Quantum Entanglement Index Overview

• Quantum Entanglement Index refers to a family of measures that assign scalar or topological invariants to capture diverse entanglement structures in quantum systems.
• They span resource-theoretic bounds, entropy scaling coefficients at conformal critical points, experimentally accessible proxies, and geometric invariants in multipartite settings.
• These indices offer practical tools to quantify entanglement for improved resource estimation, channel capacity bounds, and phase classification across various quantum domains.

Searching arXiv for recent/relevant papers on "^^^^^^^^4Quantum Entanglement Index4^^^^^^^^" and related usages. arxiv_search(query="^^^^^^^^4all:\4 entanglement index\"^^^^^^^^4 OR ti:\4^^^^^^^^"entanglement index\"^^^^^^^^4 OR abs:\4^^^^^^^^"entanglement index\"", max_results=^^^^4all:\4Quantum Entanglement Index4^^^^) “Quantum entanglement index” is used in the literature for several non-equivalent constructions that assign a scalar, a finite iteration count, or a topological integer to entanglement structure. Depending on context, the term can denote an efficiently computable upper bound on one-way distillable entanglement and quantum capacity obtained by “squeezing out” useless entanglement, a universal logarithmic coefficient in entanglement or Shannon entropies at conformal quantum critical points, an operational laboratory estimator for Negativity or Entanglement of Formation, a multipartite sharing coordinate for PRESERVED_PLACEHOLDER_^^^^4Quantum Entanglement Index4^^^^-qubit pure states, an entanglement-breaking iteration count for quantum channels, or an index-theoretic invariant built from momentum-map geometry, entanglement spectra, Uhlmann curvature, or orbifold partition functions (&&&^^^^4Quantum Entanglement Index4^^^^&&&, &&&^^^^4all:\4^^^^&&&, &&&^^^^4 OR ti:\4^^^^&&&, &&&^^^^4 OR abs:\4^^^^&&&, Lami et al., 2014, Ikeda, 6 Nov 2025). The literature therefore suggests that the phrase names a family of indices rather than a single standardized measure.

^^^^4all:\4^^^^. Terminological scope and recurrent roles

Across the cited literature, an “entanglement index” typically plays one of three roles: a quantitative upper bound on an operational resource, a universal coefficient or invariant that distinguishes phases or SLOCC families, or an experimentally accessible proxy built from a restricted measurement set. The same label is therefore attached to objects with different codomains, different invariance properties, and different task dependence.

Domain	Representative index	Primary object
Resource theory	PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^, PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^, PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^	States and channels
CQCP/CFT	$c/3$, $\alpha$, $\gamma(n)$	Entropy scaling
Optical experiments	$N$, $EF$ from PCC, MP, MI	Pure two-qutrit states
Multipartite sharing	$Y_j$	PRESERVED_PLACEHOLDER_^^^^4all:\4Quantum Entanglement Index4^^^^-qubit pure states
Channel dynamics	PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^, PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^, PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^, PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^	Quantum channels
Geometric/topological	PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^5, PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^6, PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^7	SLOCC classes, spectra, parameter families

A recurrent misconception is that any quantity called an “entanglement index” is automatically a universal entanglement monotone. The surveyed works do not support that identification. Some indices are task-specific upper bounds, some are classification invariants, some detect only particular sectors of correlation, and some are explicitly reported to fail as strict mixed-state monotones or to distinguish measures that are not monotonic functions of one another in PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^8 (&&&^^^^4 OR ti:\4^^^^&&&, Zeheiry, 2022, Guo et al., 2022).

^^^^4 OR ti:\4^^^^. Resource-theoretic and capacity-oriented indices

In the resource-theoretic formulation of “Estimate distillable entanglement and quantum capacity by squeezing useless entanglement,” the central object is the reverse divergence of resources,

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^9

with PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4Quantum Entanglement Index4^^^^ chosen to be the max-relative entropy,

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4all:\4^^^^

For one-way distillation, the free set is PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4 OR ti:\4^^^^, the set of anti-degradable (two-extendible) states, and

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4 OR abs:\4^^^^

The corresponding SDP computes

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\44^^^^

as the maximal weight of an ADG piece in a decomposition of PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^5. With the squeezed state

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^6

the paper proves

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^7

and derives the fully SDP-computable bound

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^8

where PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^9 is a spectral decomposition (&&&^^^^4Quantum Entanglement Index4^^^^&&&).

The operative interpretation is that the decomposition

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4Quantum Entanglement Index4^^^^

“squeezes out” the free component, on the premise that PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4^^^^-states carry zero useful entanglement for the task. For channels, the analogous one-number index is

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR ti:\4^^^^

computed from a channel SDP. The practical recipe is explicit: solve the state- or channel-version SDP to find PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^, form the squeezed state PRESERVED_PLACEHOLDER_^^^^4 OR abs:\44^^^^ or squeezed Choi map PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^5, evaluate a single-letter coherent-information or entanglement-of-formation bound, and multiply by PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^6. The paper states that all steps are numerically cheap in CVX/YALMIP for dimensions up to PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^7, and that the result gives one one-number index upper-bounding PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^8 or PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^9 (&&&^^^^4Quantum Entanglement Index4^^^^&&&).

The same framework yields concrete benchmarks. For $c/3$^^^^4Quantum Entanglement Index4^^^^, the new bound $c/3$^^^^4all:\4^^^^ is reported to strictly improve over the coherent-information lower bound $c/3$^^^^4 OR ti:\4^^^^, the Rains bound $c/3$^^^^4 OR abs:\4^^^^, and the continuity bounds $c/3$4. For Pauli channels $c/3$5, the explicit formula

$c/3$6

leads to the no-cloning bound

$c/3$7

which the paper states exactly recovers Cerf ^^^^4 OR ti:\4Quantum Entanglement Index4Quantum Entanglement Index4Quantum Entanglement Index4^^^^ (&&&^^^^4Quantum Entanglement Index4^^^^&&&).

^^^^4 OR abs:\4^^^^. Entropic coefficients as indices at conformal quantum critical points

In the CQCP literature, entanglement indices are universal coefficients in entropy scaling rather than direct pairwise or operational entanglement monotones. For a bipartition $c/3$8 of a pure ground state $c/3$9, the entropies are

$\alpha$^^^^4Quantum Entanglement Index4^^^^

In one-dimensional CFTs with central charge $\alpha$^^^^4all:\4^^^^, the coefficient $\alpha$^^^^4 OR ti:\4^^^^ in

$\alpha$^^^^4 OR abs:\4^^^^

is the standard entanglement index. In two dimensions, the leading term obeys an area law $\alpha$4, but universal subleading terms can appear at CQCPs: $\alpha$5 The coefficient $\alpha$6 is then the relevant entanglement index (&&&^^^^4all:\4^^^^&&&).

The paper “Logarithmic terms in entanglement entropies of ^^^^4 OR ti:\4^^^^D quantum critical points and Shannon entropies of spin chains” derives these coefficients by mapping the entanglement spectrum of a ^^^^4 OR ti:\4^^^^D strip geometry to the Shannon entropy of a ^^^^4all:\4^^^^D quantum chain with open boundaries. The probabilities

$\alpha$7

yield

$\alpha$8

with replica partition function

$\alpha$9

The logarithmic term arises from the trace anomaly at the four corners where the cut meets the external boundaries (&&&^^^^4all:\4^^^^&&&).

The model-dependent coefficients are explicit. For the compact free boson ($\gamma(n)$^^^^4Quantum Entanglement Index4^^^^) corresponding to the square-lattice quantum dimer CQCP, with Dirichlet or Neumann external boundary conditions,

$\gamma(n)$^^^^4all:\4^^^^

so that

$\gamma(n)$^^^^4 OR ti:\4^^^^

and hence $\gamma(n)$^^^^4 OR abs:\4^^^^ for Neumann or $\gamma(n)$4 for Dirichlet. For the ^^^^4 OR ti:\4^^^^D Ising CQCP ($\gamma(n)$5), numerical TEBD and finite-size scaling give

$\gamma(n)$6

Because $\gamma(n)$7 has a cusp at $\gamma(n)$8, the replica derivative is not analytic and the von Neumann entropy has no simple logarithm; for Rényi entropies, however, one obtains $\gamma(n)$9 and $N$^^^^4Quantum Entanglement Index4^^^^ (&&&^^^^4all:\4^^^^&&&).

In this usage, the “index” classifies universality classes and boundary-condition RG flows. It is not an entanglement monotone on finite-dimensional states, but a universal field-theoretic diagnostic extracted from subleading scaling.

4. Operational indices from correlations and expectation values

In higher-dimensional optical systems, a “quantum entanglement index” can be an experimentally accessible function of a small number of directly measurable correlations. For pure two-qutrit states in Schmidt form,

$N$^^^^4all:\4^^^^

“Entanglement certification and quantification in spatial-bin photonic qutrits” gives

$N$^^^^4 OR ti:\4^^^^

and relates these to two-basis measurements. In the Fourier basis, the Pearson correlation coefficient satisfies $N$^^^^4 OR abs:\4^^^^; in a mixed Fourier basis, the mutual predictability satisfies

$N$4

in the computational basis, the classical mutual information reduces to

$N$5

Experimentally, the reported values are

$N$6

$N$7

$N$8

The same work emphasizes that a maximally entangled qutrit has $N$9 and $EF$^^^^4Quantum Entanglement Index4^^^^, so the experiment exhibits a $EF$^^^^4all:\4^^^^ shortfall in Negativity but a $EF$^^^^4 OR ti:\4^^^^ shortfall in EOF. Appendix C is cited as showing classes of two-qutrit states for which $EF$^^^^4 OR abs:\4^^^^ increases while $EF$4 decreases, or vice versa, so the two measures are not monotonic functions of one another in $EF$5 (&&&^^^^4 OR ti:\4^^^^&&&).

For two-qubit mixed states, the connected-correlation construction introduces

$EF$6

and defines

$EF$7

where $EF$8 are the eigenvalues of $EF$9. For pure states, the paper states

$Y_j$^^^^4Quantum Entanglement Index4^^^^

with $Y_j$^^^^4all:\4^^^^ the concurrence. The same paper also states that in the “Class D” single-measure sector, the state is separable with classical correlation and $Y_j$^^^^4 OR ti:\4^^^^. This makes the scope of the vanishing criterion precise: $Y_j$^^^^4 OR abs:\4^^^^ vanishes exactly on product states, not on all separable states, and its monotonicity statement is formulated within classes of fixed $Y_j$4 invariants (Guo et al., 2022).

A different expectation-value construction defines, for two qubits,

$Y_j$5

with $Y_j$6 and $Y_j$7, and then

$Y_j$8

For qudits of dimension $Y_j$9, with generalized-spin spectrum PRESERVED_PLACEHOLDER_^^^^4all:\4Quantum Entanglement Index4Quantum Entanglement Index4^^^^, largest eigenvalue PRESERVED_PLACEHOLDER_^^^^4all:\4Quantum Entanglement Index4all:\4^^^^, and average

PRESERVED_PLACEHOLDER_^^^^4all:\4Quantum Entanglement Index4 OR ti:\4^^^^

the generalization is

PRESERVED_PLACEHOLDER_^^^^4all:\4Quantum Entanglement Index4 OR abs:\4^^^^

The same source states that the measure is invariant under local unitaries and continuous on pure states, but that because it depends only on local Bloch vectors it is not sensitive to all entangled mixtures and does not behave as a strict entanglement monotone for mixed states. Its Werner-state example has PRESERVED_PLACEHOLDER_^^^^4all:\4Quantum Entanglement Index44^^^^ for all PRESERVED_PLACEHOLDER_^^^^4all:\4Quantum Entanglement Index4^^^^5, which is reported as an “extreme insensitivity to PRESERVED_PLACEHOLDER_^^^^4all:\4Quantum Entanglement Index4^^^^6” (Zeheiry, 2022).

5. Multipartite sharing coordinates and geometric SLOCC invariants

For pure multiqubit systems, one prominent entanglement index is the singly-bipartitioned quantity PRESERVED_PLACEHOLDER_^^^^4all:\4Quantum Entanglement Index4^^^^7 of Qian, Alonso, and Eberly. If qubit PRESERVED_PLACEHOLDER_^^^^4all:\4Quantum Entanglement Index4^^^^8 is bipartitioned against the remaining PRESERVED_PLACEHOLDER_^^^^4all:\4Quantum Entanglement Index4^^^^9 qubits with reduced eigenvalues PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4Quantum Entanglement Index4^^^^, the Schmidt weight and normalized entanglement index are

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4all:\4^^^^

equivalently

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4 OR ti:\4^^^^

so PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4 OR abs:\4^^^^. The central theorem is the completely tight sharing inequality

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\44^^^^

valid for every PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^5. The vector

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^6

therefore occupies a convex polytope inside the unit hypercube PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^7, with Lebesgue volume

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^8

For PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^9, the triangle inequalities define the allowed solid; the GHZ family

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4Quantum Entanglement Index4^^^^

moves along the cube diagonal with

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4all:\4^^^^

while the W family

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4 OR ti:\4^^^^

fills the central triangular face and realizes maximal additivity. The area of the fixed-PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4 OR abs:\4^^^^ slice, with PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\44^^^^, is

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^5

This framework makes the “index” a coordinate of entanglement sharing rather than a scalar monotone (&&&^^^^4 OR abs:\4^^^^&&&).

A different geometric program, due to Sawicki, Oszmaniec, and Kuś, defines the total variance

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^6

for a projective pure state PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^7, and shows

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^8

with PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^9 the momentum map. Critical points are characterized by

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4Quantum Entanglement Index4^^^^

On each SLOCC orbit closure, the norm of the momentum map attains a unique minimum at a critical PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4all:\4^^^^-orbit, and the corresponding SLOCC-invariant entanglement index is

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4 OR ti:\4^^^^

The Morse index at a critical point counts the number of non-SLOCC directions in which entanglement increases. For three qubits, the paper reports Morse index PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4 OR abs:\4^^^^ for GHZ, PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\44^^^^ for W, PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^5 for each bi-separable class, and PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^6 for the fully separable critical point (&&&^^^^4all:\4^^^^9&&&).

Both constructions are geometric, but they measure different structures: PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^7 quantifies one-vs-rest sharing constraints, whereas PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^8 is a SLOCC-invariant distance from the zero-momentum-map locus.

6. Dynamical, spectral, and field-theoretic indices

For noisy channels, the phrase “entanglement index” often refers to the number of channel uses required to destroy all entanglement with an ancilla. Lami and Giovannetti define, for a channel PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^9,

PRESERVED_PLACEHOLDER_^^^^4all:\44Quantum Entanglement Index4^^^^

together with the unitary-filtered and general-filtered indices

PRESERVED_PLACEHOLDER_^^^^4all:\44all:\4^^^^

satisfying

PRESERVED_PLACEHOLDER_^^^^4all:\44 OR ti:\4^^^^

Their framework compares bare iteration with interposed filtering operations. The paper reports that in PRESERVED_PLACEHOLDER_^^^^4all:\44 OR abs:\4^^^^, non-unitary filters can be dramatically stronger than unitary ones: an explicit Werner-channel family has PRESERVED_PLACEHOLDER_^^^^4all:\444^^^^ but PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^5. By contrast, for qubits no such gap is known, and the paper formulates the conjecture that unitary filters suffice in dimension PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^6. For depolarizing channels PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^7, filtering does not help: PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^8 (Lami et al., 2014).

A related discrete-time notion is the entanglement-breaking index

PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^9

with PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^5^^^^4Quantum Entanglement Index4^^^^ if no finite power is entanglement breaking. For faithful channels and more general faithful CP maps, the theory of eventually entanglement breaking (EEB) gives structural finiteness results. The paper states: PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^5^^^^4all:\4^^^^ and more generally that any faithful CP map whose Hilbert–Schmidt adjoint is also faithful becomes entanglement breaking after finitely many iterations. Using robustness of separability and strong decoherence estimates, it obtains explicit upper bounds such as

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^5^^^^4 OR ti:\4^^^^

and for a qubit repeated-interaction channel it gives the exact formula

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^5^^^^4 OR abs:\4^^^^

(&&&^^^^4 OR ti:\4all:\4^^^^&&&).

In topological free-fermion systems, the entanglement spectrum itself carries an index. The “trace index” is defined from the jump discontinuities of PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^54, where PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^55 is the reduced correlation matrix on one side of an entanglement cut: PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^56 Each edge-state crossing of the Fermi level produces a PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^57 jump. The paper states that in a Chern insulator

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^58

while in a time-reversal-symmetric PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^59 insulator the parity of jumps on half the Brillouin zone reproduces the PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^6^^^^4Quantum Entanglement Index4^^^^ invariant. In this usage, the index is a spectral-flow invariant extracted from the single-particle entanglement spectrum (&&&^^^^4 OR ti:\4 OR ti:\4^^^^&&&).

More recent field-theoretic and high-energy papers extend the vocabulary further. In “^^^^^^^^4Quantum Entanglement Index4^^^^^^^^ in String Theory,” the indexed partition function

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^6^^^^4all:\4^^^^

defines

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^6^^^^4 OR ti:\4^^^^

and the BTZ-horizon stringy replica construction uses odd PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^6^^^^4 OR abs:\4^^^^ and opening angle PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^64. The one-loop partition function

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^65

is stated to be free of tachyons and naturally finite both in the ultraviolet and the infrared, with no analytic continuation in PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^66 required (&&&^^^^4 OR ti:\4 OR abs:\4^^^^&&&).

In “Quantum Entanglement as a Cohomological Obstruction,” the ^^^^^^^^4Quantum Entanglement Index4^^^^^^^^ is an analytic index: PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^67 with Atiyah–Singer formula

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^68

On a two-dimensional parameter surface PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^69,

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^7^^^^4Quantum Entanglement Index4^^^^

This index is explicitly topological and does not measure “amount” of entanglement like entropy or concurrence; it counts an index imbalance in the PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^7^^^^4all:\4 Dirac spectrum (Ikeda, 6 Nov 2025).

A distinct high-energy usage appears in the collider-oriented discriminant formalism. For an PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^7^^^^4 OR ti:\4^^^^-particle system, the pure-state quantum space is

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^7^^^^4 OR abs:\4^^^^

while the separable classical space is the Segre embedding

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^74

and the entanglement space is PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^75. For two fermions,

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^76

vanishes exactly on the separable locus. The paper states that PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^77 is equivalent to violation of the PPT criterion and, in the real-coefficient subcase, equivalent to violation of the CHSH criterion. Here the “index” is an algebraic discriminant reconstructed from collider angular correlations (&&&^^^^4 OR ti:\4^^^^5&&&).

Taken together, these developments show that “quantum entanglement index” has become a cross-disciplinary label for quantities that are formally quite different: SDP resource bounds, universal entropy coefficients, connected-correlation norms, multipartite sharing coordinates, entanglement-breaking times, trace discontinuities, analytic indices, and algebraic discriminants. The common thread is not a unique formula, but the attempt to compress entanglement structure into a task-adapted invariant or low-dimensional diagnostic.