---
title: Quantum Entanglement Index Overview
url: https://www.emergentmind.com/topics/quantum-entanglement-index
type: topic
---

# Quantum Entanglement Index Overview

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“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 \(N\)-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 [2303.07228] [1103.5452] [1909.01367] [1511.04354] [1411.2517] [2511.04326]. The literature therefore suggests that the phrase names a family of indices rather than a single standardized measure.

## 1. 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 | \(2^{-R_{\max,\mathrm{ADG}}}\), \(\bar E_{\mathrm{rev}}^u\), \(Q_{\mathrm{sqz}}\) | 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 | \(Y_j\) | \(N\)-qubit pure states |
| Channel dynamics | \(n(\phi)\), \(\mathcal N_U(\phi)\), \(\mathcal N(\phi)\), \(EBI(\Phi)\) | Quantum channels |
| Geometric/topological | \(E([v])\), \(I_{\rm trace}\), \(\mathrm{QEI}\) | 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 \(d>2\) [1909.01367] [2211.08981] [2211.08638].

## 2. 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,
\[
R_F(\rho) \coloneqq \min_{\sigma\in F} D(\sigma\|\rho),
\]
with \(D\) chosen to be the max-relative entropy,
\[
D_{\max}(\sigma\|\rho)\coloneqq \inf\{\lambda:\sigma\le 2^\lambda \rho\}.
\]
For one-way distillation, the free set is \(F=\mathrm{ADG}\), the set of anti-degradable (two-extendible) states, and
\[
R_{\max,\mathrm{ADG}}(\rho)\coloneqq \min_{\tau\in \mathrm{ADG}} D_{\max}(\tau\|\rho).
\]
The corresponding SDP computes
\[
s\coloneqq 2^{-R_{\max,\mathrm{ADG}}(\rho)}
\]
as the maximal weight of an ADG piece in a decomposition of \(\rho\). With the squeezed state
\[
\omega\coloneqq \frac{\rho-s\tau^*}{1-s},
\]
the paper proves
\[
E_{D,\rightarrow}(\rho)\le \bigl[1-2^{-R_{\max,\mathrm{ADG}}(\rho)}\bigr]\cdot E_F(\omega),
\]
and derives the fully SDP-computable bound
\[
\bar E_{\mathrm{rev}}^u(\rho)=\bigl[1-2^{-R_{\max,\mathrm{ADG}}(\rho)}\bigr]\cdot \sum_i \lambda_i S(B)_{\psi_i},
\]
where \(\omega=\sum_i \lambda_i |\psi_i\rangle\langle\psi_i|\) is a spectral decomposition [2303.07228].

The operative interpretation is that the decomposition
\[
\rho=(1-s)\omega+s\tau,\qquad \tau\in F,
\]
“squeezes out” the free component, on the premise that \(F\)-states carry zero useful entanglement for the task. For channels, the analogous one-number index is
\[
Q_{\mathrm{sqz}}=\bigl[1-2^{-R_{\max,\mathrm{ADG}}}(\mathcal N)\bigr]\cdot \max_\phi I_c(\phi,\mathcal N_{\mathrm{squeeze}}),
\]
computed from a channel SDP. The practical recipe is explicit: solve the state- or channel-version SDP to find \(s=2^{-R_{\max,\mathrm{ADG}}}\), form the squeezed state \(\omega\) or squeezed Choi map \(\mathcal N_{\mathrm{squeeze}}\), evaluate a single-letter coherent-information or entanglement-of-formation bound, and multiply by \((1-s)\). The paper states that all steps are numerically cheap in CVX/YALMIP for dimensions up to \(\sim 10\times 10\), and that the result gives one one-number index upper-bounding \(E_{D,\rightarrow}\) or \(Q\) [2303.07228].

The same framework yields concrete benchmarks. For \(\rho_{AB}(p,\gamma)=(AD_\gamma\otimes D_p)(|\Phi\rangle\langle\Phi|)\), the new bound \(\bar E_{\mathrm{rev}}^u\) is reported to strictly improve over the coherent-information lower bound \(I_c\), the Rains bound \(R\), and the continuity bounds \(E_{\mathrm{SCB}},E_{\mathrm{MCB}}\). For Pauli channels \(\Lambda(\rho)=\sum_{i=0}^3 p_i \sigma_i\rho\sigma_i\), the explicit formula
\[
2^{-R_{\max,\mathrm{ADG}}}(\Lambda)=(\sqrt{p_1}+\sqrt{p_2})^2+(\sqrt{p_2}+\sqrt{p_3})^2+(\sqrt{p_1}+\sqrt{p_3})^2
\]
leads to the no-cloning bound
\[
Q(\Lambda)\le 1-\bigl[(\sqrt{p_1}+\sqrt{p_2})^2+(\sqrt{p_2}+\sqrt{p_3})^2+(\sqrt{p_1}+\sqrt{p_3})^2\bigr],
\]
which the paper states exactly recovers Cerf 2000 [2303.07228].

## 3. 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 \(A\cup B\) of a pure ground state \(|\Psi\rangle\), the entropies are
\[
S_A=-\mathrm{Tr}\,\rho_A\ln\rho_A,\qquad
S_A^{(n)}=\frac{1}{1-n}\ln \mathrm{Tr}\,\rho_A^n.
\]
In one-dimensional CFTs with central charge \(c\), the coefficient \(c/3\) in
\[
S_A\approx \frac{c}{3}\ln(L/a)+\mathrm{const.}
\]
is the standard entanglement index. In two dimensions, the leading term obeys an area law \(S_A\sim aL\), but universal subleading terms can appear at CQCPs:
\[
S_A=aL-\alpha \ln L+\cdots.
\]
The coefficient \(\alpha\) is then the relevant entanglement index [1103.5452].

The paper “Logarithmic terms in entanglement entropies of 2D quantum critical points and Shannon entropies of spin chains” derives these coefficients by mapping the entanglement spectrum of a 2D strip geometry to the Shannon entropy of a 1D quantum chain with open boundaries. The probabilities
\[
p_\sigma=|\langle \sigma|0\rangle|^2
\]
yield
\[
S=-\sum_\sigma p_\sigma \ln p_\sigma,\qquad
S_n=\frac{1}{1-n}\ln\Bigl(\sum_\sigma p_\sigma^n\Bigr),
\]
with replica partition function
\[
Z(n)=\sum_\sigma p_\sigma^n=e^{-F(n)},\qquad
F(n,L)=f_1(n)L+\gamma(n)\ln(L/a)+f_0(n)+\cdots.
\]
The logarithmic term arises from the trace anomaly at the four corners where the cut meets the external boundaries [1103.5452].

The model-dependent coefficients are explicit. For the compact free boson (\(c=1\)) corresponding to the square-lattice quantum dimer CQCP, with Dirichlet or Neumann external boundary conditions,
\[
\gamma(n)=\mp \frac{n-1}{4},
\]
so that
\[
S(L)=s_1L\mp \frac14\ln(L/a)+\cdots,
\]
and hence \(\alpha=+1/4\) for Neumann or \(\alpha=-1/4\) for Dirichlet. For the 2D Ising CQCP (\(c=1/2\)), numerical TEBD and finite-size scaling give
\[
\gamma(n)=
\begin{cases}
-(1/8)n,& n<1,\\
0,& n=1,\\
+(3/8)n,& n>1.
\end{cases}
\]
Because \(\gamma(n)\) has a cusp at \(n=1\), the replica derivative is not analytic and the von Neumann entropy has no simple logarithm; for Rényi entropies, however, one obtains \(\alpha_{n<1}=+1/8\) and \(\alpha_{n>1}=-3/8\) [1103.5452].

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,
\[
|\psi\rangle=c_0|00\rangle+c_1|11\rangle+c_2|22\rangle,
\]
“Entanglement certification and quantification in spatial-bin photonic qutrits” gives
\[
N=c_0c_1+c_1c_2+c_2c_0,\qquad
EF(|\psi\rangle)=-\sum_i c_i^2\log_2(c_i^2),
\]
and relates these to two-basis measurements. In the Fourier basis, the Pearson correlation coefficient satisfies \(|C_{A_1B_1}|=N\); in a mixed Fourier basis, the mutual predictability satisfies
\[
MP=\frac{1+2N}{3}\quad\Rightarrow\quad N=\frac{3\,MP-1}{2};
\]
in the computational basis, the classical mutual information reduces to
\[
MI=EF.
\]
Experimentally, the reported values are
\[
PCC_{\mathrm{Fourier}}=0.848\pm 0.027\Rightarrow N=0.848\pm 0.027,
\]
\[
MP=0.899\pm 0.013\Rightarrow N=0.849\pm 0.020,
\]
\[
MI=EF=1.233\pm 0.012.
\]
The same work emphasizes that a maximally entangled qutrit has \(N_{\max}=1\) and \(EF_{\max}=\log_2 3\approx 1.585\), so the experiment exhibits a \(15\%\) shortfall in Negativity but a \(24\%\) shortfall in EOF. Appendix C is cited as showing classes of two-qutrit states for which \(N\) increases while \(EF\) decreases, or vice versa, so the two measures are not monotonic functions of one another in \(d>2\) [1909.01367].

For two-qubit mixed states, the connected-correlation construction introduces
\[
(R_c)_{jk}=\mathrm{Tr}(\rho\,\sigma_j\otimes \sigma_k)-\mathrm{Tr}(\rho\,\sigma_j\otimes I_2)\,\mathrm{Tr}(\rho\,I_2\otimes \sigma_k),
\]
and defines
\[
E_C(\rho)\equiv \gamma_c(\rho)=2\sqrt{u_{c,1}+u_{c,2}}\le 2\sqrt2,
\]
where \(u_{c,1}\ge u_{c,2}\ge u_{c,3}\ge 0\) are the eigenvalues of \(R_c^TR_c\). For pure states, the paper states
\[
E_C(|\psi\rangle)=2\sqrt2\,C(|\psi\rangle),
\]
with \(C(|\psi\rangle)\) the concurrence. The same paper also states that in the “Class D” single-measure sector, the state is separable with classical correlation and \(E_C=2E_4^2\). This makes the scope of the vanishing criterion precise: \(E_C\) vanishes exactly on product states, not on all separable states, and its monotonicity statement is formulated within classes of fixed \((\gamma_2,\theta)\) invariants [2211.08638].

A different expectation-value construction defines, for two qubits,
\[
M(\rho_{AB})\coloneqq \max_{n_A,n_B}\Bigl[\mathrm{Tr}\bigl(\rho_{AB}(\vec\sigma_A\cdot n_A\otimes I_B)\bigr)+\mathrm{Tr}\bigl(\rho_{AB}(I_A\otimes \vec\sigma_B\cdot n_B)\bigr)\Bigr],
\]
with \(M_{\rm sep}=2\) and \(M_{\rm ent}=0\), and then
\[
E(\rho_{AB})=1-\frac{M(\rho_{AB})}{2}.
\]
For qudits of dimension \(D\), with generalized-spin spectrum \(\{h_1,\dots,h_D\}\), largest eigenvalue \(h_{\max}=h_1\), and average
\[
\eta=\frac{1}{D}\sum_{j=1}^D h_j,
\]
the generalization is
\[
E(\rho_{AB})=\frac{2h_{\max}-M(\rho_{AB})}{2h_{\max}-2\eta}.
\]
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 \(M(\rho_W)=0\Rightarrow E(\rho_W)=1\) for all \(p>0\), which is reported as an “extreme insensitivity to \(p\)” [2211.08981].

## 5. Multipartite sharing coordinates and geometric SLOCC invariants

For pure multiqubit systems, one prominent entanglement index is the singly-bipartitioned quantity \(Y_j\) of Qian, Alonso, and Eberly. If qubit \(j\) is bipartitioned against the remaining \(N-1\) qubits with reduced eigenvalues \(\lambda_1^{(j)}+\lambda_2^{(j)}=1\), the Schmidt weight and normalized entanglement index are
\[
K_j^{-1}=(\lambda_1^{(j)})^2+(\lambda_2^{(j)})^2,\qquad
Y_j=1-\sqrt{\frac{2}{K_j}-1},
\]
equivalently
\[
Y_j=2\min\{\lambda_1^{(j)},\lambda_2^{(j)}\},
\]
so \(0\le Y_j\le 1\). The central theorem is the completely tight sharing inequality
\[
Y_j\le \sum_{k\ne j}Y_k,
\]
valid for every \(j=1,\dots,N\). The vector
\[
\mathbf Y=(Y_1,\dots,Y_N)
\]
therefore occupies a convex polytope inside the unit hypercube \([0,1]^N\), with Lebesgue volume
\[
V_N=1-\frac{1}{(N-1)!}.
\]
For \(N=3\), the triangle inequalities define the allowed solid; the GHZ family
\[
|\Psi_{GHZ}\rangle=\cos\theta\,|000\rangle+\sin\theta\,|111\rangle
\]
moves along the cube diagonal with
\[
Y_1=Y_2=Y_3=1-|\cos 2\theta|,
\]
while the W family
\[
|\Psi_W\rangle=\alpha|100\rangle+\beta|010\rangle+\gamma|001\rangle
\]
fills the central triangular face and realizes maximal additivity. The area of the fixed-\(Y_T\) slice, with \(Y_T=Y_1+Y_2+Y_3\), is
\[
A(Y_T)=\frac{\sqrt3}{2}\times
\begin{cases}
Y_T^2/4,& 0\le Y_T\le 2,\\
(3-Y_T)^2,& 2\le Y_T\le 3.
\end{cases}
\]
This framework makes the “index” a coordinate of entanglement sharing rather than a scalar monotone [1511.04354].

A different geometric program, due to Sawicki, Oszmaniec, and Kuś, defines the total variance
\[
\mathrm{Var}([v])=\frac{1}{\langle v|v\rangle}\Bigl(\sum_i \langle v|\xi_i^2|v\rangle-\frac{1}{\langle v|v\rangle}\sum_i \langle v|\xi_i|v\rangle^2\Bigr)
\]
for a projective pure state \([v]\), and shows
\[
\mathrm{Var}([v])=c-4\|\mu([v])\|^2,
\]
with \(\mu\) the momentum map. Critical points are characterized by
\[
\mu^*([v])\,v=\lambda v.
\]
On each SLOCC orbit closure, the norm of the momentum map attains a unique minimum at a critical \(K\)-orbit, and the corresponding SLOCC-invariant entanglement index is
\[
E([v])=\sqrt{\mathrm{Var}([v])_{\rm crit}}=2\,d([v]),
\qquad
d([v])=\min_{g\in G}\|\mu(g\cdot [v])\|.
\]
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 \(0\) for GHZ, \(2\) for W, \(6\) for each bi-separable class, and \(8\) for the fully separable critical point [1208.0556].

Both constructions are geometric, but they measure different structures: \(Y_j\) quantifies one-vs-rest sharing constraints, whereas \(E([v])\) 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 \(\phi:M_d\to M_d\),
\[
n(\phi)=\min\{n\ge 1:\phi^n\in \mathbf{EBt}_d\},
\]
together with the unitary-filtered and general-filtered indices
\[
\mathcal N_U(\phi),\qquad \mathcal N(\phi),
\]
satisfying
\[
n(\phi)\le \mathcal N_U(\phi)\le \mathcal N(\phi).
\]
Their framework compares bare iteration with interposed filtering operations. The paper reports that in \(d\ge 3\), non-unitary filters can be dramatically stronger than unitary ones: an explicit Werner-channel family has \(n(V_\eta)=\mathcal N_U(V_\eta)=2\) but \(\mathcal N(V_\eta)=\infty\). By contrast, for qubits no such gap is known, and the paper formulates the conjecture that unitary filters suffice in dimension \(2\). For depolarizing channels \(\Delta_\lambda\), filtering does not help:
\[
\mathcal N_U(\Delta_\lambda)=\mathcal N(\Delta_\lambda)=n(\Delta_\lambda),
\qquad
n(\Delta_\lambda)=\left\lceil \frac{\ln(d+1)}{-\ln\lambda}\right\rceil
\]
[1411.2517].

A related discrete-time notion is the entanglement-breaking index
\[
EBI(\Phi)=\min\{n\in \mathbb N:\Phi^n\in EB(\mathcal H)\},
\]
with \(EBI(\Phi)=\infty\) 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:
\[
\text{Faithful + PPT} \Rightarrow \text{EEB},
\]
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
\[
EBI(\Phi)\le \left\lceil \frac{1}{\gamma}\ln\!\Bigl(\frac{K\,d_}{\lambda_{\min}(\sigma)}\Bigr)\right\rceil,
\]
and for a qubit repeated-interaction channel it gives the exact formula
\[
EBI(\Phi)=\max\Bigl(1,\Bigl\lceil \frac{\tfrac12\ln B(g)}{\ln|\gamma|}\Bigr\rceil\Bigr)
\]
[1902.08173].

In topological free-fermion systems, the entanglement spectrum itself carries an index. The “trace index” is defined from the jump discontinuities of \(\mathrm{Tr}\,G^A(k_y)\), where \(G^A\) is the reduced correlation matrix on one side of an entanglement cut:
\[
I_{\rm trace}=\sum_\ell \Bigl[\mathrm{Tr}\,G^A(K_\ell^+)-\mathrm{Tr}\,G^A(K_\ell^-)\Bigr].
\]
Each edge-state crossing of the Fermi level produces a \(\pm 1\) jump. The paper states that in a Chern insulator
\[
I_{\rm trace}=C_1,
\]
while in a time-reversal-symmetric \(Z_2\) insulator the parity of jumps on half the Brillouin zone reproduces the \(Z_2\) invariant. In this usage, the index is a spectral-flow invariant extracted from the single-particle entanglement spectrum [1108.2907].

More recent field-theoretic and high-energy papers extend the vocabulary further. In “Quantum Entanglement Index in String Theory,” the indexed partition function
\[
\hat Z_{\rm ind}(\beta)=\mathrm{Tr}_R\bigl[e^{-\beta H_R}(-1)^{F_R}\bigr]
\]
defines
\[
S_{\rm ind}=\Bigl(-\frac{\partial}{\partial\beta}+1\Bigr)\ln \hat Z_{\rm ind}(\beta)\Big|_{\beta=2\pi}=\ln(d_{\rm ind}),
\]
and the BTZ-horizon stringy replica construction uses odd \(N\) and opening angle \(\Theta=2\pi/N\). The one-loop partition function
\[
Z(N)=\log \hat Z\!\Bigl(\frac{2\pi}{N}\Bigr)-\log \hat Z(2\pi)
\]
is stated to be free of tachyons and naturally finite both in the ultraviolet and the infrared, with no analytic continuation in \(N\) required [2507.15939].

In “Quantum Entanglement as a Cohomological Obstruction,” the Quantum Entanglement Index is an analytic index:
\[
\mathrm{QEI}=\mathrm{Ind}_S(D_X\otimes E)
=\mathrm{ind}(D_X\otimes E^+)-\mathrm{ind}(D_X\otimes E^-)\in \mathbb Z,
\]
with Atiyah–Singer formula
\[
\mathrm{QEI}
=
\Bigl\langle \widehat A(TX)\wedge \mathrm{Tr}\bigl(S\,e^{F_A/2\pi i}\bigr),[X]\Bigr\rangle.
\]
On a two-dimensional parameter surface \(\Sigma\),
\[
\mathrm{QEI}=\frac{1}{2\pi i}\int_\Sigma \mathrm{Tr}(S\,F_A)=\nu_{\rm ent}\in \mathbb Z.
\]
This index is explicitly topological and does not measure “amount” of entanglement like entropy or concurrence; it counts an index imbalance in the \(S\)-graded Dirac spectrum [2511.04326].

A distinct high-energy usage appears in the collider-oriented discriminant formalism. For an \(N\)-particle system, the pure-state quantum space is
\[
\mathcal Q=\mathbb{CP}^{J-1},\qquad J=\prod_{i=1}^N(2s_i+1),
\]
while the separable classical space is the Segre embedding
\[
\mathcal C=\mathbb{CP}^{2s_1}\times\cdots\times \mathbb{CP}^{2s_N}\subset \mathbb{CP}^{J-1},
\]
and the entanglement space is \(\mathcal E=\mathcal Q\setminus \mathcal C\). For two fermions,
\[
\Delta_{ff}
=
\alpha_{+\tfrac12,+\tfrac12}\alpha_{-\tfrac12,-\tfrac12}
-
\alpha_{+\tfrac12,-\tfrac12}\alpha_{-\tfrac12,+\tfrac12}
\]
vanishes exactly on the separable locus. The paper states that \(\Delta_{ff}\neq 0\) 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 [2505.09280].

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.

Source: https://www.emergentmind.com/topics/quantum-entanglement-index