4.1 Entanglement and Geometry: Spacetime Distance as Quantum Mutual Information

In continuous spacetime, “distance” $d(x,y)$ is a fundamental concept given by metric tensor $g_{\mu \nu }$. But in a discrete quantum network, what does “physical distance” between two nodes $x$ and $y$ mean?

If two nodes are far apart on the lattice graph (large graph distance), but have extremely strong quantum entanglement (Bell states) between them, then in the sense of quantum information, they are actually “together.”

4.1.1 Microscopic Revelation of Ryu-Takayanagi Formula

One of the most profound discoveries in the holographic principle and AdS/CFT correspondence is the Ryu-Takayanagi (RT) formula:

$$S_A=\frac{\text{Area}({\gamma }_A)}{4G}$$

It states that entanglement entropy $S_A$ on the boundary is equivalent to minimal surface area $\text{Area}({\gamma }_A)$ in the bulk. This suggests: Entanglement is the glue connecting spacetime.

If you cut entanglement between two regions, spacetime breaks. Conversely, if you increase entanglement between two regions, they are pulled closer geometrically.

4.1.2 Definition: Information Distance

In the QCA framework, we abandon the traditional “meter stick” definition and instead use quantum mutual information as a measure of distance.

Let ${\rho }_x$ and ${\rho }_y$ be the reduced density matrices of nodes $x$ and $y$ respectively, and ${\rho }_{xy}$ be their joint density matrix. Mutual information is defined as:

$$I(x:y)=S({\rho }_x)+S({\rho }_y)-S({\rho }_{xy})$$

where $S(\rho )=-\text{Tr}(\rho \ln \rho )$ is von Neumann entropy.

Mutual information measures the strength of total correlation (classical + quantum) between two systems. For QCA networks in ground state or vacuum state, mutual information decays exponentially with graph distance $d$ (due to locality axiom):

$$I(x:y)\sim e^{-d(x,y)/\xi }$$

where $\xi$ is the correlation length.

We now reverse this and use mutual information to define emergent geometric distance $D(x,y)$:

$$D(x,y)\equiv -\xi \ln \left(\frac{I(x:y)}{I_{max}}\right)$$

The physical meaning of this definition is extremely profound:

1. Stronger Entanglement, Closer Distance: When $I(x:y)\to I_{max}$ (maximum entanglement), $D\to 0$. These two points are geometrically coincident (or connected via wormhole).

2. Entanglement Vanishes, Distance Infinite: When $I(x:y)\to 0$, $D\to \infty$. These two points belong to disconnected universes.

4.1.3 Geometric Reconstruction: From Network to Manifold

With distance $D(x,y)$, we can use multidimensional scaling (MDS) to embed the discrete QCA network into a smooth Riemannian manifold $\mathcal{M}$.

If QCA is in ground state (Vacuum State), entanglement distribution across the entire network is uniform. The manifold reconstructed from this is flat Minkowski space. This explains why we see flat spacetime when there is no matter.

However, when matter excitations exist in the network, matter carries additional information and interferes with surrounding entanglement structure.

• Matter = Local Entanglement Destruction/Reorganization. A particle may be a highly entangled “knot” that consumes surrounding entanglement resources.

• If mutual information $I(A:B)$ between regions $A$ and $B$ decreases due to matter inserted in between (entanglement is “screened” or “crowded out” by matter), according to formula $D\sim -\ln I$, their physical distance $D$ increases.

This is the origin of gravity:

Matter reduces vacuum entanglement, causing space to be “stretched” or “expanded.”

This non-uniform stretching of space manifests macroscopically as curvature. Light rays bend when passing massive objects, not because light is attracted, but because light must pass through a spatial region with “lower entanglement, causing longer effective path.”

4.1.4 Summary

This section establishes a revolutionary concept:

• Geometry is Entanglement.

• Spacetime is no longer a stage, but a holographic projection of qubit correlations.

• Gravitational potential $\Phi$ is essentially a logarithmic function of mutual information $I$.

In the next section, we will quantify this qualitative picture by introducing “local information volume conservation” to derive that mysterious metric deformation formula.