---
title: 'Vertical-Distance Entanglement: Metrics & Methods'
url: https://www.emergentmind.com/topics/vertical-distance-entanglement
type: topic
---

# Vertical-Distance Entanglement: Metrics & Methods

Vertical-distance entanglement denotes, in the literature represented here, several mathematically distinct but conceptually related ways of attaching a direction- or distance-sensitive structure to entanglement. In one line of work, entanglement is quantified by restricting the Fubini–Study metric to local-unitary directions and minimizing the associated line-element density, yielding an entanglement distance for \(M\)-qubit and \(M\)-qudit hybrid systems [1908.03117] [2003.05771]. In a closely related variational-circuit setting, the same language is used for the dependence of the entanglement distance of a target qubit on qubits at graph-distance \(d\) in an \(R_Y\)–\(CZ\) ansatz [2605.00259]. Separate uses arise in emergent-spacetime scenarios, where vertical separation is encoded through mutual-information loss in a directional sector [2210.14875], and in anisotropic SPDC, where the vertical \(y\)-axis component of spatial entanglement is tuned and can even vanish under strong pump asymmetry [2207.03751].

## 1. Terminological scope and conceptual distinctions

The sources associate “vertical-distance entanglement” with three principal constructions: a local-unitary metric on state space, a directional or graph-distance dependence in many-body circuits or emergent geometry, and a literal vertical-axis component of spatial entanglement in photonic systems. These constructions are not equivalent, even though each uses the language of distance and entanglement.

| Context | Defining quantity | Vertical-distance content |
|---|---|---|
| Local-unitary entanglement distance | \(D(\psi)=\mathrm{Tr}\,\tilde g\) or \(D(\rho)\) via convex roof | Distance derived from the Fubini–Study metric restricted to local rotations |
| Emergent spacetime | \(d(A,B)=\min_P\sum_{(p\to q)\in P} w(p,q)\) with \(w(p,q)=l_{RC}\Phi(I(p:q)/I_0)\) | Vertical length recovered by tracking mutual-information loss in the \(z\)-sector |
| Anisotropic SPDC | \(K_y\), \(S_y\), and \(\gamma_y=\sigma_{y-}\Delta k_y\) | Entanglement along the physical vertical \(y\)-direction |

The distinction is substantive. In the Fubini–Study construction, “distance” is a Riemannian quantity on projective Hilbert space restricted to local-unitary orbits. In emergent spacetime, it is a shortest-path graph metric built from mutual information. In SPDC, it is the entanglement carried by one transverse coordinate direction of a biphoton wavefunction. The shared terminology therefore reflects an analogy between entanglement and distance, not a single standardized invariant [1908.03117] [2003.05771] [2210.14875] [2207.03751] [2605.00259].

## 2. Fubini–Study entanglement distance for \(M\)-qubit states

For a normalized pure state \(\ket{\psi}\) of \(M\) qubits, the starting point is the Fubini–Study line element on projective Hilbert space,
\[
ds^2=\langle d\psi|d\psi\rangle-\bigl|\langle\psi|d\psi\rangle\bigr|^2.
\]
Under infinitesimal local rotations,
\[
dU=\prod_{j=1}^M\exp\!\Bigl(-\tfrac{i}{2}\,d\xi^j\,\mathbf n^j\!\cdot\!\boldsymbol\sigma^j\Bigr),
\]
the line element can be written as
\[
ds^2=g_{\mu\nu}(\mathbf v)\,d\xi^\mu d\xi^\nu,
\]
with
\[
g_{\mu\nu}(\mathbf v)=\frac14 \Bigl\langle\psi\Bigl| \bigl(\mathbf v^\mu\!\cdot\!\boldsymbol\sigma^\mu\bigr)\bigl(\mathbf v^\nu\!\cdot\!\boldsymbol\sigma^\nu\bigr)\Bigr|\psi\Bigr\rangle
-\frac14 \Bigl\langle\psi\bigl|\mathbf v^\mu\!\cdot\!\boldsymbol\sigma^\mu\bigl|\psi\Bigr\rangle
\Bigl\langle\psi\bigl|\mathbf v^\nu\!\cdot\!\boldsymbol\sigma^\nu\bigl|\psi\Bigr\rangle.
\]
The unit vectors \(\{\mathbf v^j\}\) are then varied “in measure” to minimize \(\mathrm{Tr}\,g\). The minimizing choice defines the entanglement metric \(\tilde g\), and the entanglement distance is
\[
D(\psi)\equiv \mathrm{Tr}\,\tilde g.
\]

By construction, \(D(\psi)\) depends only on the orbit of \(\ket\psi\) under \(SU(2)^{\otimes M}\). Any local unitary rotates the optimal local directions to another optimal set and leaves \(\mathrm{Tr}\,\tilde g\) unchanged. The measure therefore has local-unitary invariance. It is also designed so that separable states have vanishing entanglement distance: for the product state \(\ket{00}\), the infimum of \(\mathrm{Tr}\,g(\mathbf v)\) is \(0\), hence \(D(00)=0\). For the Bell state \(\ket{\Phi^+}=(\ket{00}+\ket{11})/\sqrt2\), the worked example yields \(D(\Phi^+)=1/2\) [1908.03117].

The operational interpretation given in the qubit setting is geometric. The metric \(\tilde g\) captures the infinitesimal Fubini–Study cost of moving along local-unitary directions, and the trace \(D(\psi)\) is the corresponding minimal line-element density after optimization over local axes. This places entanglement directly in the differential geometry of local state deformations [1908.03117].

## 3. Generalization to arbitrary \(M\)-qudit hybrid systems

The qubit construction extends to hybrid systems with
\[
\mathcal H=\bigotimes_{\mu=0}^{M-1}\mathcal H_{d_\mu},
\]
where local perturbations are generated by \(su(d_\mu)\) generators \(T_{\mu i}\). For a pure state \(\rho=\ket\psi\bra\psi\), one introduces, for each party \(\mu\), the covariance matrix
\[
A_\mu^{ij}=\langle\psi|T_{\mu i}T_{\mu j}|\psi\rangle-\langle\psi|T_{\mu i}|\psi\rangle\langle\psi|T_{\mu j}|\psi\rangle.
\]
The minimum over local directions aligns with the principal axes of \(A_\mu\), and the minimal distance density is
\[
E(\ket\psi):=\sum_{\mu=0}^{M-1}\bigl[\mathrm{tr}\,A_\mu-2(d_\mu-1)\bigr]\ge 0.
\]
Using \(\sum_i T_{\mu i}^2=2(d_\mu^2-1)/d_\mu\), this becomes
\[
D(\ket\psi)\equiv E(\ket\psi)=\sum_{\mu=0}^{M-1}\left[\frac{2(d_\mu-1)}{d_\mu}-\sum_{i=1}^{d_\mu^2-1}\langle\psi|T_{\mu i}|\psi\rangle^2\right].
\]

For mixed states, the construction is extended by imposing four conditions: \(D(\rho)\ge 0\), vanishing on separable \(\rho\), convexity, and local-unitary invariance. The resulting extension is the convex roof,
\[
D(\rho)=\min_{\{p_j,\ket{\psi_j}\}:\,\rho=\sum_j p_j\ket{\psi_j}\bra{\psi_j}}
\sum_j p_j\,D(\ket{\psi_j}).
\]
The source states that this is the tightest extension satisfying the desiderata and that no simpler closed form exists in general, although the metric-tensor picture can be used to search for optimal decompositions.

In the specific case of \(M\)-qubit systems, the measure also acquires a direct geometric interpretation. Writing the metric tensor along local-unitary orbits as
\[
G_{\mu\nu}(\psi)=\delta_{\mu\nu}\bigl[1-\langle\psi|\sigma_\mu\!\cdot\! n_\mu|\psi\rangle^2\bigr]
\]
after minimization over \(n_\mu\in S^2\), one has
\[
ds^2=d\xi^\mu G_{\mu\nu}(\psi)\,d\xi^\nu \ge D(\ket\psi)\,\|\mathrm d\vec\xi\|^2.
\]
In this sense, \(D\) is the minimal “obstacle” to infinitesimal displacements in local-unitary directions. The paper treats Briegel–Raussendorf graph-states, GHZ-like states, hybrid qubit–qutrit states, two-qutrit generalizations, and three-qubit states with two parameters, using \(D/M\) and the spectrum of \(G\) to compare entanglement [2003.05771].

## 4. Spectral structure, robustness, and variational propagation

The metric viewpoint refines entanglement quantification through the eigenvalues of the entanglement metric. If \(\tilde g\) or \(G(\psi)\) has eigenvalues \(\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_M\ge 0\), then
\[
\mathrm{tr}\,G=\sum_i \lambda_i=D(\ket\psi).
\]
Directions with larger \(\lambda_i\) require a proportionally larger Fubini–Study distance per unit change in the local parameters. The source further states that a state is “more robust” to small mixing if all \(\lambda_i\) are large, whereas if some \(\lambda_i\approx 0\) there is a direction in which entanglement can be destroyed with negligible Fubini–Study cost. In the qubit analysis, GHZ-like states can have only one nonzero eigenvalue, while Briegel–Raussendorf states can have all \(M\) eigenvalues nonzero, which gives a multi-directional notion of rigidity [1908.03117] [2003.05771].

This spectral viewpoint is carried into variational quantum circuits in the study of \(R_Y\)-rotation and \(CZ\)-entangler ansätze. For a pure two-qubit state, the entanglement distance is written directly in terms of the reduced Bloch vector,
\[
E^{ED}=1-\langle X\rangle^2-\langle Y\rangle^2-\langle Z\rangle^2.
\]
For a target qubit \(a\) in a multi-qubit chain,
\[
E_a^{ED}=1-\langle X_a\rangle^2-\langle Y_a\rangle^2-\langle Z_a\rangle^2,
\]
and, for \(R_Y\)–\(CZ\) circuits on \(\ket{0\cdots 0}\),
\[
E_a^{ED}=1-\langle X_a\rangle^2-\langle Z_a\rangle^2
\]
because \(\langle Y_a\rangle=0\). In a closed one-dimensional ring, the one-layer expressions are
\[
\langle Z_a\rangle_1=\cos\theta_a^{(1)},\qquad
\langle X_a\rangle_1=\sin\theta_a^{(1)}\cos\theta_{a-1}^{(1)}\cos\theta_{a+1}^{(1)},\qquad
\langle Y_a\rangle_1=0.
\]
After two layers, the exact closed form for \(\langle X_a\rangle_2\) and \(\langle Z_a\rangle_2\) explicitly involves sites \(a\pm1\) and \(a\pm2\), and
\[
E_a^{ED}(n=2)=1-\langle X_a\rangle_2^2-\langle Z_a\rangle_2^2.
\]

The key structural claim is a light-cone relation between circuit depth \(p\) and graph-distance \(d\): \(p=1\) involves only nearest neighbors, \(p=2\) brings in next-nearest neighbors, and in general \(p\) layers entangle qubits out to graph-distance \(d=p\). In that setting, the source explicitly proposes “vertical-distance entanglement” as the dependence of \(E_a^{ED}\) on rotation angles at sites \(a\pm d\), together with a compact notation
\[
D(d,p)\equiv \text{“portion of }E^{ED}\text{ contributed by sites at distance }d\text{ for a depth-}p\text{ circuit.”}
\]
The 2026 study states that the analytic results agree with numerical simulations performed using quantum programming tools [2605.00259].

## 5. Emergent spacetime and vertical separation

A different notion of distance arises in emergent-spacetime scenarios, where one uses the von Neumann mutual information
\[
I(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{AB})
\]
to assign edge weights in a graph of Hilbert-space factors:
\[
w(p,q)=l_{RC}\,\Phi\!\left(\frac{I(p:q)}{I_0}\right),
\qquad I_0:=\max_{p,q} I(p:q).
\]
With the convenient choice \(\Phi(x)=-\log x\), the emergent distance is the shortest-path graph metric,
\[
d(A,B)=\min_{P}\sum_{(p\to q)\in P} w(p,q),
\]
and, in the two-vertex case,
\[
d(A,B)\simeq l_{RC}\bigl[-\log(I(A:B)/I_0)\bigr].
\]
In this framework, more mutual information implies smaller emergent distance, and if \(I(A:B)=I_0\) then \(d=0\) [2210.14875].

The paper addresses the apparent difficulty posed by a maximally entangled Bell pair that is physically separated while preserving spin entanglement. The resolution is that actual quantum systems contain multiple sectors of independent degrees of freedom. For a Bell pair, the full Hilbert space is taken as
\[
\mathcal H_A=\mathcal H_A^s\otimes\mathcal H_A^p,\qquad
\mathcal H_B=\mathcal H_B^s\otimes\mathcal H_B^p,
\]
with a momentum-uncertainty state
\[
|\Delta\pi\rangle=\sum_{n=1}^{N-1}\alpha_n\,|n,N-n\rangle.
\]
If \(|\alpha_n|^2=1/(N-1)\), then
\[
I^p(A:B)=2\log(N-1),
\]
which, for \(N\gg 2\), exceeds the spin-sector mutual information \(2\log 2\). Since
\[
I(A:B)=I^s(A:B)+I^p(A:B),
\]
the total distance is initially dominated by the momentum sector. As environmental interactions decohere successive momentum modes, \(I^p(A:B)\) falls and the emergent distance grows, even though the spin state remains maximally entangled.

Vertical separation is treated as a directional instance of the same mechanism. Because the construction assumes no preferred orientation, one can separate particles vertically and partition the momentum sector into \(x\), \(y\), and \(z\) degrees of freedom, then track mutual-information loss in the \(z\)-sector to recover vertical distance. The authors also identify several caveats: the exact form of \(\Phi(I/I_0)\) is not derived from first principles, the construction depends on the chosen tensor-product structure, finite-dimensional regulators are required so that entropy and mutual information remain finite, and the environment coupling must be designed carefully to probe specific momentum-band losses. The suggested laboratory tests include large ensembles of hyperpolarized spins probed with an NMR-based “ruler” and quantum-interferometry setups such as MAGIS-100 or related cold-atom gravitational-wave detectors [2210.14875].

## 6. Vertical spatial entanglement in anisotropic SPDC

In anisotropic spatial entanglement, the vertical direction is a physical transverse axis. For collinear Type-I SPDC in the thin-crystal, paraxial, Gaussian-pump approximation, the two-photon wavefunction along \(y\) is
\[
\Psi(y_s,y_i)=N\,
\exp\!\biggl[-\frac{(y_s+y_i)^2}{4\sigma_{y+}^2}\biggr]\,
\exp\!\biggl[-\frac{(y_s-y_i)^2}{4\sigma_{y-}^2}\biggr].
\]
Here \(\sigma_{y+}\) is the r.m.s. width of the sum coordinate \((y_s+y_i)\), inherited from the pump’s transverse profile and, for a partially coherent pump, its coherence length; \(\sigma_{y-}\) is the r.m.s. width of the difference coordinate \((y_s-y_i)\), set by the crystal thickness \(L\) and pump wavelength \(\lambda_p\). The position-difference width is
\[
\Delta(y_i\!\mid y_s)=\sigma_{y-}=\sqrt{\frac{\alpha L\lambda_p}{2\pi}},
\qquad \alpha\approx 0.455,
\]
while in the far field the momentum conditional width satisfies
\[
\Delta k_y\equiv \Delta(k_{y_i}\!\mid k_{y_s})=\frac{1}{2\sigma_{y+}},
\qquad
\frac{1}{\sigma_{y+}^2}=\frac{1}{\omega_y^2}+\frac{2}{\ell_c^2}
\]
for a Gaussian-Schell pump with coherence length \(\ell_c\) [2207.03751].

The bi-Gaussian form is diagonalized by a Hermite–Gaussian Schmidt basis. Writing
\[
r\equiv \frac{\sigma_{y-}}{\sigma_{y+}},\qquad
\xi=\frac{r-1}{r+1},
\]
the Schmidt eigenvalues are
\[
\lambda_n=(1-\xi)\xi^n,\qquad n=0,1,2,\dots,
\]
and the vertical Schmidt number is
\[
K_y=\frac{1}{\sum_n \lambda_n^2}
=\frac{1+\xi^2}{1-\xi^2}
=\frac{r^2+1}{2r}
=\frac{\sigma_{y-}^2+\sigma_{y+}^2}{2\sigma_{y-}\sigma_{y+}}.
\]
The associated entanglement entropy is
\[
S_y=-\sum_n\lambda_n\ln\lambda_n
=-\ln(1-\xi)-\frac{\xi}{1-\xi}\ln\xi.
\]

Directional anisotropy is controlled by the pump-beam asymmetry factor
\[
\beta=\frac{\omega_y}{\omega_x}.
\]
For \(\beta=1\), entanglement is isotropic and \(K_x=K_y\). As \(\beta\) decreases, \(\omega_y\) shrinks, \(\sigma_{y+}\propto 1/\omega_y\) grows, and the product
\[
\gamma_y=\Delta(y_i\!\mid y_s)\,\Delta(k_{y_i}\!\mid k_{y_s})
=\sigma_{y-}\frac{1}{2\sigma_{y+}}
\]
increases until it crosses the separability bound
\[
\gamma_y\ge \frac{\hbar}{2},
\]
at which point there is no EPR-like entanglement along \(y\). The source reports that below a critical \(\beta_c\sim 0.1\)–\(0.2\), depending on \(L\) and \(\lambda_p\), the vertical EPR correlation “dies,” while \(K_x\) remains large. In this setting, vertical-distance entanglement is not an abstract state-space metric but the experimentally controllable entanglement content of the \(y\)-direction itself. Maximizing it requires large \(\omega_y\), a spatially coherent pump with large \(\ell_c\), and crystal parameters that keep \(\gamma_y<\hbar/2\); suppressing it requires shrinking \(\omega_y\), decreasing \(\ell_c\), or increasing \(L\) so that \(\gamma_y\) rises above the bound [2207.03751].

Source: https://www.emergentmind.com/topics/vertical-distance-entanglement