Vertical-Distance Entanglement: Metrics & Methods

• The paper introduces three distinct constructions—Fubini–Study metrics, graph-distance via mutual information, and anisotropic SPDC—to quantify vertical-distance entanglement.
• It details a methodology for measuring entanglement in M-qubit and M-qudit systems by optimizing local-unitary rotations and leveraging differential geometry.
• The analysis extends to variational quantum circuits and emergent-spacetime scenarios, offering practical insights into robust, direction-sensitive entanglement measures.

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 (Cocchiarella et al., 2019, Cocchiarella et al., 2020). 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 (Gnatenko et al., 30 Apr 2026). Separate uses arise in emergent-spacetime scenarios, where vertical separation is encoded through mutual-information loss in a directional sector (Franzmann et al., 2022), and in anisotropic SPDC, where the vertical $y$-axis component of spatial entanglement is tuned and can even vanish under strong pump asymmetry (Patil et al., 2022).

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 $M$0-sector
Anisotropic SPDC	$M$1, $M$2, and $M$3	Entanglement along the physical vertical $M$4-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 (Cocchiarella et al., 2019, Cocchiarella et al., 2020, Franzmann et al., 2022, Patil et al., 2022, Gnatenko et al., 30 Apr 2026).

2. Fubini–Study entanglement distance for $M$5-qubit states

For a normalized pure state $M$6 of $M$7 qubits, the starting point is the Fubini–Study line element on projective Hilbert space,

$M$8

Under infinitesimal local rotations,

$M$9

the line element can be written as

$d$0

with

$d$1

The unit vectors $d$2 are then varied “in measure” to minimize $d$3. The minimizing choice defines the entanglement metric $d$4, and the entanglement distance is

$d$5

By construction, $d$6 depends only on the orbit of $d$7 under $d$8. Any local unitary rotates the optimal local directions to another optimal set and leaves $d$9 unchanged. The measure therefore has local-unitary invariance. It is also designed so that separable states have vanishing entanglement distance: for the product state $R_Y$0, the infimum of $R_Y$1 is $R_Y$2, hence $R_Y$3. For the Bell state $R_Y$4, the worked example yields $R_Y$5 (Cocchiarella et al., 2019).

The operational interpretation given in the qubit setting is geometric. The metric $R_Y$6 captures the infinitesimal Fubini–Study cost of moving along local-unitary directions, and the trace $R_Y$7 is the corresponding minimal line-element density after optimization over local axes. This places entanglement directly in the differential geometry of local state deformations (Cocchiarella et al., 2019).

3. Generalization to arbitrary $R_Y$8-qudit hybrid systems

The qubit construction extends to hybrid systems with

$R_Y$9

where local perturbations are generated by $CZ$0 generators $CZ$1. For a pure state $CZ$2, one introduces, for each party $CZ$3, the covariance matrix

$CZ$4

The minimum over local directions aligns with the principal axes of $CZ$5, and the minimal distance density is

$CZ$6

Using $CZ$7, this becomes

$CZ$8

For mixed states, the construction is extended by imposing four conditions: $CZ$9, vanishing on separable $y$0, convexity, and local-unitary invariance. The resulting extension is the convex roof,

$y$1

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 $y$2-qubit systems, the measure also acquires a direct geometric interpretation. Writing the metric tensor along local-unitary orbits as

$y$3

after minimization over $y$4, one has

$y$5

In this sense, $y$6 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 $y$7 and the spectrum of $y$8 to compare entanglement (Cocchiarella et al., 2020).

4. Spectral structure, robustness, and variational propagation

The metric viewpoint refines entanglement quantification through the eigenvalues of the entanglement metric. If $y$9 or $D(\psi)=\mathrm{Tr}\,\tilde g$0 has eigenvalues $D(\psi)=\mathrm{Tr}\,\tilde g$1, then

$D(\psi)=\mathrm{Tr}\,\tilde g$2

Directions with larger $D(\psi)=\mathrm{Tr}\,\tilde g$3 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 $D(\psi)=\mathrm{Tr}\,\tilde g$4 are large, whereas if some $D(\psi)=\mathrm{Tr}\,\tilde g$5 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 $D(\psi)=\mathrm{Tr}\,\tilde g$6 eigenvalues nonzero, which gives a multi-directional notion of rigidity (Cocchiarella et al., 2019, Cocchiarella et al., 2020).

This spectral viewpoint is carried into variational quantum circuits in the study of $D(\psi)=\mathrm{Tr}\,\tilde g$7-rotation and $D(\psi)=\mathrm{Tr}\,\tilde g$8-entangler ansätze. For a pure two-qubit state, the entanglement distance is written directly in terms of the reduced Bloch vector,

$D(\psi)=\mathrm{Tr}\,\tilde g$9

For a target qubit $D(\rho)$0 in a multi-qubit chain,

$D(\rho)$1

and, for $D(\rho)$2–$D(\rho)$3 circuits on $D(\rho)$4,

$D(\rho)$5

because $D(\rho)$6. In a closed one-dimensional ring, the one-layer expressions are

$D(\rho)$7

After two layers, the exact closed form for $D(\rho)$8 and $D(\rho)$9 explicitly involves sites $d(A,B)=\min_P\sum_{(p\to q)\in P} w(p,q)$0 and $d(A,B)=\min_P\sum_{(p\to q)\in P} w(p,q)$1, and

$d(A,B)=\min_P\sum_{(p\to q)\in P} w(p,q)$2

The key structural claim is a light-cone relation between circuit depth $d(A,B)=\min_P\sum_{(p\to q)\in P} w(p,q)$3 and graph-distance $d(A,B)=\min_P\sum_{(p\to q)\in P} w(p,q)$4: $d(A,B)=\min_P\sum_{(p\to q)\in P} w(p,q)$5 involves only nearest neighbors, $d(A,B)=\min_P\sum_{(p\to q)\in P} w(p,q)$6 brings in next-nearest neighbors, and in general $d(A,B)=\min_P\sum_{(p\to q)\in P} w(p,q)$7 layers entangle qubits out to graph-distance $d(A,B)=\min_P\sum_{(p\to q)\in P} w(p,q)$8. In that setting, the source explicitly proposes “vertical-distance entanglement” as the dependence of $d(A,B)=\min_P\sum_{(p\to q)\in P} w(p,q)$9 on rotation angles at sites $w(p,q)=l_{RC}\Phi(I(p:q)/I_0)$0, together with a compact notation

$w(p,q)=l_{RC}\Phi(I(p:q)/I_0)$1

The 2026 study states that the analytic results agree with numerical simulations performed using quantum programming tools (Gnatenko et al., 30 Apr 2026).

5. Emergent spacetime and vertical separation

A different notion of distance arises in emergent-spacetime scenarios, where one uses the von Neumann mutual information

$w(p,q)=l_{RC}\Phi(I(p:q)/I_0)$2

to assign edge weights in a graph of Hilbert-space factors: $w(p,q)=l_{RC}\Phi(I(p:q)/I_0)$3 With the convenient choice $w(p,q)=l_{RC}\Phi(I(p:q)/I_0)$4, the emergent distance is the shortest-path graph metric,

$w(p,q)=l_{RC}\Phi(I(p:q)/I_0)$5

and, in the two-vertex case,

$w(p,q)=l_{RC}\Phi(I(p:q)/I_0)$6

In this framework, more mutual information implies smaller emergent distance, and if $w(p,q)=l_{RC}\Phi(I(p:q)/I_0)$7 then $w(p,q)=l_{RC}\Phi(I(p:q)/I_0)$8 (Franzmann et al., 2022).

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

$w(p,q)=l_{RC}\Phi(I(p:q)/I_0)$9

with a momentum-uncertainty state

$M$00

If $M$01, then

$M$02

which, for $M$03, exceeds the spin-sector mutual information $M$04. Since

$M$05

the total distance is initially dominated by the momentum sector. As environmental interactions decohere successive momentum modes, $M$06 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 $M$07, $M$08, and $M$09 degrees of freedom, then track mutual-information loss in the $M$10-sector to recover vertical distance. The authors also identify several caveats: the exact form of $M$11 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 (Franzmann et al., 2022).

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 $M$12 is

$M$13

Here $M$14 is the r.m.s. width of the sum coordinate $M$15, inherited from the pump’s transverse profile and, for a partially coherent pump, its coherence length; $M$16 is the r.m.s. width of the difference coordinate $M$17, set by the crystal thickness $M$18 and pump wavelength $M$19. The position-difference width is

$M$20

while in the far field the momentum conditional width satisfies

$M$21

for a Gaussian-Schell pump with coherence length $M$22 (Patil et al., 2022).

The bi-Gaussian form is diagonalized by a Hermite–Gaussian Schmidt basis. Writing

$M$23

the Schmidt eigenvalues are

$M$24

and the vertical Schmidt number is

$M$25

The associated entanglement entropy is

$M$26

Directional anisotropy is controlled by the pump-beam asymmetry factor

$M$27

For $M$28, entanglement is isotropic and $M$29. As $M$30 decreases, $M$31 shrinks, $M$32 grows, and the product

$M$33

increases until it crosses the separability bound

$M$34

at which point there is no EPR-like entanglement along $M$35. The source reports that below a critical $M$36–$M$37, depending on $M$38 and $M$39, the vertical EPR correlation “dies,” while $M$40 remains large. In this setting, vertical-distance entanglement is not an abstract state-space metric but the experimentally controllable entanglement content of the $M$41-direction itself. Maximizing it requires large $M$42, a spatially coherent pump with large $M$43, and crystal parameters that keep $M$44; suppressing it requires shrinking $M$45, decreasing $M$46, or increasing $M$47 so that $M$48 rises above the bound (Patil et al., 2022).