Vacuum Entanglement
The bond coordinate \(D_{\mathrm{KL}}\), the cube cell \(Q_3\), and the per-site value \(S^{*} = 2/9\).
The bond coordinate
An elementary bond of the entanglement network is identified with one Planck area of horizon. The Bekenstein–Hawking area law \(S_{\mathrm{BH}} = A/(4\ell_P^{2})\) assigns each Planck area the entropy of one quarter of a nat, fixing the maximum entanglement entropy carried by one bond at
\[ \log\chi \;=\; \frac{S_{\mathrm{BH}}}{A}\ell_P^{2} \;=\; \tfrac14. \]
Each bond \((i,j)\) of the network carries a scalar coordinate, the Umegaki relative entropy
\[ D_{\mathrm{KL}}(\rho_{ij}\|\rho^{0}_{ij}) \;=\; \mathrm{tr}\bigl(\rho_{ij}\log\rho_{ij}-\rho_{ij}\log\rho^{0}_{ij}\bigr), \]
the reduced bond state \(\rho_{ij}\) measured against a reference \(\rho^{0}_{ij}\). The structured MERA vacuum \(\rho^{0}_{ij}\) has non-flat spectrum and per-site von Neumann entropy \(S^{*}\) (with \(0 < S^{*} < \log\chi\)); it is the reference of the dynamical coordinate, and \(D_{\mathrm{KL}}(\rho_{ij}\|\rho^{0}_{ij})\) vanishes exactly at the vacuum. The maximally mixed state \(\sigma^{0}_{ij} = \mathbb{I}_{\chi^{2}}/\chi^{2}\) gives \(D_{\mathrm{KL}}(\rho\|\sigma^{0}) = \log\chi - S(\rho)\), the entropy deficit.
The four conditions
Within the Rényi family \(D_{\alpha}(\rho\|\sigma) = \tfrac{1}{\alpha-1}\log\mathrm{tr}(\rho^{\alpha}\sigma^{1-\alpha})\), the bond coordinate is the \(\alpha\to 1\) member. It is the unique member of that family satisfying all four of:
- Positive semi-definite: \(D_{\mathrm{KL}}\geq 0\), with equality iff \(\rho_{ij}=\rho^{0}_{ij}\) (Klein's inequality).
- Thermodynamic: \(\delta D_{\mathrm{KL}} = \delta\langle K\rangle - \delta S\) with \(K = -\log\rho^{0}\) the modular Hamiltonian (quantum first law of entanglement).
- Geometric: \(\delta D_{\mathrm{KL}} = \delta A_{\min}/4\ell_P^{2}\) (the linearised Ryu–Takayanagi first law).
- Bounded: \(D_{\mathrm{KL}}\leq\log\chi\), the Bekenstein–Hawking bond capacity.
Harmonic oscillator
Each bond carries the quadratic potential \(V_{ij} = \tfrac12\kappa\,D_{\mathrm{KL}}^{2}\) at leading order. The harmonic form is derived, not assumed: the second-order expansion of \(D_{\mathrm{KL}}(\rho_{ij}\|\rho^{0}_{ij})\) is the Kubo–Mori form
\[ D_{\mathrm{KL}}(\rho_{ij}\|\rho^{0}_{ij}) \;=\; \frac{\epsilon^{2}}{2} \sum_{m,n} c_{mn}|\delta\rho_{mn}|^{2} + O(\epsilon^{3}), \qquad c_{mn} = \frac{\log\lambda_m-\log\lambda_n}{\lambda_m-\lambda_n} > 0, \]
with \(\{\lambda_m\}\) the spectrum of \(\rho^{0}_{ij}\). The Hessian at the vacuum is the positive-definite Kubo–Mori metric, so the bond minimum is at \(D_{\mathrm{KL}} = 0\): the bond is harmonic at leading order.
Partition function
With \(\log\chi = 1/4\) and \(d = 3\) as inputs, the partition function counts the microstates of one coarse-graining layer.
A layer has \(N\) input sites and one output site, total \(N+1\). Each site is an independent Hilbert space of dimension \(\chi\). The free partition function is
\[ Z_{\mathrm{free}} \;=\; \chi^{N+1}, \qquad F_{\mathrm{free}} \;=\; -(N+1)\log\chi. \]
The coarse-graining map \(w:\mathcal{H}^{\otimes N}\to\mathcal{H}\) satisfies \(w^{\dagger}w = \mathbb{I}\). The output is deterministic in the inputs: \(S(\mathrm{output}\mid i_1,\ldots,i_N) = 0\). The effective site count drops from \(N+1\) to \(N\):
\[ Z \;=\; \chi^{N}, \qquad F \;=\; -N\log\chi. \]
The per-site entropy is
\[ s \;=\; -\frac{\partial}{\partial\beta}\frac{F}{N+1}\bigg|_{\beta=1} \;=\; \frac{N\log\chi}{N+1}. \]
With \(d = 3\) fixed, the simplest geometry is the three-cube \(Q_3\): 8 vertices, 12 edges, 6 square faces. The 8 vertices are the input sites of one layer; the centre is the output site, so \(N = 8\). By \(B_3\) symmetry of \(Q_3\) all input sites carry equal entanglement; by the RG fixed-point property the output of one layer is an input of the next, so it carries the same per-site value. Substituting \(N = 8\) and \(\log\chi = 1/4\),
\[ S^{*} \;=\; s \;=\; \frac{N\log\chi}{N+1} \;=\; \frac{8 \cdot \tfrac14}{9} \;=\; \tfrac{2}{9}. \]
Eigenvalue scan and selection of MERA
The multi-scale entanglement renormalization ansatz (MERA) is a network of bonds in which short-range entanglement between neighbouring sites is factored out step by step (disentangled), with sites grouped together at each step into a coarser layer. Each MERA layer takes the \(N\) input sites of the defined unit cell — the vertices of the cell, connected pairwise by bonds — factors out their short-range entanglement via disentanglers on those bonds, then coarse-grains the cell isometrically (\(w^{\dagger}w = \mathbb{I}\)) to one output site. The branching factor \(b\) is the linear scale of this coarse-graining and \(n\) the cell's spatial dimension. With \(b = 2\) (binary MERA) and \(n = 3\) (observed spatial dimension), the cell is \(Q_3\) with \(N = 2^{3} = 8\) vertices.
The disentangling step at each layer is the mechanism HET proposes for generating the force carriers of that scale — gauge fields on the edges of the cell, and the graviton on the faces.
The binary MERA on \(Q_3\) is selected through a scan of eigenvalues across candidate tensor-network structures on the cube. With \(S^{*} = 2/9\) from the partition function and the PDG-derived strong coupling \(g_s^{2}(m_P) \approx 0.23613\) run up to the Planck scale, the dimensionless ratio is
\[ \frac{g_s^{2}(m_P)}{S^{*}} \;=\; \frac{0.23613}{2/9} \;\approx\; 1.0626 \;=\; \frac{17}{16} \;=\; 1 + (\log\chi)^{2} \quad(\text{to within } 10^{-4}), \]
tying the ratio to the Bekenstein–Hawking ceiling \(\log\chi = 1/4\). Multiplying through by \(S^{*}\) gives the central identity
\[ \langle D_{\mathrm{KL}}\rangle_{\square} \;=\; S^{*}\bigl(1+(\log\chi)^{2}\bigr) \;=\; \tfrac{2}{9}\cdot\tfrac{17}{16} \;=\; \tfrac{17}{72}. \]
\(\langle D_{\mathrm{KL}}\rangle_{\square}\) is the plaquette-averaged committed entanglement: the relative-entropy deformation of the bond density matrices from the maximally mixed reference, averaged over the four edges of a square face of \(Q_3\) at the Bekenstein–Hawking-saturated fixed point. It measures how far the structured entanglement of the cube's vacuum sits from the unstructured reference on each plaquette — the committed entanglement carried per face of the cell.