Consistency Conditions for an AdS/MERA Correspondence

TL;DR

The paper critically tests the viability of an AdS/MERA correspondence by deriving geometric and entropic consistency conditions. It shows that a conventional MERA can only describe geometry at or above the AdS radius and cannot simultaneously reproduce bulk physics and entropy bounds; even with the Ryu–Takayanagi framework, the central charge bound demands an exponentially large bond dimension χ for large c. By applying the Bousso bound, the authors demonstrate that no standard MERA can satisfy all holographic constraints, though generalized tensor networks with entangled ancillae or alternative architectures may salvage the correspondence for certain $k$ values. The work also connects these bounds to BTZ/thermal states and discusses broader implications for holographic tensor networks beyond MERA-based constructions.

Abstract

The Multi-scale Entanglement Renormalization Ansatz (MERA) is a tensor network that provides an efficient way of variationally estimating the ground state of a critical quantum system. The network geometry resembles a discretization of spatial slices of an AdS spacetime and "geodesics" in the MERA reproduce the Ryu-Takayanagi formula for the entanglement entropy of a boundary region in terms of bulk properties. It has therefore been suggested that there could be an AdS/MERA correspondence, relating states in the Hilbert space of the boundary quantum system to ones defined on the bulk lattice. Here we investigate this proposal and derive necessary conditions for it to apply, using geometric features and entropy inequalities that we expect to hold in the bulk. We show that, perhaps unsurprisingly, the MERA lattice can only describe physics on length scales larger than the AdS radius. Further, using the covariant entropy bound in the bulk, we show that there are no conventional MERA parameters that completely reproduce bulk physics even on super-AdS scales. We suggest modifications or generalizations of this kind of tensor network that may be able to provide a more robust correspondence.

Figures (9)

• Figure 1: (a) Basic construction of a $k=2$ MERA (2 sites renormalized to 1). (b) The squares represent disentanglers: unitary maps that, from the moving-upward perspective, remove entanglement between two adjacent sites. (c) The triangles represent isometries: linear maps that, again from the moving-upward perspective, coarse-grain two sites into one. Moving downward, we may think of isometries as unitary operators that, in the MERA, map a state in $V \otimes |0\rangle$ into $V \otimes V$. The $i$ and $j$ labels in (b) and (c) represent the tensor indices of the disentangler and isometry.
• Figure 2: (a) A $k=2$ MERA, and (b) the same MERA with its disentanglers and isometries suppressed. The horizontal lines in the graph on the right indicate lattice connectivity at different renormalization depths, and the vertical lines indicate which sites at different depths are related via coarse-graining due to the isometries. Each site, represented by a circle, is associated with a Hilbert space $V$ with bond dimension $\chi$. In the simplest case, a copy of the same Hilbert space is located at each site. When assigning a metric to the graph on the right, translation and scale invariance dictate that there are only two possible length scales: a horizontal proper length $L_1$ and a vertical proper length $L_2$.
• Figure 3: A horizontal line ($\gamma_1$) and a geodesic ($\gamma_2$) in a spatial slice of $\mathrm{AdS}_3$.
• Figure 4: Causal cone (shaded) for a set of $\ell_0 = 6$ sites in a MERA with $k=2$. The width $\ell_m$ of the causal cone at depth $m$ is $\ell_1 = 4$, $\ell_2 = 3$, $\ell_3 = 3$, $\ell_4 = 3$, etc. The crossover scale for this causal cone occurs at $\bar{m} = 2$. Between the zeroth and first layer, $n_1^\mathrm{tr} = 2$ bonds are cut by the causal cone. Similarly, $n_2^\mathrm{tr} = 2$, $n_3^\mathrm{tr} = 3$, etc.
• Figure 5: A pair of isometries with their ancillae explicitly indicated for a MERA with (a) $k=2$ and (b) general $k$. The thick bonds below the isometries, the state of which is denoted by $\rho_1$, are unitarily related to the bonds that exit the isometries and the ancillae, the state of which is denoted by $\rho_2$.