟩}i does not exceed r. In contrast, if no decomposition of the form (Eq. 1) exists, the simulation of ρ via classical randomness and bipartite quantum states then requires that at least one of the ensemble states σS∣S¯ has a Schmidt number exceeding r. This naturally leads us to our main quantity of interest, namely, the largest r found over all the bipartitions, in the least “dimension-expensive” ensemble realizations of ρ. We call this the GME-dimension.

Definition 1 (GME-dimension). For an arbitrary n-partite state ρ, its GME-dimension is

𝒟GME(ρ)=min{qS∣S¯},{σS∣S¯}rmax:ρ=∑S∣S¯‍qS∣S¯ σS∣S¯and rmax=max{S∣S¯} rS∣S¯ (2)

where rS∣S¯ is the Schmidt number of σS∣S¯ across {S,S¯}.

Notably, setting n = 2 reduces the GME-dimension to the Schmidt number, as there is only one possible bipartiton. Furthermore, setting 𝒟GME = 1 instead reduces it to the definition of biseparability. Thus, all GME states have 𝒟GME > 1, but the precise value of the GME-dimension additionally reveals knowledge of their dimensionality. For fixed (n, d), we write 𝒢dGME for the set of states obeying 𝒟GME(ρ) ≤ dGME. The GME-dimension can also be viewed as the smallest element of the so-called Schmidt number vector (50). We note that while Schmidt number vectors cannot, in general, be compared, the GME-dimension forms an ordered hierarchy of nested convex sets, ranging from states that fail to be GME (𝒟GME = 1) up to GME states that are genuinely d-dimensionally entangled (𝒟GME = d). This is illustrated in Fig. 3.

Fig. 3. Hierarchy of GME-dimension states.

States with 𝒟GME = 1 are biseparable. Any state with 𝒟GME > 1 is GME. States with 𝒟GME = d cannot be simulated by any lower-dimensional entanglement.

RESULTS

Fidelity witnesses

Measuring the fidelity, Fψ(ρ) = 〈ψ∣ρ∣ψ〉, between a pure target state ψ and a mixed laboratory state ρ is a standard way to detect GME in the vicinity of ψ. Therefore, we begin with identifying how the fidelity can also be used to detect the GME-dimension.

Result 1 (Fidelity witness). For any n-partite pure target state ψ with local dimension d, its fidelity with any state ρ with GME-dimension no larger than dGME is bounded asmaxρ∈𝒢dGMEFψ(ρ)≤max{S,S¯}∑i=1dGME‍λi(ψS)(3)

where the maximization is over all bipartitions of the subsystems and λ(ψS) is the spectrum of the reduced state ψS=tr S¯(ψ), ordered nonincreasingly.

To prove this, a straightforward argument builds directly on extending previously known properties of the bipartite fidelity to the multipartite case. For n = 2, for which there is only one bipartition, it is known that Eq. 3 holds (66). Because of the linearity of the fidelity function and the convexity of the set dS∣S¯𝒢dGME, Fψ(ρ) is optimized for a pure state ρ = ∣ϕ⟩⟨ϕ∣ ∈ 𝒢dGME. Thus, we may separately consider each bipartition and select the maximal value; that is Eq. 3. An alternative proof method is possible without relying on directly extending the bipartite case. This is based on using the strong duality of linear programming and it is detailed in section SI. Notice that choosing dGME = 1 reduces Result 1 to the standard fidelity witness for GME (14). Moreover, note that if we instead were to assign a separate Schmidt number to each bipartition, namely, dS∣S¯, the only change in Eq. 1 will be that the demarcation of the sum becomes dS∣S¯ instead of dGME.

It is important to consider Result 1 in practically relevant cases. We have evaluated it explicitly, for arbitrary choices of (n, d), for three different seminal families of states, namely, GHZ states, cluster states, and absolutely maximally entangled states [assuming they exist, see (67, 68)]. In all three cases, the fidelity witness for the GME-dimension is identical; it readsFψ(ρ)≤dGMEd(4)

which is notably independent of n. Our proof for the former and latter state is straightforward, but less so for cluster states (see details in section SII). While Eq. 4 provides a simple criterion, it also means that fidelity methods cannot distinguish between these three classes of states. We note that Result 1 does not reduce to Eq. 4 for arbitrary choices of ψ.

Minimal fidelity witnesses

With multipartite high-dimensional states, it is often practically relevant to probe the system using as few measurements as possible. Therefore, we now aim to detect the GME-dimension via fidelity estimation from only two complementary basis measurements. This is the minimal setting for detecting the GME-dimension.

First, we target states in the vicinity of the GHZ state∣ghzn,d⟩=1d∑i=0d−1‍∣i⟩⊗n(5)

for arbitrary (n, d). This state is natural to focus on because it has been the goal of most multiparticle entanglement experiments in the literature. Let {∣j⟩}j=0d−1 be the computational basis and {∣ej⟩}j=0d−1 the Fourier basis, ∣ej⟩=1d∑k=0d−1‍ωjk∣k⟩, where ω=e2πid. Consider that we (i) measure all subsystems in the computational basis and compute the total probability of all n local outcomes being identical, and (ii) measure all subsystems in the Fourier basis and compute the total probability that the sum over all local outcomes is divisible by d. The Hermitian operator describing the sum of these two events takes the formOn,dghz=∑j=0d−1‍∣j⟩⟨j∣⊗n+∑j1,…,jn=0d−1‍⊗l=1n∣ejl⟩⟨ejl∣δj1⊕…⊕jn,0(6)

where ⊕ denotes addition modulo d. The entanglement witness is the expectation value, 𝒲n,dghz(ρ)=(ρOn,dghz). It is immediate that the GHZ state has perfect correlations for the event (i) and a direct calculation shows the same also for event (ii). Hence, 𝒲n,dghz(ghzn,d)=2. Our next result shows how this witness detects the GME-dimension.

Result 2 (Minimal GHZ state witness). For any n-partite state ρ of local dimension d with GME-dimension no larger than dGME𝒲n,dghz(ρ)≤1+dGMEd(7)

Moreover, any observed value of 𝒲n,dghz implies a GHZ-fidelity bound Fghz(ρ)≥𝒲n,dghz(ρ)−1.

The proof is based on the observation that the witness operator (Eq. 6) can be decomposed as a sum of a projector and the GHZ state. See section SIII for details.

Naturally, because we use only two bases, this criterion is weaker than the exact fidelity criterion, but it is practically advantageous. It can perform well for the two most relevant noise models, namely, depolarization and dephasing. Take first depolarizing (white) noise, corresponding to ρvghz=v∣ghzn,d⟩⟨ghzn,d∣+1−vdn𝟙, where v ∈ [0,1] is the visibility. The critical visibility for violating inequality (Eq. 7) isvcrit=1−dn−2(dGME+d−1)1−dn−2(2d−1)(8)

For instance, take a system of four qutrits; the threshold for detecting dGME = 3 becomes vcrit = 79.5%, which is in the regime relevant for state-of-the-art experiments. Next, consider instead dephasing noise, corresponding to τvghz=v∣ghzn,d⟩⟨ghzn,d∣+1−vd∑i=0d−1‍∣i⟩⟨i∣⊗n. The critical visibility from Result 2 becomesvcrit=dGME−1d−1(9)

independently of n. We observe that this is equal to what is obtained from Eq. 4 by using complete fidelity measurements. We prove in section SIV that vcrit is the exact threshold for the GME-dimension of τvghz; i.e., our minimal witness is actually necessary and sufficient.

Cluster states are another important class of states. Interest in them draws mainly from that they are a universal resource for one-way quantum computing (69, 70). High-dimensional cluster states are generated in lattices of n qudits with Ising-type interaction; in a linear lattice, they take the form∣Cn,d⟩=1dn⊗a=1n(∑k=0d−1‍∣k⟩aZa+1k)(10)

with Z=∑k=1d‍ωk∣k⟩⟨k∣ and the convention Zn+1=𝟙 (71). Using only two global product bases, we construct a GME-dimension witness targeting Cn,d.

Consider that we (i) measure the odd subsystems in the Fourier basis and the even ones in the computational basis and (ii) measure the odd subsystems in the computational basis and the even ones in the Fourier basis. The relevant outcome combinations are different from those used before for GHZ. Specifically, they correspond to the Hermitian operatorOn,dcluster=⊗ℓoddn ∑ql,pl+1=0d−1‍∣eqlpl+1⟩⟨eqlpl+1∣δpl+1⊕pl−1⊖ql,0+⊗ℓoddn ∑pl,ql+1=0d−1‍∣pleql+1⟩⟨pleql+1∣δpl⊕pl+2⊖ql+1,0(11)

The total probability associated with observing these outcomes in events (i) and (ii) corresponds to the entanglement witness, i.e., 𝒲n,dcluster(ρ)=(ρOn,dcluster). The witness is constructed so that the cluster state exhibits perfect correlations for both (i) and (ii), and hence, 𝒲n,dcluster(Cn,d)=2. The next result shows how this witness detects the GME-dimension.

Result 3 (Minimal cluster state witness). For any n-partite state ρ of local dimension d with GME-dimension no larger than dGME𝒲n,dcluster(ρ)≤1+dGMEd(12)

Moreover, any observed value of 𝒲n,dcluster implies a cluster state fidelity bound Fcluster(ρ)≥𝒲n,dcluster(ρ)−1.

The proof of this result is given in section SVI and largely parallels the ideas used to derive Result 2.

In analogy with the analysis of the GHZ witness, we apply Result 3 to cluster states with depolarizing and dephasing noise, namely, ρvcluster=v∣Cn,d⟩⟨Cn,d∣+1−vdn𝟙 and ρvcluster=v∣Cn,d⟩⟨Cn,d∣+1−vdn11−vd∑i=0d−1‍∣i⟩⟨i∣⊗n, respectively. The two critical noise thresholds turn out to be identical, namelyvcrit=b−dn2+a−dn2+a−1dGMEb−2dn2+a(13)

where b = 1 + d2a and a = [1 + (−1)n+1]/4. Continuing the example of four qutrits from earlier, with a maximal GME-dimension, this becomes vcrit = 81.3%, which is reasonable for practical purposes.

Last, we note that if instead of the GME-dimension, we assign a separate Schmidt number, {dS∣S¯}, to each bipartition and determine whether the state is compatible with this hypothesis, we obtain necessary criteria directly by modifying Result 2 and Result 3. The reason is that both of these ultimately rely on fidelity bounds, as previously mentioned in the context of Result 1. For example, Result 2 will remain unchanged because only the largest element of {dS∣S¯} will be relevant.

Convex programming method

We now go beyond fidelity-based criteria and detect the GME-dimension via efficiently computable convex programming relaxations (72). To this end, let Λ be an r-positive trace-preserving map. Such maps are positive when applied to one share of every bipartite state with a Schmidt number at most r but nonpositive for some states with a larger Schmidt number (33). For any r-positive map, we define the semidefinite programmaxʋ,σ˜ʋs.t.ʋρ+1−ʋdn𝟙=∑S∣S¯‍σ˜S∣S¯,σ˜S∣S¯≥0 ∀(S∣S¯),(ΛS⊗ 𝟙S¯)[σ˜S∣S¯]≥0 ∀(S∣S¯)(14)

where σ˜S∣S¯=qS∣S¯σS∣S¯ are unnormalized states. Here, we have chosen to introduce depolarizing noise on the state ρ. This serves as one of several possible quantifiers of the separation of ρ with respect to the selected relaxation of 𝒢r. Thus, obtaining any value v < 1 implies that ρ ∉ 𝒢r and, hence, 𝒟GME(ρ) > r. For example, selecting Λ as the partial transpose map, which is 1-positive, reduces Eq. 14 to the approach first outlined in (73) for GME. This partial transpose map can be adapted for our higher-dimensional analysis by introducing auxiliary Hilbert spaces of dimension r for each qudit (37, 39). However, this is a costly approach for the GME-dimension because the extra dimensions accumulate exponentially in n when r > 1, causing a large computational overhead. Therefore, we instead propose to use the generalized reduction map, Λ(X) = tr X 𝟙 − αX, which is known to be r-positive when α=1r (33, 74). Note that applying the map on the system S, as in Eq. 14, is not necessarily equivalent to applying it on system S¯. They correspond to different relaxations, and one can even apply it to both subsystems separately to obtain a more accurate relaxation of 𝒢r.

The program (Eq. 14) can many times be considerably reduced. If ρ is a pure state and we use the generalized reduction map, we can express Eq. 14 as just a linear program (LP) by representing it in a basis in which ρ is diagonal [if ρ is a graph state, similar reductions are also possible for GME tests based on the partial transpose map (75); see section SVIII]. This speeds up computations. Furthermore, depending on the choice of ρ, additional symmetries may be available, which can be used to further reduce the program. For instance, for the GHZ state, one can also exploit that the state is invariant under permutations of its particle labels. This reduces the exponentially many diagonal matrix variables to just n−1.

We have evaluated these programs explicitly. On a standard computer, we could evaluate the LP up to, e.g., (n, d) = (5,4) for the GHZ state and (n, d) = (4,4) for the cluster state. Our implementation is available in (76). In Table 1, we display some of the resulting bounds on the critical visibility; more extensive results are given in section SVIII. The main observation is that the visibilities are considerably smaller than those obtained from the fidelity witness (Eq. 4). This showcases the relevance of this method. Moreover, with small modifications, the program (Eq. 14) can also be adapted to the scenario with individual Schmidt numbers for each bipartition. To this end, it suffices, for each bipartition (S∣S¯), to appropriately choose the value of α in the definition of Λ as α=1dS∣S¯. Further details are given in section SIX together with case studies for GHZ states.

Table 1. Results from the convex programming method.

Critical visibility, v, for the GME-dimension computed by reformulating the SDP in Eq. 14 as a linear program (LP). Case study presented for target GHZ states and cluster states under white noise. Note that these states are local-unitary equivalent for n = 3 and hence the results are identical (marked by “*” in the table). The results are compared with the visibility obtained from the fidelity bound (Eq. 4).

(n,d)	𝒟GME(ρ)	vghz (LP)	vcluster (LP)	v (fidelity)	
(3,3)	1	0.2500	*	0.3077	
(3,3)	2	0.5909	*	0.6538	
(4,3)	1	0.2203	0.2174	0.3250	
(4,3)	2	0.6029	0.5129	0.6625	
(4,4)	3	0.6503	0.534	0.7490	

Furthermore, the method can also be used to give resource-efficient criteria that are more noise robust than the minimal fidelity estimation method discussed earlier. To this end, we no longer simulate the whole state ρ but only its statistics when a small number of measurements are performed on it. In section SX, we show how to adapt the program (Eq. 14) for this purpose. To showcase its usefulness, consider that we measure just two or three global product measurements, where the local bases are mutually unbiased bases (MUBs) (77). We denote by Ei the set of constraints on the statistics for each choice of global product measurements. The results are displayed in Table 2 based on the GHZ state and a maximal GME-dimension. We see that using two bases (the computational basis and the Fourier basis, EC + EF) is more noise robust than the minimal fidelity witnesses, although the measurements are identical. The noise tolerance considerably improves by just imposing one additional set of constraints, EM, that involve a specific choice of local MUBs (see section SX). By adding further appropriately selected global product MUBs, one can further improve the visibility.

Table 2. Results from SDP with constraints on the statistics.

Comparison between values of critical visibility, v, for different GME-dimensions obtained from fidelity criterion (Eq. 4), minimal fidelity witness (Eq. 8), and convex programming relaxations based on simulating statistics from two (EC + EF), and three global product measurements (EC + EF + EM). Case study presented for target GHZ states under white noise.

(n,d)	𝒟GME(ρ)	v (fidelity)	v (min fid)	v (EC + EF)	v (ECF + EM)	
(3,3)	2	0.6538	0.7857	0.7500	0.6667	
(3,4)	3	0.7460	0.8518	0.8222	0.7576	
(4,3)	2	0.6625	0.7954	0.7750	0.7097	

DISCUSSION

We have used the GME-dimension as a quantity for benchmarking genuinely high-dimensional forms of genuine multiparticle entanglement. This naturally extends the established entanglement concepts of GME and Schmidt number into the multiparticle high-dimensional regime (see Fig. 2). To detect the GME-dimension of initially unknown states, we have put forward three classes of criteria.

First, we use the fidelity with a target state. This has the advantage of being applicable for any dimension and particle number and to arbitrary target states. As we have shown for several of the most relevant families of quantum states, the fidelity method offers promising, yet not optimal, noise robustness. The fidelity is also in itself a natural quantifier of the quality of a state preparation. In addition, measuring the fidelity is also far more sparse in terms of the number of necessary measurements than is quantum state tomography. Nevertheless, it is likely to be time expensive, for instance, on optical platforms where the multiphoton coincidence rate decreases with the particle number and visibility (21).

This motivated our second class of criteria, namely, those using the smallest possible number of measurements (two) to estimate the fidelity. We have constructed such criteria for arbitrary dimension and particle number, for both the seminal GHZ states and the cluster states. Despite the resource-minimal approach, our criteria are considerably, and sometimes even optimally, noise tolerant. They perform particularly well for high-quality sources. They are much less time expensive to estimate and can readily be applied to experiments on any physical platform.

Third, we develop convex programming criteria that can approximate the GME-dimension of arbitrary states. We show that this method is effectively computable for systems of a few particles with low dimension and that it can considerably outperform fidelity criteria in terms of noise robustness. This method is therefore practically relevant, because most experiments (57–63) presently concern choices of (n, d > 2) that we are able to treat explicitly, even without enhanced computing resources. This method can also systematically produce resource-efficient detection criteria, using any measurements considered convenient by the experimenter. Using the duality theory of semidefinite programming, one can also extract explicit witnesses for the GME-dimension from this method.

It is an interesting open problem how much noise various states tolerate before their GME-dimension is reduced. It is well known that high-dimensional GME can persist under diverging noise rates (78–80), but it is an open problem whether anything similar is possible for larger, or even maximal, GME-dimension. Connected to this is also the question of which classes of states are most strongly entangled in terms of the GME-dimension. For instance, Table 1 already suggests that the cluster state is more strongly high-dimensionally GME than the GHZ state. From this point of view, there may be even more interesting high-dimensional multiparticle states to study. Last, we remark that there may also be other interesting ways to approach the characterization of genuinely high-dimensional genuine multiparticle entanglement. Exploring the alternative paths is of evident interest.

Acknowledgments

We thank O. Gühne and M. Huber for useful comments and E. Zambrini Cruzeiro for computation support.

Funding: This work was supported by the Wenner-Gren Foundation and by the Knut and Alice Wallenberg Foundation through the Wallenberg Center for Quantum Technology (WACQT).

Author contributions: A.T. had the idea and proposed the basic concept. A.T. and G.C. developed the theory. Both authors participated in the writing of the manuscript.

Competing interests: The authors declare that they have no competing interests.

Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Our implementation of the linear programs is available at https://doi.org/10.5281/zenodo.13123607.

Supplementary Materials

This PDF file includes:

Supplementary Text

Tables S1 and S2

References
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