---
title: Negativity Percolation Theory in Quantum Networks
url: https://www.emergentmind.com/topics/negativity-percolation-theory-negpt
type: topic
---

# Negativity Percolation Theory in Quantum Networks

Searching arXiv for the core NegPT paper and closely related precursor strands.
arXiv search query: "Negativity Percolation Theory continuous-variable quantum networks ratio negativity"
Negativity Percolation Theory (NegPT) is the percolation-theoretic description of deterministic entanglement distribution in continuous-variable quantum networks introduced for Gaussian resources, especially two-mode squeezed vacuum states, in a Gaussian-to-Gaussian deterministic entanglement transmission scheme. In this formulation, the relevant link variable is the bounded **ratio negativity** \(\chi\in[0,1]\), and the relevant macroscopic connectivity observable is the **sponge-crossing ratio negativity** \(\mathrm{X}_{\mathrm{SC}}\), defined between two macroscopic node sets \(S\) and \(T\) [2507.16417]. The broader literature that informs NegPT also includes earlier work on negative-weight percolation as a disorder-driven geometric critical phenomenon [1003.1591], directed and two-dimensional variants of that phenomenon [1701.04726] [1205.1412], and several quantum-negativity formalisms in which negativity functions as a geometric, spectral, dynamical, or resource-theoretic variable [1206.3092] [1412.7568] [1607.02992] [1809.07772] [2105.01685] [2012.00031].

## 1. Core concept and formal scope

NegPT is formulated on a quantum network represented by a graph in which nodes are spatially separated parties and edges are shared bipartite entangled resource states. The elementary resource is the two-mode squeezed vacuum state
\[
|\psi^{r}\rangle = \sqrt{1-\tanh^2 r}\sum_{n=0}^{\infty}\tanh^n r\,|nn\rangle, \qquad r\in[0,\infty),
\]
and in homogeneous networks every link is assigned the same entanglement weight
\[
\chi \equiv \tanh r.
\]
The paper introducing NegPT presents this variable as the continuous-variable analogue of the bond occupation probability in classical percolation and of concurrence in discrete-variable deterministic entanglement transmission [2507.16417].

The defining claim of NegPT is that large-scale continuous-variable entanglement transport is not naturally organized by occupation probability or concurrence, but by the bounded ratio negativity \(\chi\). The network-level observable is the sponge-crossing ratio negativity \(\mathrm{X}_{\mathrm{SC}}\), which plays the role of a global connectivity or order parameter between two macroscopic node sets \(S\) and \(T\). In this sense, NegPT is the statistical theory generated by deterministic continuous-variable entanglement-composition rules rather than by stochastic bond activation [2507.16417].

Several model classes appear in the formal development. The theory is developed most explicitly for series-parallel networks, for some non-series-parallel motifs treated approximately by star-mesh transforms, and for the Bethe lattice, where exact self-consistent equations can be written. This suggests that NegPT is presently most mature as a renormalization-style theory of effective entanglement connectivity on recursively reducible graph families [2507.16417].

## 2. Conceptual antecedents and precursor lineages

One precursor lineage comes from **negative-weight percolation** (NWP), which treats percolation as a global optimization problem over nonintersecting negative-weight loops, or one path plus loops, rather than as independent local occupation. On hypercubic lattices, the configurational energy is
\[
\mathcal E=\sum_{\mathcal L\in\mathcal C}\omega_{\mathcal L},
\]
and the transition is characterized by observables such as the percolation probability \(P_L(\rho)\), the order parameter \(P_\infty=\ell/L^d\), the susceptibility-like quantity \(\chi_L=L^d\,{\rm var}(P_\infty)\), the loop fractal dimension \(d_f\), and the Fisher-type exponent \(\tau\). Numerical evidence on \(d=2,\dots,7\) is reported as consistent with an upper critical dimension \(d_u=6\) [1003.1591]. On random regular graphs, a mean-field formulation gives \(\nu^*=3\), \(\beta=2\), \(d_f=2\), and \(\gamma\approx -1\), again supporting \(d_u=6\) [1107.1742].

In two dimensions, a further NWP result is that the critical spanning-path geometry is incompatible with Schramm–Loewner evolution. At the critical disorder
\[
\rho_c = 0.3413(9),
\]
the path-length scaling
\[
\langle \ell \rangle \sim L^{d_f}
\]
gives \(d_f\approx 1.282\), while the two extracted \(\kappa\) values,
\[
\kappa_{d_f}=2.26(2), \qquad \kappa_{\mathrm{SLPF}}=3.34(8),
\]
do not agree [1205.1412]. A directed variant on the two-dimensional periodic square lattice shows a continuous phase transition at
\[
\rho_c=0.3789(3),
\]
with anisotropic exponents
\[
\nu_\parallel=1.18(10),\qquad \nu_\perp=0.75(6),\qquad \beta=1.42(21),\qquad \gamma=0.00(5),\qquad \tau=0.780(2),
\]
and the paper emphasizes a strong change of universality class with respect to both standard directed percolation and undirected NWP [1701.04726].

This earlier NWP literature is not identical to NegPT in the continuous-variable quantum-network sense. It concerns negative-weight geometry rather than entanglement negativity. Its relevance is structural: it established a percolation vocabulary in which globally optimized negative objects, exact minimum-weight perfect matching constructions, nonlocal constraints, and finite-size scaling can define a critical theory distinct from ordinary occupation-based percolation [1003.1591] [1701.04726] [1205.1412] [1107.1742].

## 3. Gaussian-to-Gaussian deterministic transport and the link variable \(\chi\)

The operational substrate of NegPT is the **Gaussian-to-Gaussian deterministic entanglement transmission** scheme. Its first elementary rule is a continuous-variable entanglement-swapping operation on a series chain. For two links with squeezings \(r_1,r_2\), the output is again a two-mode squeezed vacuum state with
\[
\tanh r = \tanh r_1 \tanh r_2.
\]
For a chain of \(N\) links,
\[
\tanh r = \prod_{j=1}^N \tanh r_j,
\]
so in ratio-negativity variables
\[
\chi = \prod_{j=1}^N \chi_j.
\]
The paper denotes this series rule by
\[
\mathrm{seri}(\chi_1,\chi_2,\dots)=\prod_{j=1}^N \chi_j.
\]
This rule is commutative [2507.16417].

The second elementary rule is deterministic concentration of parallel Gaussian resources. For \(K\) parallel two-mode squeezed vacuum states ordered as
\[
r_1\ge r_2\ge \cdots \ge r_K>0,
\]
iterated concentration gives
\[
\sinh r = \sinh r_1 \prod_{k=2}^{K}\cosh r_k.
\]
In ratio-negativity variables, the exact formula is
\[
\frac{\chi^2}{1-\chi^2} = \frac{\max_k \chi_k^2}{\prod_{k=1}^K (1-\chi_k^2)},
\]
equivalently
\[
\mathrm{para}(\chi_1,\chi_2,\dots) = \frac{\max_k \chi_k}{\sqrt{\max_k \chi_k^2+\prod_{k=1}^K(1-\chi_k^2)}}.
\]
For \(K\) identical parallel links,
\[
\mathrm{X}_{\mathrm{SC}} = \frac{\chi}{\sqrt{\chi^2+(1-\chi^2)^K}}.
\]
The paper stresses that this rule is not commutative in the same sense as the series rule, because the largest squeezing must occupy the distinguished \(\sinh r_1\) position for optimal output entanglement [2507.16417].

The Gaussian formalism is also given explicitly. For a two-mode squeezed vacuum state with squeezing \(r_j\), the covariance matrix is
\[
\mathbf{\Gamma}_{j}= 
\begin{pmatrix} 
\cosh 2r_j\, \mathbf{I}_2 & \sinh 2r_j\, \mathbf{Z}\\
\sinh 2r_j\, \mathbf{Z} & \cosh 2r_j\, \mathbf{I}_2 
\end{pmatrix},
\]
with
\[
\mathbf{I}_2= 
\begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}, \qquad
\mathbf{Z}= 
\begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.
\]
For a general Gaussian measurement with seed covariance matrix \(\mathbf{\Sigma}\), if
\[
\mathbf{\Sigma}_{\rho}= 
\begin{pmatrix} 
\mathbf{A} & \mathbf{C}\\
\mathbf{C}^{\mathrm T} & \mathbf{B} 
\end{pmatrix},
\]
then the conditional covariance matrix on subsystem \(B\) is
\[
\mathbf{\Gamma}_{\alpha}^{B} = \mathbf{B}-\mathbf{C}^{\mathrm T}(\mathbf{A}+\mathbf{\Sigma})^{-1}\mathbf{C}.
\]
These formulas establish that the swapping step is deterministic and Gaussian-output, whereas the concentration step is deterministic but relies on non-Gaussian LOCC [2507.16417].

The boundedness
\[
0\le \chi\le 1
\]
is not merely a normalization choice. It underlies one of the theory’s key distinctions from concurrence-based percolation: there is no nontrivial saturation point, and in the Bethe-lattice solution
\[
1-\mathrm{X}_{\mathrm{SC}} \sim (1-\chi)^k \qquad (\chi\to 1^-),
\]
so \(\mathrm{X}_{\mathrm{SC}}=1\) iff \(\chi=1\) [2507.16417].

## 4. Bethe-lattice formulation and mixed-order criticality

On the Bethe lattice of degree \(k\), NegPT admits exact recursive equations. If \(\chi_l^{(1)}\) is the sponge-crossing ratio negativity for one branch at depth \(l\), and \(\chi_l^{(k-1)}\) is the parallel combination of \(k-1\) such branches, then
\[
\chi_l^{(1)} = \chi\, \chi_{l-1}^{(k-1)},
\]
and
\[
\frac{\left(\chi_l^{(k-1)}\right)^2}{1-\left(\chi_l^{(k-1)}\right)^2} = \frac{\left(\chi_l^{(1)}\right)^2}{\left[1-\left(\chi_l^{(1)}\right)^2\right]^{k-1}}.
\]
In the thermodynamic limit these become
\[
\mathrm{X}_{\mathrm{SC}}^{(1)} = \chi\, \mathrm{X}_{\mathrm{SC}}^{(k-1)},
\]
\[
\frac{\left(\mathrm{X}_{\mathrm{SC}}^{(k-1)}\right)^2}{1-\left(\mathrm{X}_{\mathrm{SC}}^{(k-1)}\right)^2} = \frac{\left(\mathrm{X}_{\mathrm{SC}}^{(1)}\right)^2}{\left[1-\left(\mathrm{X}_{\mathrm{SC}}^{(1)}\right)^2\right]^{k-1}},
\]
and the full order parameter obeys
\[
\frac{\left(\mathrm{X}_{\mathrm{SC}}\right)^2}{1-\left(\mathrm{X}_{\mathrm{SC}}\right)^2} = \frac{\left(\mathrm{X}_{\mathrm{SC}}^{(1)}\right)^2}{\left[1-\left(\mathrm{X}_{\mathrm{SC}}^{(1)}\right)^2\right]^k}.
\]
Combining the branch equations yields
\[
\chi^2 = \left(\mathrm{X}_{\mathrm{SC}}^{(1)}\right)^2 + \left[1-\left(\mathrm{X}_{\mathrm{SC}}^{(1)}\right)^2\right]^{k-1}, \qquad \mathrm{X}_{\mathrm{SC}}^{(1)}>0.
\]
This is the key exact self-consistency equation of NegPT on the tree [2507.16417].

The central result is that the transition is **mixed order**. As \(\chi\) increases, \(\mathrm{X}_{\mathrm{SC}}\) remains zero until a threshold \(\chi_{\mathrm{th}}\), where it jumps discontinuously to a finite \(\mathrm{X}_{\mathrm{SC}}^{+}\), while a characteristic length diverges continuously. The threshold is obtained by minimizing \(\chi^2\) with respect to \((\mathrm{X}_{\mathrm{SC}}^{(1)})^2\), yielding
\[
T(k)=(k-1)^{-1/(k-2)}, \qquad \mathrm{X}_{\mathrm{SC}}^{(1),+}=\sqrt{1-T(k)},
\]
and the macroscopic jump is
\[
\mathrm{X}_{\mathrm{SC}}^{+} = \sqrt{\frac{1-T(k)}{T(k)^k-T(k)+1}}.
\]
For \(k=3\),
\[
\chi_{\mathrm{th}}=\frac{\sqrt 3}{2}\approx 0.866025, \qquad \mathrm{X}_{\mathrm{SC}}^{+}\approx 0.894.
\]
These values are reported from the exact Bethe-lattice solution and finite-depth convergence studies [2507.16417].

The continuous part of the transition is encoded in the characteristic depth \(l^*\), defined operationally as the depth where \(\mathrm{X}_{\mathrm{SC}}=0.5\). Below threshold, \(\mathrm{X}_{\mathrm{SC}}(l)\) exhibits a plateau near \(\mathrm{X}_{\mathrm{SC}}^{+}\) and then drops to zero once \(l\) exceeds \(l^*\). The scaling law is
\[
l^* \sim |\chi-\chi_{\mathrm{th}}|^{-z\nu},
\]
with
\[
z\nu \approx \frac12.
\]
The order-parameter singularity is
\[
|\mathrm{X}_{\mathrm{SC}}-\mathrm{X}_{\mathrm{SC}}^{+}| \sim |\chi-\chi_{\mathrm{th}}|^{1/2},
\]
so
\[
\beta=\frac12.
\]
The coexistence of a discontinuous jump with a diverging characteristic length is the precise reason the transition is classified as mixed order [2507.16417].

This places NegPT in a universality class distinct from both classical percolation and discrete-variable concurrence percolation. The paper emphasizes three distinctions: a jump in the order parameter, the exponent \(z\nu\approx 1/2\) rather than \(z\nu\approx 1\), and the absence of a nontrivial saturation point [2507.16417].

## 5. Relation to other negativity-based frameworks

Several other literatures are described as directly useful for a broader NegPT framework because they show how negativity can behave as a geometric, spectral, or resource-like observable. In \(1+1\)-dimensional conformal field theory, logarithmic entanglement negativity for adjacent intervals obeys
\[
\mathcal E = \frac{c}{4}\ln\!\left(\frac{\ell_1\ell_2}{\ell_1+\ell_2}\right) +\mathrm{const},
\]
while for disjoint intervals it depends only on the conformal cross ratio and is therefore scale invariant [1206.3092]. In nonequilibrium conformal field theory after a local quench, negativity exhibits a sharp light-cone onset, exact regime decompositions, logarithmic early-time growth, and distinct pre-steady and steady-state plateaus [1412.7568]. These results show that negativity can be organized by geometry, causal structure, and universal scaling variables rather than only by a single scalar summary.

A second cluster of results concerns the fine structure of negativity itself. The **negativity spectrum** of adjacent intervals in one-dimensional critical systems is shown to be universal and to depend only on the central charge, with sign dependence weak in the bulk and strong at the spectrum edges [1607.02992]. A perturbative calculus for negativity based on patterned matrix calculus yields explicit first- and second-order derivatives of
\[
\mathcal N(\rho)=\frac12\left(\|\rho^{T_B}\|_1-1\right),
\]
together with a convenient vectorized representation of partial transpose,
\[
\operatorname{vec}\rho^{T_B}=K_B\, \operatorname{vec}\rho,
\]
which is directly suited to studying growth and decay of entanglement under arbitrary perturbations [1809.07772]. This suggests a route from scalar negativity observables to local response coefficients on graphs or lattices.

A third cluster concerns negativity as a quantified nonclassical resource. In quasiprobabilistic hidden-variable models for CHSH,
\[
S_{\max}=2+\mathcal N, \qquad \mathcal N_{\min}(S)=S-2,
\]
so Bell-violation overhead is exactly priced by a negativity witness [2105.01685]. In an exactly solvable measurement-only circuit that maps to two-dimensional critical percolation, mutual negativity is associated with a double-arc event and has scaling dimension
\[
h_{\mathrm{MN}}=2,
\]
whereas in several hybrid circuits the reported value is
\[
h_{\mathrm{MN}}\approx 3,
\]
even while other critical exponents vary [2012.00031]. These results do not define NegPT in the continuous-variable network sense, but they reinforce a common theme: negativity can organize threshold behavior, connectivity observables, and universality classes that differ from those of more conventional correlation measures.

## 6. Limits, instability, and open directions

The present formulation of NegPT has clear scope conditions. Exact transport rules are available on series-parallel architectures and on the Bethe lattice, while non-series-parallel motifs such as the Wheatstone and Kelvin bridges are treated by approximate star-mesh transforms rather than exact continuous-variable transport laws [2507.16417]. The concentration step is deterministic in the theoretical construction, but the paper notes that it relies on non-Gaussian LOCC and that, unlike in discrete-variable systems, the continuous-variable parallel rule is not known to be optimal [2507.16417].

Noise and loss are not part of the core percolation equations. Instead, they enter later through a phenomenological feedback model in which link entanglement decays as
\[
\chi(t)\sim e^{-t/\tau},
\]
or
\[
\frac{d\chi(t)}{dt} = -\tau^{-1}\chi(t)+u(t-T_0),
\]
with a PID-type controller
\[
u(t)=K_P\epsilon(t)+K_I\int_0^t \epsilon(t')\,dt'+K_D\frac{d\epsilon(t)}{dt},
\qquad
\epsilon(t)=\mathrm{X}^{\mathrm{target}}_{\mathrm{SC}}-\mathrm{X}_{\mathrm{SC}}(t).
\]
Under the same feedback settings, the paper reports smooth stabilization for discrete-variable concurrence percolation and persistent on/off oscillatory instability for continuous-variable NegPT near threshold. The reason given is that a small downward fluctuation of \(\chi(t)\) below \(\chi_{\mathrm{th}}\) can drive
\[
\mathrm{X}_{\mathrm{SC}}\to 0
\]
abruptly, so delayed control overreacts or reacts too late [2507.16417].

The paper also introduces a generalized NegPT with transport coefficients \(\eta_s\) and \(\eta_p\),
\[
\tanh r = \eta_s \tanh r_1 \tanh r_2 \cdots,
\qquad
\sinh r = \eta_p \sinh r_1 \cosh r_2 \cosh r_3\cdots,
\]
leading on the Bethe lattice to
\[
\mathrm{X}_{\mathrm{SC}}^{(1)}=\eta_s\chi\, \mathrm{X}_{\mathrm{SC}}^{(k-1)},
\qquad
\frac{\left(\mathrm{X}_{\mathrm{SC}}^{(k-1)}\right)^2}{1-\left(\mathrm{X}_{\mathrm{SC}}^{(k-1)}\right)^2}
=
\eta_p^2
\frac{\left(\mathrm{X}_{\mathrm{SC}}^{(1)}\right)^2}{\left[1-\left(\mathrm{X}_{\mathrm{SC}}^{(1)}\right)^2\right]^{k-1}}.
\]
The reported criterion is that if \(\eta_p<\sqrt{k-1}\), second-order transition is forbidden and mixed-order can occur, whereas if \(\eta_p\ge \sqrt{k-1}\), mixed-order is forbidden and the transition is second-order [2507.16417]. This identifies the nonlinear parallel-composition rule, rather than Gaussianity alone, as a decisive ingredient of the mixed-order fixed point.

Several open directions remain explicit. Universality beyond the Bethe lattice is not settled; the strongest analytical results are tree-based. The full general definition of ratio negativity is imported rather than redeveloped in the NegPT paper itself. The feedback treatment is illustrative rather than a full nonlinear stability theory. A plausible implication is that future NegPT work will need to connect exact transport laws, nonsmooth threshold behavior, and experimentally realistic noise models on non-recursive graph topologies [2507.16417].

Source: https://www.emergentmind.com/topics/negativity-percolation-theory-negpt