Quantum-Connected Fraction (QCF)

• Quantum-Connected Fraction (QCF) is a thresholded metric that quantifies the fraction of node pairs in a quantum network with effective entanglement quality above a specified task threshold.
• It distinguishes functional quantum connectivity from mere physical links by assessing whether end-to-end entanglement meets the quality requirements for quantum information processing tasks.
• QCF is influenced by the entanglement distribution protocol, link quality statistics, and network topology, leading to distinct step-like transitions as network parameters cross the threshold.

Quantum-Connected Fraction (QCF) is a thresholded global metric for quantum networks. It is the normalized number of node pairs whose effective end-to-end entanglement quality exceeds the minimum level required for a specified quantum information processing task. Introduced as a quantity derived from the Quantum Connectivity Measure (QCM), QCF characterizes functional quantum connectivity rather than merely physical or graph-theoretic connectivity. In that sense, it is designed to answer a task-dependent question: how much of a network is actually usable for entanglement-enabled operations, given a concrete entanglement distribution protocol and a required quality threshold (Mondal et al., 31 Mar 2026).

1. Formal definition in quantum networks

The network-theoretic definition of QCF starts from the Quantum Connectivity Measure for a node set $\mathcal N \subseteq V$,

$$\mathcal{Q}_{\mathcal N}(\{\mathcal S_{ij}\})= \frac{2}{|\mathcal N|(|\mathcal N|-1)} \sum_{i,j\in \mathcal N} \mathcal S_{ij}\,\Theta(\mathcal S_{ij}-\epsilon),$$

where $\mathcal S_{ij}$ is the effective connection strength between nodes $i$ and $j$, $\Theta$ is the Heaviside step function, and $\epsilon\in[0,1]$ is the minimum connection quality required by the intended task. QCF is then obtained by differentiating QCM with respect to the connection strengths and taking the $1$-norm of that gradient. Operationally,

$$\mathcal F_{\mathcal N}(\{\mathcal S_{ij}\}) :=\sum_{i,j\in\mathcal N}\frac{\partial \mathcal Q_{\mathcal N}}{\partial \mathcal S_{ij}} = \frac{2}{|\mathcal N|(|\mathcal N|-1)} \sum_{i,j\in\mathcal N}\Theta(\mathcal S_{ij}-\epsilon).$$

This makes QCF a threshold-only quantity: it records whether each pair is usable, but not by how much it exceeds the threshold (Mondal et al., 31 Mar 2026).

A central structural feature is that $\mathcal S_{ij}$ is not defined directly from graph adjacency. Rather, it is the effective strength produced by the optimal entanglement distribution protocol or path $$\mathcal{Q}_{\mathcal N}(\{\mathcal S_{ij}\})= \frac{2}{|\mathcal N|(|\mathcal N|-1)} \sum_{i,j\in \mathcal N} \mathcal S_{ij}\,\Theta(\mathcal S_{ij}-\epsilon),$$0,

$$\mathcal{Q}_{\mathcal N}(\{\mathcal S_{ij}\})= \frac{2}{|\mathcal N|(|\mathcal N|-1)} \sum_{i,j\in \mathcal N} \mathcal S_{ij}\,\Theta(\mathcal S_{ij}-\epsilon),$$1

For the paper’s main example, entanglement swapping with pure links whose quality is concurrence, the effective strength is multiplicative,

$$\mathcal{Q}_{\mathcal N}(\{\mathcal S_{ij}\})= \frac{2}{|\mathcal N|(|\mathcal N|-1)} \sum_{i,j\in \mathcal N} \mathcal S_{ij}\,\Theta(\mathcal S_{ij}-\epsilon),$$2

Accordingly, QCF is protocol-sensitive from the outset: it is a property of a functional entanglement layer built over the physical network, not a purely combinatorial invariant.

2. Functional meaning and relation to other connectivity notions

QCF measures the prevalence of usable quantum links at the scale of the entire network. If $$\mathcal{Q}_{\mathcal N}(\{\mathcal S_{ij}\})= \frac{2}{|\mathcal N|(|\mathcal N|-1)} \sum_{i,j\in \mathcal N} \mathcal S_{ij}\,\Theta(\mathcal S_{ij}-\epsilon),$$3, no pair in $$\mathcal{Q}_{\mathcal N}(\{\mathcal S_{ij}\})= \frac{2}{|\mathcal N|(|\mathcal N|-1)} \sum_{i,j\in \mathcal N} \mathcal S_{ij}\,\Theta(\mathcal S_{ij}-\epsilon),$$4 can support the target task. If $$\mathcal{Q}_{\mathcal N}(\{\mathcal S_{ij}\})= \frac{2}{|\mathcal N|(|\mathcal N|-1)} \sum_{i,j\in \mathcal N} \mathcal S_{ij}\,\Theta(\mathcal S_{ij}-\epsilon),$$5, every pair in $$\mathcal{Q}_{\mathcal N}(\{\mathcal S_{ij}\})= \frac{2}{|\mathcal N|(|\mathcal N|-1)} \sum_{i,j\in \mathcal N} \mathcal S_{ij}\,\Theta(\mathcal S_{ij}-\epsilon),$$6 meets the required quality. The quantity therefore summarizes a binary notion of task feasibility across all node pairs, whereas QCM summarizes average successful connection strength (Mondal et al., 31 Mar 2026).

This distinction is the main reason QCF is not reducible to a classical connectivity statistic. Classical metrics ask whether nodes are linked in the graph. QCF asks whether they are linked well enough for a quantum task. A pair may be functionally quantum-connected even without a direct edge, provided that some path and entanglement-distribution protocol produce $$\mathcal{Q}_{\mathcal N}(\{\mathcal S_{ij}\})= \frac{2}{|\mathcal N|(|\mathcal N|-1)} \sum_{i,j\in \mathcal N} \mathcal S_{ij}\,\Theta(\mathcal S_{ij}-\epsilon),$$7. Conversely, a graph can be topologically connected yet functionally disconnected if end-to-end entanglement never crosses the task threshold.

The same framework also introduces a local analogue, the Quantum Clustering Coefficient (QCC), defined for a node $$\mathcal{Q}_{\mathcal N}(\{\mathcal S_{ij}\})= \frac{2}{|\mathcal N|(|\mathcal N|-1)} \sum_{i,j\in \mathcal N} \mathcal S_{ij}\,\Theta(\mathcal S_{ij}-\epsilon),$$8 by

$$\mathcal{Q}_{\mathcal N}(\{\mathcal S_{ij}\})= \frac{2}{|\mathcal N|(|\mathcal N|-1)} \sum_{i,j\in \mathcal N} \mathcal S_{ij}\,\Theta(\mathcal S_{ij}-\epsilon),$$9

with $\mathcal S_{ij}$0 the neighbor set of $\mathcal S_{ij}$1. QCF is thus global and pair-fraction-based, whereas QCC is local and neighbor-based. The distinction matters because quantum routing and entanglement swapping can create functional interconnectivity among a node’s neighbors even when the classical clustering coefficient is zero.

The paper further notes that $\mathcal S_{ij}$2 acts as a quantum analogue of the giant component fraction, but not in the classical sense. The classical giant component counts only the largest physically connected component. QCF counts all functionally connected node pairs, including those inside smaller components if the entanglement quality is sufficient. The analogue is therefore operational rather than structural.

3. Threshold structure, discontinuities, and controlling parameters

Because QCF is built from Heaviside functions, it changes discontinuously as network parameters cross the task threshold: $\mathcal S_{ij}$3 When a change in edge concurrence or path quality moves a given pair from $\mathcal S_{ij}$4 to $\mathcal S_{ij}$5, that pair begins contributing to QCF abruptly. This produces step-like transitions in QCF. By contrast, QCM varies smoothly because it retains the magnitude of $\mathcal S_{ij}$6 for successful pairs (Mondal et al., 31 Mar 2026).

Three dependencies are structurally decisive. First, the entanglement distribution protocol determines the map $\mathcal S_{ij}$7, and therefore which pairs qualify as functionally connected. Second, the edge concurrence distribution controls how likely pathwise end-to-end quality is to exceed the threshold. Higher average concurrence generally increases QCF, but the transition depends on the variance of the edge-quality distribution. Third, the network topology controls the available path set and the path-length distribution. Since end-to-end concurrence under swapping takes the product form

$\mathcal S_{ij}$8

longer optimal paths generally require higher edge concurrence to maintain the same QCF.

A direct implication is that topological density does not by itself imply large QCF. A dense network with weak links can have low or vanishing QCF, while a sparser network can exhibit higher functional connectivity if its protocol, edge statistics, and path structure favor threshold crossing. The metric is therefore best understood as a task-conditioned network response function.

4. Ensemble averages and representative network families

For a statistically defined family of quantum networks, the mean QCF is defined by

$\mathcal S_{ij}$9

and, under i.i.d. edge parameters and a path-length distribution $i$0,

$i$1

where the region $i$2 is constrained by the task condition

$i$3

For concurrence under swapping this becomes

$i$4

The mean QCF is therefore the average fraction of node pairs whose end-to-end concurrence exceeds the task threshold (Mondal et al., 31 Mar 2026).

The complete graph provides the simplest special case because the optimal path length is $i$5. For homogeneous edge concurrence, $i$6,

$i$7

This is the sharpest illustration of the distinction between topological and functional connectivity: a fully connected graph can still be functionally disconnected if the direct links are below threshold.

For an inhomogeneous uniform concurrence distribution,

$i$8

the paper gives

$i$9

Here the discontinuous transition of the homogeneous case broadens into a linear transition window controlled by the concurrence variance.

For random networks, the shortest graph path is used as the optimal path approximation when concurrence variance is small. In the homogeneous case, QCF then rises in discrete steps as progressively longer path lengths cross the task threshold; in the inhomogeneous case, those steps smooth out. These examples reinforce that QCF is jointly determined by protocol, path statistics, and link-quality distribution.

5. State-space analogue on isospectral density-matrix orbits

A distinct but related usage appears in the study of correlations on isospectral sets of density matrices. There the basic object is the unitary orbit

$j$0

equivalently $j$1 for $j$2 and $j$3. Mixed non-correlated states are defined as the convex hull $j$4 of a chosen class $j$5 of pure non-correlated states. The framework applies to mixed-state entanglement for distinguishable particles, particle entanglement for bosons, particle entanglement for fermions, and fermionic Gaussian correlations (Oszmaniec et al., 2013).

In that setting, the paper does not foreground the acronym QCF, but it identifies a directly analogous quantity: the measure of correlated states inside a fixed isospectral class. One can write

$j$6

with the complementary fraction given by the measure of $j$7. The main asymptotic result is expressed through the purity

$j$8

and the critical purity

$j$9

If

$\Theta$0

then

$\Theta$1

so the non-correlated fraction is exponentially suppressed in the Hilbert-space dimension $\Theta$2.

This produces a state-space typicality statement rather than a network-connectivity statement. Above the critical purity, a randomly chosen isospectral state is asymptotically correlated. A plausible implication is that QCF can denote, in a broader interpretive sense, the fraction of a constrained quantum state space that exhibits the relevant form of quantum correlation. Even so, this usage is conceptually separate from the thresholded node-pair metric defined for quantum networks.

6. Terminological ambiguity and distinct acronym usages

The label “QCF” is not uniform across the literature. In the quantum-network setting it denotes Quantum-Connected Fraction, but older and unrelated work uses the same acronym for quantum coin flipping. Other papers study fraction-like resource measures that are close in form but not in terminology. This ambiguity is substantive, because the objects being quantified are different: node-pair task feasibility, state overlap with resource states, contextual decomposition weights, or cheating bias in cryptographic protocols.

Usage	Meaning	Source
QCF	Quantum-Connected Fraction in quantum networks	(Mondal et al., 31 Mar 2026)
QCF	Quantum coin flipping, including semi-loss-tolerant and strong protocols	(Yang et al., 2012, He, 2016)
FEF	Fully entangled fraction, not quantum-connected fraction	(Zhu et al., 25 Feb 2026)
CF	Contextual fraction in contextuality and MBQC	(Okay et al., 2018)
Coherence fraction	Maximal overlap with maximally coherent states	(Karmakar et al., 2019)

In the cryptographic literature, QCF is a protocol class rather than a connectivity metric. One paper presents a semi-loss-tolerant strong quantum coin-flipping protocol using QND measurement and reports a best bias of $\Theta$3 (Yang et al., 2012). Another claims unconditionally secure strong QCF with arbitrarily small bias (He, 2016). These uses are acronymically identical but conceptually unrelated to network QCF.

Likewise, the fully entangled fraction

$\Theta$4

is a state-resource quantity for $\Theta$5 systems rather than a connectivity fraction (Zhu et al., 25 Feb 2026). The contextual fraction

$\Theta$6

measures the contextual component in an empirical model (Okay et al., 2018). The coherence fraction

$\Theta$7

measures proximity to maximally coherent states (Karmakar et al., 2019). These are mathematically analogous in being fraction-like quantifiers, but they are not definitions of Quantum-Connected Fraction.

Taken in its strictest contemporary sense, QCF denotes the fraction of node pairs in a quantum network that are functionally quantum-connected for a specified task. More broadly, the literature also supports QCF-like interpretations as fractions of correlated states within constrained state spaces. The term therefore has a precise technical meaning in quantum-network theory, but it sits within a wider and sometimes conflicting vocabulary of fraction-based quantum resource measures.