Nonlinear Monogamy Bound in Quantum Entanglement

• Nonlinear monogamy-of-entanglement bounds are precise polynomial constraints that govern how quantum entanglement is shared among multiple subsystems.
• They replace the traditional linear relations by applying power-law inequalities to measures like negativity, concurrence, and entanglement of formation.
• These bounds are critical for optimizing quantum protocols and for understanding the structure of multipartite correlations in both qubit and qudit systems.

Nonlinear monogamy-of-entanglement bounds represent a set of precise quantitative constraints governing the distribution of quantum entanglement among multipartite quantum systems. Unlike the original linear monogamy relations, these bounds involve nonlinear (typically polynomial or power-law) constraints on entanglement measures such as negativity, concurrence, and entanglement of formation. Their characterization is essential both for the fundamental structure of multipartite quantum correlations and for operational tasks in quantum information theory. The most advanced results provide both necessary and sufficient polynomial inequalities for physically allowed sets of bipartite entanglements and define the minimal “monogamy power” required for various entanglement monotones to enforce exclusive sharing.

1. Formulation and Key Definitions

For a tripartite pure state $\rho_{ABC}$, consider a computable bipartite entanglement measure $E$ (e.g., negativity, concurrence, or entanglement of formation). The standard (linear) monogamy relation attempts to bound the entanglement across the partition $A|BC$ by the sum of the entanglements across $A|B$ and $A|C$: $E_{A|BC} \geq E_{A|B} + E_{A|C}.$ However, this inequality fails for most measures except in trivial or highly restricted situations. Nonlinear monogamy bounds resolve this by imposing a higher degree polynomial or power-law constraint, typically involving $E^\alpha$ for a suitable $\alpha > 1$, or, more precisely, by introducing a polynomial boundary surface in the space of achievable bipartite entanglement values (Allen et al., 2015).

For entanglement negativity in three-qubit systems, the physically achievable triple $$(x, y, z) \equiv (\mathcal{N}_{A|C}, \mathcal{N}_{A|B}, \mathcal{N}_{A|BC})$$ must satisfy a necessary and sufficient nonlinear polynomial inequality: $$f(x,y,z) \equiv (z^2 - x^2 - y^2 + xy)(1+\sqrt{1-z^2}) - xy (2 + x + y) \ge 0,$$ where

$$\mathcal{N}_{X|Y} = 2 \sum_{\lambda_i < 0} \lambda_i(\rho_{XY}^{T_X}),$$

and $\rho_{XY}^{T_X}$ denotes the partial transpose over $X$, normalized so that $\mathcal{N}=1$ for a maximally entangled qubit pair (Allen et al., 2015).

2. Tight Polynomial Bounds for Negativity

The nonlinear monogamy constraint for negativity is given by a degree-6 polynomial in $x, y, z$, after clearing the radical: \begin{align*} &z^6 - 2z^4(x^2 - x y + y^2) \ &+ z^2\left(x^4 + y^4 - 2x y[x(x-1) + y(y-1) - \tfrac32 x y + 2]\right) \ &+ x y (2 y^2 + x y^2 + x^2 y + 2 x^2)(x + y + 2) = 0 \end{align*} The physically valid set is the region $$\{(x, y, z) \mid 0 \le x, y \le z \le 1, \, f(x, y, z) \ge 0 \}$$.

This bound is both necessary and sufficient; every physically realizable triple arises as a point in this region, and every boundary point can be saturated by explicit quantum states. The only states that saturate $f(x, y, z) = 0$ are locally equivalent to the W-family ($c=0$, $\omega=0$ in the three-qubit Acín form) (Allen et al., 2015).

3. Linear Monogamy Failure and Boundary States

The standard linear monogamy bound $z \geq x + y$ follows trivially from monotonicity of negativity under LOCC (convexity), but is only saturated when either $x = 0$ or $y = 0$. For many states, notably those in the three-qubit W-class, physically valid points appear strictly below the plane $z = x + y$. In these cases, $z \approx x + y - \delta$ for some $\delta > 0$. This exposes substantial “holes” in the parameter region excluded by linear monogamy, which are filled precisely by the degree-6 polynomial boundary (Allen et al., 2015).

Explicitly, W-class states,

$$|\Psi_W\rangle = d|000\rangle + a|101\rangle + b|110\rangle,$$

with $d,a,b \geq 0$ and $d^2 + a^2 + b^2 = 1$, yield tuples $(x, y, z)$ parametrized as

$$(x^2, y^2, z^2) = \left( (b^2 - \sqrt{b^4 + 4a^2 d^2})^2,\, (a^2 - \sqrt{a^4 + 4b^2 d^2})^2,\, 4 d^2 (a^2 + b^2) \right).$$

These states, under elimination via Gröbner basis, precisely fill the polynomial bounding surface (Allen et al., 2015).

4. Generalization to Qudit Systems and Asymptotic Behavior

The non-linear monogamy bound generalizes to $D$-dimensional qudits using W-type boundary states,

$$|\Psi_{\mathrm{W}}^{(D)}\rangle = d |0,0,0\rangle + \sum_{j=1}^{D-1} a |j,0,j\rangle + b |j, j, 0\rangle,$$

with $c=0$ (and normalization). For large $D$,

$$x \approx D^2 a^2, \quad y \approx D^2 b^2, \quad z \approx D^2(a^2 + b^2),$$

and the polynomial bound collapses in the $D \to \infty$ limit to $z \geq x + y$ up to $O(1/D)$ corrections. Thus, asymptotically, negativity can be distributed "at least linearly," and, supported by explicit examples and numerics, it is conjectured to be "at most linear" for large $D$ (Allen et al., 2015).

5. Nonlinear Power-law Monogamy for General Measures

A broad class of nonlinear monogamy inequalities are power-law relations of the form

$$E^\alpha_{A|BC} \geq E^\alpha_{A|B} + E^\alpha_{A|C}$$

for some minimal $\alpha > 1$ specific to each entanglement measure $E$ (Guo, 2017, Luo et al., 2015, Zhu et al., 2023). For negativity and concurrence, $\alpha = 2$ suffices in qubit systems; for entanglement of formation, $\alpha = 2$ or at least $\sqrt{2}$ is required. These power-law exponents define the "monogamy power" $\alpha(E)$ of $E$ and quantify the departure from linear separability in the sharing structure of entanglement.

For generic quantum measures, the existence of such an $\alpha$ is both necessary and sufficient for global monogamy in that measure, and tightness is realized typically on extremal states with only one nonzero bipartite marginal (Guo, 2017, Zhu et al., 2023).

6. Methodologies: Algebraic Construction and Boundary Characterization

The derivation of sharp nonlinear bounds, such as the degree-6 polynomial for negativity, relies on:

• Parametric reduction to canonical forms (e.g., Acín decomposition for pure states)
• Explicit calculation and extremization of negativity for all bipartite cuts
• Systematic elimination of state parameters (e.g., via Gröbner basis techniques) to yield a minimal algebraic relation for observables
• Examination of extremal classes (e.g., W-class, GHZ) to prove tightness, saturation, and boundary structure

The general methodology is to identify the full parameter set of tripartite or multipartite entanglement measures, determine all algebraic constraints between them for pure states, and project these constraints onto lower-dimensional convex sets or algebraic surfaces.

7. Physical Consequences and Open Problems

Nonlinear monogamy-of-entanglement bounds establish exact multipartite constraints that are operationally significant in quantum network architecture, entanglement distillation protocols, entanglement catalysis, and as diagnostic tools for multipartite entanglement structure.

The only states saturating the non-linear polynomial bound for negativity are locally equivalent to the W-family. All other physically allowed points inside the region are interior points corresponding to mixing with separable components (Allen et al., 2015). Notably, the linear monogamy relation proved too stringent, and only the nonlinear (polynomial or power-law) refinement provides necessary and sufficient constraints.

Outstanding open questions include:

• Full algebraic characterizations for higher-dimensional qudit multipartite systems (beyond $D=2$ and generic states), hindered by complexity of elimination in the parameter space
• Extension of exact polynomial monogamy bounds to other entanglement monotones with algebraic (rather than mere power-law) relations
• Analytical determination of the minimal monogamy exponent $\alpha(E)$ for composite or higher-dimensional monotones

These non-linear, often polynomial, relations provide both sharp operational restrictions and a refined taxonomy of multipartite entangled states, filling the gaps left by earlier linear monogamy conjectures and characterizing physically valid configurations in multipartite quantum networks (Allen et al., 2015, Guo, 2017, Zhu et al., 2023).