Polynomial W Entanglement Measure

• The polynomial W measure is a cubic, local unitary–invariant function that precisely quantifies genuine tripartite W-type entanglement in three-qubit systems.
• It distinguishes W, bipartite, and GHZ entanglement by leveraging unique algebraic constructions and distinct polynomial degrees for each entanglement class.
• This measure underpins operational protocols in quantum information science, facilitating the detection, classification, and quantification of entangled pure and mixed states.

A polynomial measure of W entanglement quantifies genuine tripartite W-type entanglement in three-qubit systems using a local unitary–invariant polynomial function of the pure-state amplitudes. It provides a mathematical tool for distinguishing and ordering different kinds of entanglement—bipartite, W, and GHZ—in terms of both degree and operational significance. Recent advances have clarified the algebraic, geometric, and operational structure of such measures, offering precise methods for detection, classification, and quantification of W-class entanglement in pure and mixed states.

1. Algebraic Construction of the Polynomial W Measure

The canonical polynomial W measure for three-qubit pure states is the cubic SL(2, ℂ)³-invariant function ω(ψ), distinct from the bipartite concurrence (degree 2) and the three-tangle (GHZ invariant, degree 4). Given a three-qubit pure state in computational basis,

$$|\psi\rangle = \sum_{i,j,k=0}^1 \psi^{ijk} |i\,j\,k\rangle \in \mathcal{H}_1\otimes\mathcal{H}_2\otimes\mathcal{H}_3,$$

one defines the antisymmetric tensor

$$\epsilon = \sum_{i,i'=0}^1 \epsilon_{ii'}|i\rangle\otimes|i'\rangle,\quad\epsilon_{01}=+1,\ \epsilon_{10}=-1,$$

satisfying $\epsilon(A\otimes A)=\det(A)\epsilon$ for any 2×2 matrix A. The cubic tensor

$$T(\psi)^{ijk} = -\sum_{l,l',m,m',n,n'=0}^1 \epsilon_{ll'}\epsilon_{mm'}\epsilon_{nn'}\, \psi^{imn}\psi^{l m'n'}\psi^{l' j k}$$

encodes the essential polynomial contraction, symmetric under relabeling of qubits. The polynomial W measure is then

$$\boxed{ \omega(\psi) = 2\|T(\psi)\| = 2\sqrt{ \sum_{i,j,k=0}^1 |T(\psi)^{ijk}|^2 } }$$

where $\|\cdot\|$ is the Euclidean norm in $\mathbb{C}^8$.

Under local unitaries and invertible SL transformations, ω(ψ) is invariant: each $\epsilon$-contraction yields one power of $\det(A)$ per party, ensuring homogeneous transformation properties and establishing ω as a genuine LU and SLOCC invariant of degree 3 in amplitudes.

2. Classification and Ordering of Entanglement

In three-qubit systems, the relevant polynomial entanglement measures and their vanishing loci delineate the classes of entanglement:

• Bipartite concurrence ($c_{a|bc}(\psi)$): Degree 2, vanishes on states separable across the $a|bc$ partition.
• W measure ($\omega(\psi)$): Degree 3, vanishes on biseparable and fully separable states; nonzero for W and GHZ classes.
• Three-tangle ($\tau(\psi)$): Degree 4, nonzero only for GHZ-class states; vanishes for W and biseparable classes.

These measures satisfy the strict ordering chain for all pure three-qubit states: $$0 \leq \tau(\psi) \leq \omega(\psi) \leq c_{a|bc}(\psi) \leq 1$$ with normalization such that the GHZ state achieves $\omega=1$ and the W state achieves $\omega(W)=\frac{4}{3\sqrt{3}} \approx 0.770$.

This structure reflects the inclusion of equivalence classes under SLOCC:

• Bipartite class ⊂ W class ⊂ GHZ class, and operational strength: bipartite entanglement is weaker than W, which is weaker than GHZ (Szalay, 10 Nov 2025).

3. Geometric and Operational Interpretations

The measure ω(ψ) is naturally interpreted within the Freudenthal triple system (FTS) formalism:

• FS-rank 1: Fully separable
• FS-rank 2: Biseparable
• FS-rank 3: W-class
• FS-rank 4: GHZ-class

The cubic-norm form $T(\psi)$ and its norm ω(ψ) quantify the geometric "distance" to biseparability inside the W stratum. Operationally, although there is no single-shot LOCC protocol that yields ω(ψ) as a distillation rate, the measure increases strictly with SLOCC rank, aligning with resource conversion rates in nonlocal games distinguishing SLOCC types (e.g., the GHZ game of Mermin–Watrous). Thus, ω quantifies a resource stronger than bipartite entanglement but not as strong as GHZ-type nonlocality.

4. Connection to Other Polynomial and Operational Measures

W entanglement can also be captured using polynomial invariants distinct from ω, including those arising from:

• Pairwise squared-concurrences: $$M_W(|\psi\rangle) = |d_{AB}|^2 + |d_{BC}|^2 + |d_{CA}|^2$$, degree 4, provides a W-monotone strictly positive on W-class states and vanishing on GHZ and fully separable states (Gadde et al., 25 Jun 2024).
• Local purity deficits: For n-qubit W states, $1 - \operatorname{Tr}\rho_t^2$ is a degree 4 LU-invariant polynomial, proportional to the sum of two-tangles (pairwise squared-concurrence) in the system, establishing closed-form relations among W entanglement, the k-ME concurrence, negativity, and tangle (Zhang et al., 2020).
• Entanglement polynomials via operator size: The polynomial $E(\psi;x)$ from the operator size formalism yields a unique classification of W-type entanglement via the generating function $E(W; x) = 1 + 6x + x^2$ for three qubits, providing a SLOCC-invariant and factorization-based approach that distinguishes the W class from GHZ and other classes (Wu, 2021).
• LOCC monotones for W-class states: Symmetric polynomial monotones such as η(⃗x) and κ(⃗x) provide analytic LOCC monotones for arbitrary W-class states with operational significance for EPR distillation probabilities (Chitambar et al., 2011).

5. Explicit Example: Evaluation on the Canonical W State

Consider the standard three-qubit W state,

$$|W\rangle = \frac{1}{\sqrt{3}} (|100\rangle + |010\rangle + |001\rangle)$$

The cubic tensor T(W) yields only a small set of nonzero components, leading to: $\|T(W)\|^2 = \frac{1}{18}$ so

$$\omega(W) = 2 \times \frac{1}{\sqrt{18}} = \frac{4}{3\sqrt{3}} \approx 0.770$$

The single-qubit reduced density matrices have eigenvalues $\{2/3, 1/3\}$, yielding

$$c_{a|bc}(W) = 2\sqrt{(2/3)(1/3)} = \frac{2\sqrt{2}}{3} \approx 0.943$$

and

$\tau(W) = 0$

Exemplifying the ordering: $$0 = \tau(W) < \omega(W) = \frac{4}{3\sqrt{3}} < c_{a|bc}(W) = \frac{2\sqrt{2}}{3}$$ This confirms that ω(W) quantifies precisely genuine W-type entanglement, not reducible to bipartite or GHZ-type invariants.

6. Extensions to Mixed States and Higher Qubit Number

For polynomial entanglement measures E (including ω), mixed-state entanglement is characterized via the convex roof construction: $$E(\rho) = \min_{\{p_i, |\psi_i\rangle\}} \sum_i p_i E(|\psi_i\rangle)$$ Algorithms for upper bounds use best-zero-E decompositions ("BEA") and rank-reducing S-decompositions to efficiently compute bounds for arbitrary mixed states (Rodriques et al., 2013). For larger systems, generalizations include polynomials capturing higher-order tangles (e.g., threetangle for four-qubit W-analogues) (Gartzke et al., 2017), and polynomial invariants arising from sums of pairwise concurrences or local purity deficits (Zhang et al., 2020).

The operator-size approach (entanglement polynomial) yields for the three-qubit W state the unique polynomial $E(W;x) = 1+6x+x^2$, whose factorization uniquely signals genuine W-type entanglement among the SLOCC orbit classes (Wu, 2021).

7. Significance for Quantum Information Science

The polynomial W measure ω(ψ) completes the classification of three-qubit entanglement in terms of LU- and SLOCC-invariant polynomials. It distinguishes W-type entanglement, which is the minimal genuinely multiparty entanglement not reducible to bipartite splits, from both biseparable and GHZ-type entanglement. This enables:

• Precise detection of the W class in experimental or simulated three-qubit states.
• Quantitative comparison and ordering of tripartite entanglement resources.
• Foundation for operational protocols (e.g., nonlocal games, distillation) that exploit or distinguish SLOCC class resources.
• A consistent extension to polynomial entanglement monotones for multipartite systems beyond three qubits, as demonstrated via generalizations of the cubic and quartic invariants and analytic monotones for higher W and GHZ analogues.

The polynomial measure of W entanglement thus forms a cornerstone of multipartite entanglement theory, enabling both structural understanding and practical quantification within quantum information science.