Global Concurrence Measure in Quantum and Temporal Networks

• Global concurrence measure is a framework that aggregates pairwise or bipartition concurrences into a unified system-level scalar for multipartite entanglement analysis.
• It employs methods such as minima over bipartitions, averages over single-party reductions, geometric means, and area-based combinations to quantify entanglement in various quantum states and temporal networks.
• Operational approaches leverage convex-roof extensions, computable lower bounds, and observable-based metrics to assess both quantum coherence and synchronized interaction dynamics.

Global concurrence measure denotes a family of non-equivalent constructions that aggregate pairwise or bipartition-level concurrence into a system-level scalar. In quantum information, the term refers to extensions of bipartite concurrence to multipartite pure states, mixed states, higher-dimensional systems, and parameterized families of entanglement monotones. These constructions include minima over bipartitions, averages over single-party reductions, geometric means over cuts, area-based combinations of concurrence triangles, and total-concurrence functionals built from spectral data of reduced states (Hong et al., 2012, Xuan et al., 30 Dec 2025). In a separate temporal-network literature, a similarly named global quantity is defined by averaging pairwise edge-event correlation over edges or hyperedges, thereby quantifying synchronized interaction activity rather than quantum entanglement (Kang et al., 23 May 2026).

1. Foundational notion of concurrence

The common quantum starting point is the bipartite pure-state concurrence

$$C(|\psi\rangle_{AB})=\sqrt{2\bigl(1-\operatorname{Tr}(\rho_A^2)\bigr)},$$

where $$\rho_A=\operatorname{Tr}_B(|\psi\rangle\langle\psi|)$$. For two qubits,

$$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle, \qquad C(|\psi\rangle)=2|ad-bc|.$$

For mixed states, concurrence is extended by the convex roof, namely the infimum of the average pure-state concurrence over all ensemble decompositions of the density operator (Guo et al., 2012, Yu et al., 2013).

Several formulations appearing in the literature are equivalent at the pure-state level. The concurrence vector construction defines components $$C_{\alpha\beta}(\psi)=\langle\psi|S_{\alpha\beta}|\psi^*\rangle$$ and takes their Euclidean norm; this norm is exactly the same quantity as I-concurrence, i.e. $\sqrt{2(1-\operatorname{Tr}\rho_A^2)}$ (Akhtarshenas, 2010). For infinite-dimensional bipartite systems, the same formula remains well defined, continuous in the trace norm, and monotone under LOCC, and the associated PHC measure coincides with concurrence (Guo et al., 2012).

The measure also admits operational reinterpretations. For bipartite pure states, concurrence can be written as the maximal difference between localizable coherence and local coherence of a reduced subsystem,

$C^2(|\psi\rangle_{AB})=D_L^2(\rho_A)-D^2(\rho_A),$

for qubit $A$, and more generally

$$C^2(|\psi\rangle_{AB})=D_{FL}^2(\rho_A)-D_F^2(\rho_A),$$

for higher-dimensional $A$ (Yu et al., 2013). This places concurrence within a broader class of resource-difference measures derived from reduced-state structure.

2. Multipartite extensions and genuinely global constructions

Once concurrence is lifted from a single bipartition to a multipartite system, there is no unique canonical globalization. The cited literature instead develops several distinct constructions, each emphasizing a different aspect of multipartite entanglement.

Measure	Defining aggregation	Typical scope
Average determinant / Meyer–Wallach form	$$\displaystyle E_{AD}=\frac{4}{n}\sum_{i=1}^n \det(\rho_i)=\frac{1}{n}\sum_i \tau_{i|\bar i}$$	Pure $$\rho_A=\operatorname{Tr}_B(|\psi\rangle\langle\psi|)$$0-qubit global entanglement (Li, 6 Oct 2025)
$$\rho_A=\operatorname{Tr}_B(|\psi\rangle\langle\psi|)$$1-ME concurrence	$$\rho_A=\operatorname{Tr}_B(|\psi\rangle\langle\psi|)$$2	$$\rho_A=\operatorname{Tr}_B(|\psi\rangle\langle\psi|)$$3-nonseparability in arbitrary dimensions (Hong et al., 2012)
GME concurrence	$$\rho_A=\operatorname{Tr}_B(|\psi\rangle\langle\psi|)$$4	Genuine multipartite entanglement (Li et al., 19 Apr 2025)
Concurrence-triangle measure	Normalized Heron area of triangles built from one-to-rest concurrences	GME via geometric aggregation (Jin et al., 2022)
Minimum pairwise concurrence	$$\rho_A=\operatorname{Tr}_B(|\psi\rangle\langle\psi|)$$5 with $$\rho_A=\operatorname{Tr}_B(|\psi\rangle\langle\psi|)$$6	Genuine tripartite entanglement (Dong et al., 2024)
$$\rho_A=\operatorname{Tr}_B(|\psi\rangle\langle\psi|)$$7	Geometric mean of bipartite $$\rho_A=\operatorname{Tr}_B(|\psi\rangle\langle\psi|)$$8-concurrences over all nontrivial cuts	Genuine multipartite entanglement (Wang et al., 9 May 2025)

The average-determinant construction

$$\rho_A=\operatorname{Tr}_B(|\psi\rangle\langle\psi|)$$9

is algebraically identical to the Meyer–Wallach global entanglement and equals the average $$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle, \qquad C(|\psi\rangle)=2|ad-bc|.$$0-tangle $$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle, \qquad C(|\psi\rangle)=2|ad-bc|.$$1. For two qubits it reduces to the square of concurrence, while for $$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle, \qquad C(|\psi\rangle)=2|ad-bc|.$$2-qubit GHZ states it equals $$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle, \qquad C(|\psi\rangle)=2|ad-bc|.$$3, and for $$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle, \qquad C(|\psi\rangle)=2|ad-bc|.$$4 states it equals $$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle, \qquad C(|\psi\rangle)=2|ad-bc|.$$5 (Li, 6 Oct 2025).

The $$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle, \qquad C(|\psi\rangle)=2|ad-bc|.$$6-ME concurrence generalizes concurrence to $$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle, \qquad C(|\psi\rangle)=2|ad-bc|.$$7-nonseparability. Its mixed-state version is again a convex roof, and the special case $$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle, \qquad C(|\psi\rangle)=2|ad-bc|.$$8 is a genuine multipartite entanglement measure. The global case $$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle, \qquad C(|\psi\rangle)=2|ad-bc|.$$9 is especially notable: for pure $$C_{\alpha\beta}(\psi)=\langle\psi|S_{\alpha\beta}|\psi^*\rangle$$0-qubit states it becomes the root-mean-square of the one-versus-rest negativities, and for mixed $$C_{\alpha\beta}(\psi)=\langle\psi|S_{\alpha\beta}|\psi^*\rangle$$1-qubit states it admits the corresponding lower bound (Zhang et al., 2020). Closely related is the GME concurrence

$$C_{\alpha\beta}(\psi)=\langle\psi|S_{\alpha\beta}|\psi^*\rangle$$2

which selects the weakest bipartition and therefore vanishes as soon as one cut becomes separable (Li et al., 19 Apr 2025).

Tripartite globalizations often incorporate genuine three-body structure more explicitly. The minimum pairwise concurrence defines

$$C_{\alpha\beta}(\psi)=\langle\psi|S_{\alpha\beta}|\psi^*\rangle$$3

so that pairwise concurrence and three-tangle contribute on the same footing. In this normalization, $$C_{\alpha\beta}(\psi)=\langle\psi|S_{\alpha\beta}|\psi^*\rangle$$4 and $$C_{\alpha\beta}(\psi)=\langle\psi|S_{\alpha\beta}|\psi^*\rangle$$5 (Dong et al., 2024).

A different genuinely multipartite construction is the geometric-mean measure $$C_{\alpha\beta}(\psi)=\langle\psi|S_{\alpha\beta}|\psi^*\rangle$$6, defined for $$C_{\alpha\beta}(\psi)=\langle\psi|S_{\alpha\beta}|\psi^*\rangle$$7 by taking the geometric mean of

$$C_{\alpha\beta}(\psi)=\langle\psi|S_{\alpha\beta}|\psi^*\rangle$$8

over all nontrivial bipartitions $$C_{\alpha\beta}(\psi)=\langle\psi|S_{\alpha\beta}|\psi^*\rangle$$9. For $\sqrt{2(1-\operatorname{Tr}\rho_A^2)}$0-qubit GHZ states, every factor equals $\sqrt{2(1-\operatorname{Tr}\rho_A^2)}$1, while for $\sqrt{2(1-\operatorname{Tr}\rho_A^2)}$2 states the factor for a cut of size $\sqrt{2(1-\operatorname{Tr}\rho_A^2)}$3 is $\sqrt{2(1-\operatorname{Tr}\rho_A^2)}$4; within the stated parameter range, the GHZ value is larger for finite $\sqrt{2(1-\operatorname{Tr}\rho_A^2)}$5 (Wang et al., 9 May 2025).

3. Parameterized and total-concurrence families

A major line of work replaces the single concurrence functional by parameterized families. The unified $\sqrt{2(1-\operatorname{Tr}\rho_A^2)}$6-concurrence is defined for bipartite pure states by

$\sqrt{2(1-\operatorname{Tr}\rho_A^2)}$7

with a convex-roof extension to mixed states. It contains the standard concurrence and $\sqrt{2(1-\operatorname{Tr}\rho_A^2)}$8-concurrence as special cases, supports analytical lower bounds from PPT and realignment criteria, obeys monogamy inequalities for suitable qubit-domain parameters, and satisfies entanglement polygon inequalities for multipartite qudit pure states (Bai et al., 30 Jan 2026).

The $\sqrt{2(1-\operatorname{Tr}\rho_A^2)}$9-concurrence defines

$C^2(|\psi\rangle_{AB})=D_L^2(\rho_A)-D^2(\rho_A),$0

again with convex-roof extension. For two qubits and $C^2(|\psi\rangle_{AB})=D_L^2(\rho_A)-D^2(\rho_A),$1, it is related analytically to the standard concurrence by $C^2(|\psi\rangle_{AB})=D_L^2(\rho_A)-D^2(\rho_A),$2, where $C^2(|\psi\rangle_{AB})=D_L^2(\rho_A)-D^2(\rho_A),$3 is an explicit increasing convex function. In multiqubit systems, $C^2(|\psi\rangle_{AB})=D_L^2(\rho_A)-D^2(\rho_A),$4 itself is not monogamous in that parameter range, but $C^2(|\psi\rangle_{AB})=D_L^2(\rho_A)-D^2(\rho_A),$5 is (Li et al., 2024).

The total-concurrence framework augments concurrence by adding the complementary dual contribution of $C^2(|\psi\rangle_{AB})=D_L^2(\rho_A)-D^2(\rho_A),$6. For $C^2(|\psi\rangle_{AB})=D_L^2(\rho_A)-D^2(\rho_A),$7,

$C^2(|\psi\rangle_{AB})=D_L^2(\rho_A)-D^2(\rho_A),$8

and the normalized $C^2(|\psi\rangle_{AB})=D_L^2(\rho_A)-D^2(\rho_A),$9-concurrence is obtained by dividing by the pure-state maximum. The same paper introduces a complementary family $A$0 for $A$1 (Xuan et al., 30 Dec 2025). In qubit systems, $A$2 equals the square of the usual concurrence, and $A$3 satisfies CKW-type monogamy for $A$4 (Xuan et al., 30 Dec 2025).

Some constructions globalize concurrence over independent degrees of freedom rather than over subsystems or cuts. For polarization–momentum hyperentanglement, the total concurrence is defined additively,

$A$5

with each degree-of-freedom concurrence extracted directly from odd-parity success probabilities in a two-step QND protocol (Sheng et al., 2014).

4. Geometric, algebraic, and operator-theoretic interpretations

Several papers recast concurrence into geometric or algebraic objects. In the stereographic-projection framework, a two-qubit state

$A$6

is mapped to a quaternionic spinor $A$7, and the quaternionic stereographic projection satisfies

$A$8

The standard concurrence is then

$A$9

The same program extends to $$C^2(|\psi\rangle_{AB})=D_{FL}^2(\rho_A)-D_F^2(\rho_A),$$0, three-qubit octonionic representations, and $$C^2(|\psi\rangle_{AB})=D_{FL}^2(\rho_A)-D_F^2(\rho_A),$$1 bipartitions, where norms of non-complex projection components assemble into generalized concurrence expressions (Najarbashi et al., 2015).

A different operator-algebraic reinterpretation arises from Tomita–Takesaki theory. For bipartite pure states in a standard-form von Neumann algebra setting, concurrence is written as

$$C^2(|\psi\rangle_{AB})=D_{FL}^2(\rho_A)-D_F^2(\rho_A),$$2

where $$C^2(|\psi\rangle_{AB})=D_{FL}^2(\rho_A)-D_F^2(\rho_A),$$3 is the modular conjugation associated with the local algebras of the two subsystems. In two-qubit language, this antiunitary is unitarily equivalent to the familiar spin-flip operation; the construction is worked out explicitly for a supersymmetric quantum-mechanical model and for two Unruh–DeWitt detectors coupled to a scalar field (Chatterjee, 2024).

The concurrence-triangle program supplies a geometric multipartite interpretation. For three parties, with side lengths $$C^2(|\psi\rangle_{AB})=D_{FL}^2(\rho_A)-D_F^2(\rho_A),$$4, $$C^2(|\psi\rangle_{AB})=D_{FL}^2(\rho_A)-D_F^2(\rho_A),$$5, and $$C^2(|\psi\rangle_{AB})=D_{FL}^2(\rho_A)-D_F^2(\rho_A),$$6, the normalized triangle-area measure is

$$C^2(|\psi\rangle_{AB})=D_{FL}^2(\rho_A)-D_F^2(\rho_A),$$7

General $$C^2(|\psi\rangle_{AB})=D_{FL}^2(\rho_A)-D_F^2(\rho_A),$$8-partite versions are obtained by geometric means of normalized areas over families of concurrence triangles indexed by different partitions (Jin et al., 2022).

5. Computability, lower bounds, and operational access

Because most global concurrence measures are convex-roof constructions, computability is a central issue. One strategy is to derive observables on two copies of the state. For pure multipartite states, the GME concurrence across a fixed cut can be written as

$$C^2(|\psi\rangle_{AB})=D_{FL}^2(\rho_A)-D_F^2(\rho_A),$$9

where $A$0 depends only on the bipartition. This yields direct pure-state observables and analytic lower bounds for mixed states that bypass full tomography (Chen et al., 2012).

The $A$1-ME concurrence admits two computable lower bounds based on products of local populations and coherences in suitably chosen product bases. These bounds require no convex-roof optimization and no eigenvalue computation, and the paper states that they are experimentally implementable without quantum state tomography (Hong et al., 2012).

PPT- and realignment-based lower bounds recur across parameterized families. The unified $A$2-concurrence satisfies analytical lower bounds involving $A$3 and $A$4, with explicit formulas for isotropic and Werner states (Bai et al., 30 Jan 2026). The total-concurrence family $A$5 likewise admits lower bounds in terms of partial-transpose and realignment norms, together with exact analytic expressions for isotropic and Werner states in several parameter regimes (Xuan et al., 30 Dec 2025).

Operational interpretations often accompany these computable formulas. In the coherence-based approach, concurrence is the amount of extra local coherence that can be induced by LOCC on the other subsystem (Yu et al., 2013). In the infinite-dimensional PHC construction, concurrence is the Hilbert–Schmidt distance between a pure state and its partial Hermitian conjugate, and the PHC measure is exactly equal to concurrence (Guo et al., 2012).

6. Representative applications and distinct temporal-network usage

The quantum constructions have been applied in several nonstandard physical settings. In three-flavor neutrino oscillation, entanglement can be expressed entirely through appearance and disappearance probabilities. Within the single-excitation three-mode description, pairwise and bipartition concurrences become explicit functions of $A$6, the three-tangle vanishes identically, and the state is therefore W-type rather than GHZ-type. The paper defines a symmetric invariant

$A$7

and the concurrence fill $A$8, the normalized area of the concurrence triangle, as global concurrence-type diagnostics of distributed flavor-mode entanglement (Banerjee et al., 12 Dec 2025).

A separate, non-quantum usage appears in temporal network and hypergraph analysis. There, event-count time series $A$9 are assigned to edges or hyperedges, and pairwise concurrence is quantified by edge-event correlation,

$$\displaystyle E_{AD}=\frac{4}{n}\sum_{i=1}^n \det(\rho_i)=\frac{1}{n}\sum_i \tau_{i|\bar i}$$0

a cosine similarity of windowed event counts. A principled global concurrence measure is then the mean pairwise EEC,

$$\displaystyle E_{AD}=\frac{4}{n}\sum_{i=1}^n \det(\rho_i)=\frac{1}{n}\sum_i \tau_{i|\bar i}$$1

with restricted variants such as $$\displaystyle E_{AD}=\frac{4}{n}\sum_{i=1}^n \det(\rho_i)=\frac{1}{n}\sum_i \tau_{i|\bar i}$$2, $$\displaystyle E_{AD}=\frac{4}{n}\sum_{i=1}^n \det(\rho_i)=\frac{1}{n}\sum_i \tau_{i|\bar i}$$3, $$\displaystyle E_{AD}=\frac{4}{n}\sum_{i=1}^n \det(\rho_i)=\frac{1}{n}\sum_i \tau_{i|\bar i}$$4, and $$\displaystyle E_{AD}=\frac{4}{n}\sum_{i=1}^n \det(\rho_i)=\frac{1}{n}\sum_i \tau_{i|\bar i}$$5 defined by pair type. In the reported datasets, intersecting pairs, especially those in closed local structures such as triangles, show systematically elevated concurrence (Kang et al., 23 May 2026).

Taken together, these literatures show that global concurrence measure is best understood not as a single invariant but as a design pattern. In quantum information, it is any concurrence-based scalar that preserves the entanglement-theoretic role of concurrence while extending it to larger state spaces, richer partitions, or alternative operational formalisms. In temporal-network analysis, it is an aggregate of pairwise temporal co-activation scores. The shared terminology reflects aggregation; the underlying mathematics and semantics remain domain specific (Xuan et al., 30 Dec 2025, Kang et al., 23 May 2026).