Entanglement Distance: Concepts & Frameworks

• Entanglement distance is a family of measures defining how close an entangled state is to separability or quantifying the geometric spread in various quantum systems.
• It encompasses different methodologies including Fubini–Study metrics, Hilbert–Schmidt distances, and emergent spatial constructions from mutual information.
• These approaches inform both theoretical insights and practical implementations in many-body physics and long-distance quantum networking experiments.

Entanglement distance denotes several distinct, non-equivalent quantities in quantum information, condensed-matter theory, and quantum-network research. In the surveyed literature, it may mean a geometric distance from an entangled state to the separable-state manifold, a Fubini–Study–derived measure attached to local-unitary orbits, a distance between reduced density matrices of different many-body eigenstates, an emergent spatial distance defined from mutual information, or the literal physical separation across which entanglement is distributed and verified (Carrington et al., 2015, Cocchiarella et al., 2019, Vaezi et al., 2016, Franzmann et al., 2022, Inagaki et al., 2013). This suggests that the expression is best understood as a family of constructions rather than a single standardized observable.

1. Terminological scope

Several mathematically different notions appear under closely related names.

Usage	Representative definition	Representative papers
Distance to separability	$E_G(|\psi\rangle)=1-\Lambda_{\max}^2$, with $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$	(Carrington et al., 2015, Streltsov et al., 2010)
Hilbert–Schmidt distance to separable states	$$\mathcal E(\rho_e)=\min_{\rho_s\in\mathcal S}\|\rho_e-\rho_s\|_{\rm HS}^2$$	(Athira et al., 2023)
Fubini–Study–based entanglement distance	$E(|s\rangle)=\mathrm{Tr}\,\tilde g$ or closed forms in local expectation values	(Cocchiarella et al., 2019, Cocchiarella et al., 2020)
Emergent-space distance from entanglement	$d(A,B)\approx \ell_{\rm RC}\Phi(I(A:B)/I_0)$	(Franzmann et al., 2022)
Distance between eigenstates’ entanglement structures	$$\epsilon_1^{(a,b)}\propto \mathrm{Tr}[(\rho_L^{(a)}-\rho_L^{(b)})^2]$$	(Vaezi et al., 2016)
Physical separation of entangled systems	fiber, free-space, and network link length	(Inagaki et al., 2013, Krutyanskiy et al., 2022)

A recurrent misconception is to treat these usages as interchangeable. They are not. A Bell pair can have maximal bipartite entanglement, zero emergent distance in a spin-only toy model, nonzero distance in a multi-sector emergent-space construction, and a laboratory separation of meters or hundreds of kilometers, depending on which notion is being used (Franzmann et al., 2022, Krutyanskiy et al., 2022).

2. Distance to the separable-state manifold

A central line of work defines entanglement through proximity to separable states. For pure multipartite qubit states, the geometric measure is built from the maximal separable overlap

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|, \qquad E_G(|\psi\rangle)=1-\Lambda_{\max}^2, \qquad \theta(|\psi\rangle)=\arccos(\Lambda_{\max}),$$

with equivalent distance conventions such as $D_c^2=1-\Lambda_{\max}^2$ or, for normalized closest product states, $D_c^2=2(1-\Lambda_{\max})$ (Carrington et al., 2015). In permutation-symmetric families this admits analytic solutions. For example, $E_G(GHZ_q)=1/2$, $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$0, and for the Dicke state $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$1 one has

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$2

(Carrington et al., 2015).

For mixed states, the same program is tied to fidelity. Defining the fidelity of separability

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$3

the geometric measure satisfies

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$4

and the Bures entanglement becomes

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$5

For two qubits, using concurrence $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$6,

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$7

The same analysis also yields a Carathéodory bound on optimal decompositions and stationarity conditions for candidate optimal ensembles (Streltsov et al., 2010).

A closely related but operationally narrower construction uses the Hilbert–Schmidt distance to the convex set $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$8 of separable states. In the thermal two-qubit Heisenberg dimer studied for copper acetate,

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$9

and this exactly coincides with Wootters’ concurrence for the block-structured thermal state considered in that work (Athira et al., 2023). The same paper expresses $$\mathcal E(\rho_e)=\min_{\rho_s\in\mathcal S}\|\rho_e-\rho_s\|_{\rm HS}^2$$0 through magnetic susceptibility and magnetic specific heat: $$\mathcal E(\rho_e)=\min_{\rho_s\in\mathcal S}\|\rho_e-\rho_s\|_{\rm HS}^2$$1

$$\mathcal E(\rho_e)=\min_{\rho_s\in\mathcal S}\|\rho_e-\rho_s\|_{\rm HS}^2$$2

Experimentally, the resulting bipartite entanglement was reported to persist to room temperature and to vanish only near $$\mathcal E(\rho_e)=\min_{\rho_s\in\mathcal S}\|\rho_e-\rho_s\|_{\rm HS}^2$$3 according to the susceptibility analysis (Athira et al., 2023).

Another pure-state distance-style measure is Gudder’s entanglement number. For Schmidt coefficients $$\mathcal E(\rho_e)=\min_{\rho_s\in\mathcal S}\|\rho_e-\rho_s\|_{\rm HS}^2$$4,

$$\mathcal E(\rho_e)=\min_{\rho_s\in\mathcal S}\|\rho_e-\rho_s\|_{\rm HS}^2$$5

equivalently the Euclidean distance of the Schmidt-probability vector from an extremal separable distribution. It vanishes on product states and reaches $$\mathcal E(\rho_e)=\min_{\rho_s\in\mathcal S}\|\rho_e-\rho_s\|_{\rm HS}^2$$6 for maximally entangled Schmidt-rank-$$\mathcal E(\rho_e)=\min_{\rho_s\in\mathcal S}\|\rho_e-\rho_s\|_{\rm HS}^2$$7 states (Gudder, 2020).

3. Fubini–Study geometry and local-unitary orbits

A separate literature derives entanglement distance from the Fubini–Study metric on projective Hilbert space. For pure $$\mathcal E(\rho_e)=\min_{\rho_s\in\mathcal S}\|\rho_e-\rho_s\|_{\rm HS}^2$$8-qubit states, one considers infinitesimal displacements generated by local $$\mathcal E(\rho_e)=\min_{\rho_s\in\mathcal S}\|\rho_e-\rho_s\|_{\rm HS}^2$$9 rotations, defines a metric $E(|s\rangle)=\mathrm{Tr}\,\tilde g$0 on the local-unitary orbit, and then minimizes its trace over local axes to obtain an entanglement metric $E(|s\rangle)=\mathrm{Tr}\,\tilde g$1. The entanglement distance is

$E(|s\rangle)=\mathrm{Tr}\,\tilde g$2

which is invariant under local unitaries and whose eigenvalues encode directional robustness of entanglement (Cocchiarella et al., 2019). In that normalization, GHZ-like states yield

$E(|s\rangle)=\mathrm{Tr}\,\tilde g$3

with one nonzero eigenvalue and $E(|s\rangle)=\mathrm{Tr}\,\tilde g$4 zero eigenvalues, whereas W-like or cluster-like states can have full-rank $E(|s\rangle)=\mathrm{Tr}\,\tilde g$5 (Cocchiarella et al., 2019).

The construction was generalized to arbitrary $E(|s\rangle)=\mathrm{Tr}\,\tilde g$6-qudit hybrid systems. For a pure state on $E(|s\rangle)=\mathrm{Tr}\,\tilde g$7, using traceless $E(|s\rangle)=\mathrm{Tr}\,\tilde g$8 generators $E(|s\rangle)=\mathrm{Tr}\,\tilde g$9,

$d(A,B)\approx \ell_{\rm RC}\Phi(I(A:B)/I_0)$0

and for mixed states the measure is extended by a convex roof (Cocchiarella et al., 2020). In the qubit specialization,

$d(A,B)\approx \ell_{\rm RC}\Phi(I(A:B)/I_0)$1

The same work interprets $d(A,B)\approx \ell_{\rm RC}\Phi(I(A:B)/I_0)$2 as a lower bound on the minimal infinitesimal Fubini–Study distance density under local-unitary motion, and introduces an entanglement metric tensor $d(A,B)\approx \ell_{\rm RC}\Phi(I(A:B)/I_0)$3 whose eigenvalues quantify robustness (Cocchiarella et al., 2020). The published formulas therefore use different normalizations across related Fubini–Study constructions.

This framework admits unusually explicit results for graph and circuit states. For directed graph states built from commuting controlled-unitary gates

$d(A,B)\approx \ell_{\rm RC}\Phi(I(A:B)/I_0)$4

the one-qubit Bloch-vector length depends only on total degree $d(A,B)\approx \ell_{\rm RC}\Phi(I(A:B)/I_0)$5,

$d(A,B)\approx \ell_{\rm RC}\Phi(I(A:B)/I_0)$6

and the entanglement distance per qubit becomes

$d(A,B)\approx \ell_{\rm RC}\Phi(I(A:B)/I_0)$7

This dependence only on the degree distribution yields invariance under vertex relabeling and insensitivity to the distinction between incoming and outgoing edges at fixed total degree (Simone et al., 15 May 2025).

For variational states generated by layers of $d(A,B)\approx \ell_{\rm RC}\Phi(I(A:B)/I_0)$8 rotations and $d(A,B)\approx \ell_{\rm RC}\Phi(I(A:B)/I_0)$9 gates, the single-qubit reduced-state entanglement distance is defined as

$$\epsilon_1^{(a,b)}\propto \mathrm{Tr}[(\rho_L^{(a)}-\rho_L^{(b)})^2]$$0

In the two-qubit one-layer case,

$$\epsilon_1^{(a,b)}\propto \mathrm{Tr}[(\rho_L^{(a)}-\rho_L^{(b)})^2]$$1

and in a closed one-dimensional chain the dependence of local expectation values expands outward with circuit depth, so after two layers a target qubit already depends on sites $$\epsilon_1^{(a,b)}\propto \mathrm{Tr}[(\rho_L^{(a)}-\rho_L^{(b)})^2]$$2 (Gnatenko et al., 30 Apr 2026). This is presented there as a direct manifestation of correlation spreading in structured parametrized circuits.

4. Reduced-density-matrix distances and emergent-space distances

In many-body theory, entanglement distance can also compare the entanglement structures of two different eigenstates rather than quantify entanglement inside one state. Let $$\epsilon_1^{(a,b)}\propto \mathrm{Tr}[(\rho_L^{(a)}-\rho_L^{(b)})^2]$$3 and $$\epsilon_1^{(a,b)}\propto \mathrm{Tr}[(\rho_L^{(a)}-\rho_L^{(b)})^2]$$4 be eigenstates with reduced density matrices $$\epsilon_1^{(a,b)}\propto \mathrm{Tr}[(\rho_L^{(a)}-\rho_L^{(b)})^2]$$5 and $$\epsilon_1^{(a,b)}\propto \mathrm{Tr}[(\rho_L^{(a)}-\rho_L^{(b)})^2]$$6 on a subsystem $$\epsilon_1^{(a,b)}\propto \mathrm{Tr}[(\rho_L^{(a)}-\rho_L^{(b)})^2]$$7. Vaezi and Vaezi define

$$\epsilon_1^{(a,b)}\propto \mathrm{Tr}[(\rho_L^{(a)}-\rho_L^{(b)})^2]$$8

and a DMRG-oriented truncation-error distance

$$\epsilon_1^{(a,b)}\propto \mathrm{Tr}[(\rho_L^{(a)}-\rho_L^{(b)})^2]$$9

For $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|, \qquad E_G(|\psi\rangle)=1-\Lambda_{\max}^2, \qquad \theta(|\psi\rangle)=\arccos(\Lambda_{\max}),$$0D critical systems they obtain scaling

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|, \qquad E_G(|\psi\rangle)=1-\Lambda_{\max}^2, \qquad \theta(|\psi\rangle)=\arccos(\Lambda_{\max}),$$1

with $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|, \qquad E_G(|\psi\rangle)=1-\Lambda_{\max}^2, \qquad \theta(|\psi\rangle)=\arccos(\Lambda_{\max}),$$2 and $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|, \qquad E_G(|\psi\rangle)=1-\Lambda_{\max}^2, \qquad \theta(|\psi\rangle)=\arccos(\Lambda_{\max}),$$3 for Abelian $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|, \qquad E_G(|\psi\rangle)=1-\Lambda_{\max}^2, \qquad \theta(|\psi\rangle)=\arccos(\Lambda_{\max}),$$4 theories, while for non-Abelian theories $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|, \qquad E_G(|\psi\rangle)=1-\Lambda_{\max}^2, \qquad \theta(|\psi\rangle)=\arccos(\Lambda_{\max}),$$5 and $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|, \qquad E_G(|\psi\rangle)=1-\Lambda_{\max}^2, \qquad \theta(|\psi\rangle)=\arccos(\Lambda_{\max}),$$6; the $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|, \qquad E_G(|\psi\rangle)=1-\Lambda_{\max}^2, \qquad \theta(|\psi\rangle)=\arccos(\Lambda_{\max}),$$7 parafermion example gives $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|, \qquad E_G(|\psi\rangle)=1-\Lambda_{\max}^2, \qquad \theta(|\psi\rangle)=\arccos(\Lambda_{\max}),$$8 (Vaezi et al., 2016). In $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|, \qquad E_G(|\psi\rangle)=1-\Lambda_{\max}^2, \qquad \theta(|\psi\rangle)=\arccos(\Lambda_{\max}),$$9D topological phases, different topological sectors have maximal entanglement distance, while degenerate ground states related by confined anyon excitations such as genons have minimal entanglement distance (Vaezi et al., 2016).

A conceptually different use appears in emergent-spacetime scenarios, where distance is constructed from mutual information. Starting from a factored Hilbert space $D_c^2=1-\Lambda_{\max}^2$0, one defines a graph with edge weights

$D_c^2=1-\Lambda_{\max}^2$1

and emergent distance

$D_c^2=1-\Lambda_{\max}^2$2

With $D_c^2=1-\Lambda_{\max}^2$3, a nearest-link approximation gives

$D_c^2=1-\Lambda_{\max}^2$4

Franzmann and collaborators use a toy model with independent spin and momentum sectors,

$D_c^2=1-\Lambda_{\max}^2$5

to address the tension between a maximally entangled Bell pair and finite laboratory separation. In the pure spin Bell sector, $D_c^2=1-\Lambda_{\max}^2$6; in the momentum sector, $D_c^2=1-\Lambda_{\max}^2$7; and total mutual information is $D_c^2=1-\Lambda_{\max}^2$8 (Franzmann et al., 2022). Momentum decoherence can reduce $D_c^2=1-\Lambda_{\max}^2$9 while leaving $D_c^2=2(1-\Lambda_{\max})$0 unchanged, so a pair can remain spin-entangled yet acquire nonzero emergent distance. This directly separates entanglement as a quantum-information property from distance as an emergent-geometric assignment.

5. Physical separation and long-distance entanglement distribution

A further usage is literal: how far entanglement can be distributed and verified between distant systems. Recent experiments span solid-state spins, trapped ions, fiber-based photonic links, and integrated photonics.

Platform	Separation	Key reported result
NV-center spin qubits (Bernien et al., 2012)	$D_c^2=2(1-\Lambda_{\max})$1	Heralded fidelities $D_c^2=2(1-\Lambda_{\max})$2, $D_c^2=2(1-\Lambda_{\max})$3
Trapped-ion qubits (Krutyanskiy et al., 2022)	$D_c^2=2(1-\Lambda_{\max})$4 line-of-sight, $D_c^2=2(1-\Lambda_{\max})$5 fiber	Fidelity up to $D_c^2=2(1-\Lambda_{\max})$6, success probability $D_c^2=2(1-\Lambda_{\max})$7
Silicon-chip time-energy pairs (Zhao et al., 2024)	$D_c^2=2(1-\Lambda_{\max})$8 fiber	Franson visibility $D_c^2=2(1-\Lambda_{\max})$9 after transmission
Four-dimensional time-bin pairs (Ikuta et al., 2018)	$E_G(GHZ_q)=1/2$0 fiber	Fidelity $E_G(GHZ_q)=1/2$1, conditional entropy $E_G(GHZ_q)=1/2$2 bits
Time-bin entangled photons (Inagaki et al., 2013)	$E_G(GHZ_q)=1/2$3 fiber	Raw visibility $E_G(GHZ_q)=1/2$4, $E_G(GHZ_q)=1/2$5, Bell violation by $E_G(GHZ_q)=1/2$6 standard deviations

The $E_G(GHZ_q)=1/2$7 fiber experiment used a continuous-wave diode laser at $E_G(GHZ_q)=1/2$8, PPLN waveguides for SHG and SPDC, superconducting single-photon detectors, and planar-lightwave-circuit Mach–Zehnder interferometers with $E_G(GHZ_q)=1/2$9 delay (Inagaki et al., 2013). The time-bin entangled state was approximated by

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$00

with coherence over $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$01 pulses. After $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$02 of dispersion-shifted fiber in each arm, the experiment reported total transmission loss $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$03, raw two-photon visibility $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$04, and a CHSH parameter

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$05

obtained after $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$06 hours of data acquisition repeated three times (Inagaki et al., 2013).

The $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$07 silicon-chip experiment instead used spontaneous four-wave mixing in a silicon racetrack ring resonator with $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$08, $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$09, and photon coherence time $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$10. The measured visibility dropped only slightly, from $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$11 before fiber to $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$12 after $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$13, while the coincidence-to-accidental ratio reached $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$14 at low pump power (Zhao et al., 2024).

High-dimensional transmission has also been demonstrated. In the four-dimensional time-bin experiment, each photon traversed $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$15 of dispersion-shifted fiber, and tomography with cascaded Mach–Zehnder interferometers yielded fidelity $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$16, trace distance $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$17, and conditional entropy $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$18 bits, corresponding to coherent information above $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$19 bit per entangled pair (Ikuta et al., 2018).

Matter-node experiments emphasize heralding and network compatibility rather than raw distance. The $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$20 NV-center experiment used a two-round Barrett–Kok protocol with spin–photon entanglement and beam-splitter interference, reaching an entanglement rate of about $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$21 with per-round detection efficiencies $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$22 (Bernien et al., 2012). The trapped-ion experiment used cavity-mediated ion–photon entanglement and Hong–Ou–Mandel interference between photons routed through about $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$23 of single-mode fiber per half-link (Krutyanskiy et al., 2022).

6. Distance limits, scaling laws, and network extensions

Direct entanglement distribution through lossy channels exhibits sharply unfavorable scaling. In the $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$24 fiber analysis, coincidence rate scales as $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$25 with $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$26, and although accidental coincidences remain negligible out to about $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$27 with ultra-low-dark-count SSPDs and narrow coincidence windows, count rates fall below $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$28 once the distance exceeds $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$29 (Inagaki et al., 2013). That work therefore identifies about $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$30 as the practical upper limit for direct entanglement distribution in standard telecom fiber without quantum repeaters, while also noting that near-unity detector efficiency and sub-Hertz dark counts could make $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$31–$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$32 tests feasible on day-to-month timescales (Inagaki et al., 2013).

Repeater architectures address this exponential decay. In cavity-based repeaters employing two-photon heralded entanglement generation and a heralded CZ gate for swapping, the end-to-end rate is approximated by

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$33

with

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$34

Because the heralded gate transfers dissipation from infidelity to gate-failure probability, the conditional swap fidelity remains approximately unity, and for $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$35 the scheme was reported to support $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$36 with secret-key rates of order $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$37 (Borregaard et al., 2015).

Network-theoretic approaches can even target distance-independent final fidelity. In two-dimensional networks built from rank-2 or rank-3 mixed states, global error correction and entanglement percolation were shown to generate long-distance entanglement whose final fidelity does not decay with separation once the relevant thresholds are met. On the infinite square lattice, the binary-state global-error-correction threshold is $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$38, while bond percolation requires $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$39 on the square lattice and $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$40 on the triangular lattice (Broadfoot et al., 2010).

Distance can also enter dynamically through coupling to a common quantum field. For two directly coupled harmonic-oscillator detectors at separation $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$41, the late-time entanglement criterion is

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$42

The analysis shows a direct-coupling-dominated regime with an outer critical distance $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$43, a non-Markovian near-zone window $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$44, and the conclusion that without direct coupling entanglement does not survive beyond more than a few inverse high-frequency cutoff scales, whereas with direct coupling it can sustain over a finite distance (Hsiang et al., 2015).

A much more speculative limit comes from quantum-gravitational considerations. A proposed “utmost distance” for entanglement is

$$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$45

with $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$46 suggested by UV–IR consistency arguments and $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$47 by holographic entropy arguments (Xiao, 2017). For visible photons with $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$48, the proposal gives $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$49 for $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$50 and $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$51 for $$\Lambda_{\max}=\max_{|\phi_{\rm sep}\rangle}|\langle\phi_{\rm sep}|\psi\rangle|$$52, far beyond any laboratory or near-Earth experiment (Xiao, 2017). This proposal is explicitly presented as very weak and hard to detect in foreseeable experiments.

Taken together, these results show that “entanglement distance” is a layered concept. In one layer it quantifies proximity to separability; in another it measures geometric obstruction under local-unitary motion; in another it compares reduced states across eigenstates; in another it defines emergent geometry from mutual information; and in experimental quantum networking it refers to the actual separation over which entanglement survives loss, decoherence, and heralding constraints. The principal technical challenge is therefore not to choose a single definition, but to specify which distance is under discussion and which physical question it is meant to answer.