Global Tripartite Negativity

• Global Tripartite Negativity (gTN) is defined as the geometric mean of bipartite negativities, providing a clear criterion for full tripartite entanglement.
• It applies to diverse systems such as impurity models, mixed-spin clusters, and dynamical quantum settings, offering insights into block-level entanglement.
• gTN reveals critical scaling behavior at quantum phase transitions and evidences universality while highlighting its limitations in capturing all entanglement features.

Global Tripartite Negativity (gTN) is a negativity-based tripartite entanglement measure that, in its most common usage, is defined as the geometric mean of the three bipartite negativities associated with the three one-versus-two bipartitions of a tripartite system. In this form it functions as a global diagnostic of whether entanglement is shared across all three parties rather than being confined to a single bipartition. The terminology is not fully uniform across the literature: in impurity quantum criticality the same construction appears as the quantity $E_1$, while in later spin-cluster work it is named “global tripartite negativity” or “genuine tripartite negativity” (Bayat, 2016, Vargová, 19 Jun 2025).

1. Definition and normalization conventions

In the geometric-mean convention, gTN is written as

$${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$

or, in the notation of the impurity-physics literature,

$E_1=\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}.$

The construction is identical in structure across several subfields, even when notation differs (Bayat, 2016, Vargová, 19 Jun 2025).

Source	Name used	Formula or role
(Bayat, 2016)	$E_1$	$\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}$
(Vargová, 19 Jun 2025)	global tripartite entanglement / ${\cal N}_{ABC}$	$$\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3}$$
(Vargová et al., 2023)	genuine tripartite negativity	same geometric mean on a three-spin reduced state
(Kumar et al., 2022)	tripartite negativity	same geometric mean for three quantum memristors

The underlying bipartite negativity is not normalized identically in all papers. In the impurity setting,

$N_{A,B}=\sum_k |\lambda_k|-1,$

with $\lambda_k$ the eigenvalues of the partial transpose (Bayat, 2016). In mixed-spin trimer and tetramer studies, the Vidal–Werner form is used,

${\cal N}_{A|BC}=\sum_{\lambda_i<0}|\lambda_i|,$

while the quantum-memristor paper writes

$${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$0

with $${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$1 the negative eigenvalues of the partial transpose (Vargová et al., 1 Sep 2025, Vargová et al., 2023, Kumar et al., 2022). These are normalization differences at the bipartite layer; the tripartite layer remains the same geometric-mean construction.

For ground states in the impurity analysis, the three factors entering $${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$2 are monotonic functions of von Neumann entropies, but negativity is retained for generality and for consistency with the alternative measure $${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$3 (Bayat, 2016).

2. Operational meaning and relation to other tripartite negativity measures

The defining operational feature of the geometric-mean gTN is that it vanishes if any one subsystem is disentangled from the other two. In the impurity formulation this is stated explicitly: if one factor vanishes, then $${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$4, regardless of entanglement between the remaining two subsystems (Bayat, 2016). In the mixed-spin tetramer literature the same logic is expressed as full inseparability of the three-spin reduced state: $${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$5 is nonzero only when all three one-versus-two cuts have nonzero negativity (Vargová et al., 2023).

This property makes gTN a block-level criterion of shared tripartite entanglement, but not a complete descriptor of multipartite structure. The impurity paper contrasts $${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$6 with

$${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$7

which is a monogamy-residual measure inspired by the inequality

$${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$8

Accordingly, $${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$9 is a product or geometric-mean measure, whereas $E_1=\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}.$0 is a residual measure built from squared negativities (Bayat, 2016).

A further distinction appears in neutrino oscillation studies. There the paper does not use “Global Tripartite Negativity,” but instead adopts the three-$E_1=\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}.$1 negativity

$E_1=\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}.$2

with residual negativities defined from one-versus-pair and pairwise negativities. The paper presents this as the genuine-tripartite measure appropriate for W-class states, for which the three-tangle vanishes (Banerjee et al., 12 Jun 2026). This suggests that “global tripartite negativity” and “negativity-based genuine tripartite entanglement” are closely related but not universally synonymous across all subliteratures.

3. Coarse-grained gTN at impurity quantum phase transitions

The most influential use of the geometric-mean construction in many-body impurity physics is the coarse-grained measure $E_1=\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}.$3 in the study of impurity quantum phase transitions (Bayat, 2016). The central methodological step is the partition into three physically natural subsystems rather than microscopic particles. In the two-impurity Kondo model (2IKM), $E_1=\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}.$4 is the two-impurity sector, $E_1=\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}.$5 the left bulk, and $E_1=\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}.$6 the right bulk. In the two-channel Kondo model (2CKM), $E_1=\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}.$7 is the single impurity spin and $E_1=\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}.$8 are the left and right bulks. The quantity therefore measures entanglement among the impurity sector and the two bulks, not among individual spins (Bayat, 2016).

Within this coarse-grained partition, the paper reports that tripartite entanglement diverges at criticality in the thermodynamic limit and obeys finite-size scaling. At the critical points,

$E_1=\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}.$9

and

$E_1$0

The scaling ansatz is given as

$E_1$1

with

$E_1$2

For the global measure $E_1$3, both models yield

$E_1$4

The associated divergent length scale is

$E_1$5

and the paper states that $E_1$6 is in perfect agreement with conformal field theory and Schmidt-gap analyses (Bayat, 2016).

The main interpretive claim is universality. Because the 2IKM and 2CKM exhibit the same $E_1$7 and the same $E_1$8 exponent $E_1$9, the paper presents the near identity of the scaling behavior as evidence that the two models belong to the same universality class (Bayat, 2016).

The same study also delineates the limitations of gTN. In the 2IKM for $\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}$0, both $\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}$1 and $\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}$2 vanish, yet the bulk-bulk negativity $\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}$3 remains significant because the impurities mediate an effective coupling $\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}$4. Thus gTN is sensitive to a specific coarse-grained notion of genuine tripartite sharing and does not exhaust all entanglement present in the system (Bayat, 2016).

4. Mixed-spin trimers, tetramers, and molecular realizations

In finite spin clusters, gTN is used as an explicit entanglement observable rather than only as a scaling diagnostic. For the mixed-spin $\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}$5 Heisenberg trimer, the quantity

$\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}$6

is the primary global witness of full three-party entanglement. Because the two spin-1 sites are equivalent, the trimer reduces to

$\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}$7

The 2025 anisotropy study shows that the uniaxial single-ion anisotropy $\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}$8 can either increase or decrease gTN depending on which ground state is stabilized, can shift the magnetic-field values where ground-state transitions occur, and can alter entanglement indirectly through the eigenvector coefficients rather than only through energies (Vargová et al., 1 Sep 2025).

Several phase-resolved statements are explicit. In the phase $\left[N_{A,BC}N_{B,AC}N_{C,AB}\right]^{1/3}$9, gTN is independent of both ${\cal N}_{ABC}$0 and ${\cal N}_{ABC}$1, reaches

${\cal N}_{ABC}$2

at approximately ${\cal N}_{ABC}$3, and then decreases at larger ${\cal N}_{ABC}$4 because ${\cal N}_{ABC}$5 is reduced (Vargová et al., 1 Sep 2025). In the phase ${\cal N}_{ABC}$6, the maximum is

${\cal N}_{ABC}$7

at

${\cal N}_{ABC}$8

In the phase ${\cal N}_{ABC}$9, the $$\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3}$$0 ground state can be biseparable for $$\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3}$$1, but once $$\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3}$$2, anisotropy turns on gTN even for arbitrarily small $$\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3}$$3 (Vargová et al., 1 Sep 2025).

A complementary trimer study emphasizes robustness and thermal activation. For the same mixed-spin $$\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3}$$4 model, gTN is reported at zero temperature in the antiferromagnetic regime $$\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3}$$5 when

$$\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3}$$6

and in a real compound $$\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3}$$7 it is predicted to persist up to approximately $$\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3}$$8 K and magnetic fields approaching $$\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3}$$9 T (Vargová, 19 Jun 2025). The same paper highlights thermally induced activation in a biseparable ground-state region, where the maximum activated gTN can reach

$N_{A,B}=\sum_k |\lambda_k|-1,$0

described as nearly $N_{A,B}=\sum_k |\lambda_k|-1,$1 (Vargová, 19 Jun 2025).

In the mixed-spin $N_{A,B}=\sum_k |\lambda_k|-1,$2 triangular trimer, the same geometric-mean construction appears under the name “tripartite negativity.” The paper states that non-uniform isotropic $N_{A,B}=\sum_k |\lambda_k|-1,$3-factors make the magnetic moment operator fail to commute with the Hamiltonian, so ground-state entanglement can vary continuously with field inside a fixed branch. In the fully antiferromagnetic case, the $N_{A,B}=\sum_k |\lambda_k|-1,$4 branch attains the maximum

$N_{A,B}=\sum_k |\lambda_k|-1,$5

at $N_{A,B}=\sum_k |\lambda_k|-1,$6, while the $N_{A,B}=\sum_k |\lambda_k|-1,$7 branch is biseparable with $N_{A,B}=\sum_k |\lambda_k|-1,$8 despite enhanced bipartite entanglement (Adamyan et al., 2024).

The mixed spin-$N_{A,B}=\sum_k |\lambda_k|-1,$9 Heisenberg tetramer extends the construction by tracing out one spin from a four-spin density matrix and evaluating gTN on the resulting trimer. Two inequivalent reduced trimers occur, $\lambda_k$0 and $\lambda_k$1, and the paper reports that the $\lambda_k$2 subsystem generally exhibits stronger and more thermally robust gTN than the $\lambda_k$3 subsystem (Vargová et al., 2023).

5. Thermal, dynamical, and non-equilibrium settings

The geometric-mean tripartite negativity is also used for explicitly mixed and time-dependent states. In the inhomogeneous spin-star system, the tripartite negativity of the three outer spins is computed after tracing out the central spin from a thermal Gibbs state. In the homogeneous case the three partial negativities are equal, so the geometric mean reduces to any one of them. The paper reports that the negativity decreases with increasing temperature, shows low-temperature jumps at level crossings, and can attain a maximum away from the homogeneous point: for type-A inhomogeneity the maximum occurs at

$\lambda_k$4

for $\lambda_k$5 and low $\lambda_k$6, while for type-B inhomogeneity it occurs at

$\lambda_k$7

under the same parameter choice (Anzà et al., 2010). The authors emphasize that concurrence can vanish while tripartite negativity remains nonzero, because $\lambda_k$8 probes entanglement between one spin and the pair of the other two jointly rather than pairwise entanglement after tracing (Anzà et al., 2010).

In three coupled quantum memristors, the same formula is called “tripartite negativity” and is evaluated on the full time-dependent density matrix $\lambda_k$9. The principal dynamical result is that “the tripartite negativity for ${\cal N}_{A|BC}=\sum_{\lambda_i<0}|\lambda_i|,$0 is always different from zero,” in both triangular and linear coupling geometries and for identical, partially identical, and fully non-identical devices (Kumar et al., 2022). The paper uses this together with monogamy analysis to argue that the three-memristor state is in a genuine tripartite entangled quantum state at any time, even when some pairwise entanglement measures exhibit sudden death or zeros (Kumar et al., 2022).

In oscillating neutrinos, the closest counterpart is not the geometric-mean gTN but the three-${\cal N}_{A|BC}=\sum_{\lambda_i<0}|\lambda_i|,$1 negativity. The neutrino flavor state is treated as a three-qubit state in the ${\cal N}_{A|BC}=\sum_{\lambda_i<0}|\lambda_i|,$2 occupation-number basis, and the paper states that the neutrino belongs to the W class. Because the CKW equation in terms of negativities is not saturated, the residual quantity

${\cal N}_{A|BC}=\sum_{\lambda_i<0}|\lambda_i|,$3

remains nonzero and quantifies genuine tripartite entanglement, which the paper reports to survive even in the decoherence limit and to approach a constant asymptotic value at large baselines (Banerjee et al., 12 Jun 2026). This is not the standard geometric-mean gTN, but it occupies the same conceptual role of a negativity-based tripartite entanglement diagnostic for a W-class system.

6. Conceptual extensions and terminological caveats

The term “gTN” does not denote a unique quantity across all current arXiv usage. A particularly important divergence appears in high-energy spin tomography, where the relevant object is the genuine multipartite negativity (GMN) rather than the geometric mean of one-versus-rest negativities. For ${\cal N}_{A|BC}=\sum_{\lambda_i<0}|\lambda_i|,$4 in ${\cal N}_{A|BC}=\sum_{\lambda_i<0}|\lambda_i|,$5, GMN is defined by the semidefinite optimization

${\cal N}_{A|BC}=\sum_{\lambda_i<0}|\lambda_i|,$6

with ${\cal N}_{A|BC}=\sum_{\lambda_i<0}|\lambda_i|,$7 fully decomposable across all three cuts. The paper proves

${\cal N}_{A|BC}=\sum_{\lambda_i<0}|\lambda_i|,$8

and for the ${\cal N}_{A|BC}=\sum_{\lambda_i<0}|\lambda_i|,$9 system gives the bound

$${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$00

Numerically, differential states show $${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$01, partially inclusive states peak near $${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$02, and fully inclusive states yield $${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$03 for all energies (Gonçalves et al., 9 Jun 2026). GMN is therefore a stricter mixed-state certification of genuine multipartite entanglement than the geometric-mean gTN and should not be identified with it.

A second extension is the “third-order negativity” $${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$04 in the separability framework

$${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$05

That paper proves

$${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$06

for tripartite pure states and constructs convex-roof mixed-state measures whose vanishing is equivalent to full separability (Ma et al., 3 May 2026). This is again a negativity-based tripartite quantity, but not the conventional gTN.

A terminological caveat is also necessary for “GTN.” In the Schwarzschild-spacetime decoherence paper, GTN means genuine tripartite nonlocality detected by Svetlichny inequality violation, not negativity-based global tripartite entanglement (Liu et al., 2024). The acronym overlap can be substantial in quantum-information and relativistic-QI contexts.

Taken together, these usages support a precise but narrow definition: in the conventional sense, Global Tripartite Negativity is the geometric mean of the three one-versus-two negativities. It is most naturally interpreted as a symmetric block-level witness of full tripartite inseparability, especially effective in coarse-grained impurity systems, mixed-spin clusters, and other three-body reduced states. Beyond that core usage, the broader literature contains related but distinct negativity-based tripartite diagnostics—residual three-$${\cal N}_{ABC}=\left({\cal N}_{A|BC}{\cal N}_{B|AC}{\cal N}_{C|AB}\right)^{1/3},$$07 measures, GMN witness optimizations, and higher-order replica negativities—that address different questions about genuine multipartite entanglement, separability, or nonlocality (Bayat, 2016, Gonçalves et al., 9 Jun 2026).