Tripartite Negativity: Quantum Entanglement Measure

• Tripartite negativity is a multipartite entanglement measure defined as the geometric mean of bipartite negativities computed via partial transposes of a three-party density matrix.
• It signals genuine three-way inseparability in quantum systems by yielding a nonzero value only for fully inseparable states, with efficient computability under LOCC.
• This measure is applied in multimode cavity QED, spin systems, and open quantum dynamics, aiding in the design of quantum networks and error correction protocols.

Tripartite negativity is a multipartite entanglement measure constructed as the geometric mean of bipartite negativities over all possible single-versus-pair bipartitions of a three-party quantum system. It extends the widely employed bipartite negativity (defined via eigenvalues of the partial transpose of the density matrix) to provide a symmetric, operationally accessible indicator of genuine three-way inseparability, particularly in systems of finite-dimensional qudits as well as in continuous-variables, hybrid, and even indistinguishable-particle settings.

1. Definition and General Properties

Consider a state $\rho_{ABC}$ of three quantum subsystems $A$, $B$, $C$. For each partition $X|YZ$, define the bipartite negativity as

$$N_{X|YZ} = \sum_i \left(|\lambda_i(\rho^{T_X}_{ABC})| - \lambda_i(\rho^{T_X}_{ABC})\right),$$

where $\rho^{T_X}_{ABC}$ denotes the partial transpose of $\rho_{ABC}$ with respect to subsystem $X$, and $\lambda_i(\cdot)$ its eigenvalues. This quantity is equal to twice the sum of the negative eigenvalues of the partially transposed density matrix.

The tripartite negativity is then given by the geometric mean

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

A nonzero $N_{ABC}$ is a sufficient condition for genuine three-partite entanglement—i.e., inseparability across every bipartition—although it may not distinguish certain mixtures of biseparable states from fully inseparable ones.

Tripartite negativity

• vanishes on all fully separable and biseparable states,
• is invariant under local unitaries,
• is monotonic under LOCC,
• is efficiently computable for any state where the reduced density matrix can be constructed and diagonalized.

2. Computation and Implementation Strategy

To compute the tripartite negativity for a general mixed state $\rho_{ABC}$:

1. For each partition $X|YZ$ ($X=A,B,C$), construct the partial transpose $\rho^{T_X}_{ABC}$ of the $8\times8$ (qubit) or higher-dimensional density matrix.
2. Diagonalize $\rho^{T_X}_{ABC}$, extract its negative eigenvalues $\{\lambda_i^- \}$, and evaluate $N_{X|YZ} = 2\sum_i |\lambda_i^-|$.
3. Compute the geometric mean over all three bipartitions.

For open-system dynamics or reduced models (e.g., tracing out cavity modes (Amgain et al., 16 Jul 2025)), one first traces over environmental degrees of freedom to obtain the relevant three-partite reduced density matrix as a function of time and system parameters, then proceeds as above.

3. Physical Contexts and Representative Results

Tripartite negativity has been deployed across a wide variety of physical platforms:

a) Multimode Cavity QED

In multimode cQED with three initially unentangled two-level qubits in a triangle-shaped optical cavity, the time-evolved state under a multimode Jaynes–Cummings-like Hamiltonian is used to generate multipartite entanglement (Amgain et al., 16 Jul 2025). Key steps include:

• Numerically solving the Schrödinger equation (three-excitation manifold) for up to $N_m\simeq 31$ modes;
• Tracing out all field modes to obtain $\rho_{ABC}(t)$;
• Computing $N_{ABC}(t)$ as described above.

Main findings:

• In the single-mode case, tripartite negativity exhibits regular collapse–revival oscillations with maximum $N_{ABC}^{(\text{max})} \approx 0.68$ (lossless).
• In the multimode case, $N_{ABC}(t)$ displays additional kinks and retardation effects linked to photon round trip and inter-qubit propagation times, with maximum $N_{ABC}$ saturating for $N_m \gtrsim 15$.
• Losses (mirror leakage, spontaneous emission) suppress peak $N_{ABC}$, yielding $N_{ABC}^{(\text{max})}\sim 0.1$–$0.2$ in typical dissipative scenarios.

b) Quantum Impurity and Spin Systems

In spin trimers (e.g., $(1,1/2,1)$ and $(1/2,1/2,1)$ Heisenberg triangle models (Vargová, 19 Jun 2025, Adamyan et al., 2 Sep 2024)), as well as mixed-spin tetramers (Vargová et al., 2023), tripartite negativity provides signatures of various entanglement-type "phases” and their robustness against external field, temperature, and exchange-coupling inhomogeneity. In particular:

• At zero temperature, nonzero $N_{ABC}$ identifies parameter regimes with fully inseparable ground states.
• At finite temperature, $N_{ABC}$ quantifies thermal resilience of genuine tripartite entanglement, often showing threshold temperatures above which $N_{ABC}$ vanishes.

c) Disordered and Topological Systems

In many-body settings where partitioning is by spatial regions (e.g., free fermion chains, topologically ordered media), variants such as logarithmic tripartite negativity, as well as symmetry- or sector-resolved negativities, provide scaling data (e.g., area laws, critical exponents (Bayat, 2016)) or reveal universal contributions distinguishing Abelian from non-Abelian order (Sohal et al., 2023).

d) Open Quantum Systems and Decoherence

The time dependence of $N_{ABC}$ under various Markovian and non-Markovian decoherence models has been analyzed for canonical states (GHZ, W, Werner-class), revealing collapse–revival dynamics, sudden death of entanglement, and, in gradient environments, regimes of "freezing" or delayed disentanglement (Hu, 2012, Magare et al., 2021).

4. Role in Quantum Technologies

Tripartite negativity functions as both a diagnostic and a performance metric for multipartite entanglement in engineered quantum systems:

• In cavity/circuit QED, controlling the timing and geometry to maximize $N_{ABC}$ enables on-demand generation of GHZ-like entangled resources for quantum networking and secret sharing.
• In quantum memories, mapping the temporal structure of tripartite negativity allows for synchronization and robust storage of three-party entangled states.
• In higher-spin or molecular implementations (e.g., Ni–Cu complexes (Vargová, 19 Jun 2025)), experimental proxies of $N_{ABC}$ can be constructed from measurable observables such as magnetization or neutron scattering structure factors.
• In quantum information protocols, $N_{ABC}>0$ is a necessary resource for multipartite teleportation or error correction routines that cannot be performed with only bipartite entanglement.

5. Limitations and Interpretational Cautions

While $N_{ABC}>0$ suffices for witnessing genuine three-way inseparability, it is not a full entanglement monotone. Specifically:

• It cannot distinguish among inequivalent entanglement classes (e.g., GHZ-like vs. W-like) for mixed states.
• There exist biseparable mixtures with $N_{ABC}>0$, so false positives are possible if more stringent detection is required. For rigorous verification of full inseparability, witnesses based on the Huber–Ghira criteria or symmetry-resolved entropies may be preferable (Militello et al., 2010).
• In large systems or nontrivial partitions, computational cost can grow rapidly, but block structure and symmetry can often be exploited for tractable calculation.

6. Extensions and Theoretical Generalizations

Generalizations of tripartite negativity include:

• Monogamy-based residual entanglement measures (e.g., tangle-inspired, $\pi_{ABC}$), which quantify the "excess" inseparability beyond that present in any pair (Jha et al., 2020).
• Symmetry- and sector-resolved negativities (via partial time-reversal), enabling finer analysis of entanglement sharing among conserved quantities (Murciano et al., 2021, Travaglino et al., 11 Jun 2025).
• Application to indistinguishable-particle scenarios (fermionic or bosonic), where superselection rules require projection onto fixed local particle number sectors prior to calculation (Buscemi et al., 2011).

7. Summary Table: Technical Features of Tripartite Negativity

Feature	Description	Physical/Mathematical Implementation
Definition	Geometric mean of three bipartite negativities	$N_{ABC} = [N_{A|BC}N_{B|AC}N_{C|AB}]^{1/3}$
Applicability	General quantum systems, pure or mixed states	Finite-qudit, hybrid, or CV systems
Robustness to Loss	Sensitive to decoherence, can exhibit revivals	Dynamics controlled by system–bath couplings
Operational Meaning	Witnesses genuine multipartite inseparability	$N_{ABC} > 0$ iff not biseparable
Experimental Protocols	Density matrix tomography, observable reconstruction	Magnetometry, neutron scattering, state tomography
Theoretical Extensions	Monogamy residuals, symmetry-resolved, indistinguishable particles	$\pi_{ABC}$, sector negativities, projected negativities

Tripartite negativity therefore serves as a central quantitative tool for diagnosing, quantifying, and manipulating genuine three-way entanglement in contemporary quantum science, bridging theoretical analysis, numerical simulation, and experimental implementation.