Overlapping qubits from non-isometric maps and de Sitter tensor networks

TL;DR

This work addresses the tension between local effective field theory locality and the finite information capacity suggested by quantum gravity and holography. It introduces overlapping qubits built from non-isometric maps $V:\mathcal{H}_N\to\mathcal{H}_n$, enabling $N>n$ qubits to be approximately realized within $\mathcal{H}_n$ while preserving an approximate algebra of observables; locality becomes state-dependent through the commutator defect $T_p^{[Q,Q']}$. The authors develop a concrete framework, including conditions for spoofing via $P=V^{\dagger}V$ and an analysis of $T^{[Q,Q']}$, and illustrate the construction with de Sitter MERA tensor networks in global and local variants. They show that de Sitter expansion can be mimicked by overlapping qubits with a fundamental Hilbert space of size $\dim\mathcal{H}_n\sim O(e^{S_{dS}})$, preserving approximate locality up to a timescale $T\sim S_{dS}$, and that horizon entanglement in the local model approaches maximality, linking entanglement structure to gravitational constraints. Overall, the work provides a concrete bridge between quantum information constraints and gravitational holography, offering new perspectives on locality, gravity dressing, and the emergence of spacetime in cosmological settings.

Abstract

We construct approximately local observables, or "overlapping qubits", using non-isometric maps and show that processes in local effective theories can be spoofed with a quantum system with fewer degrees of freedom, similar to our expectation in holography. Furthermore, the spoofed system naturally deviates from an actual local theory in ways that can be identified with features in quantum gravity. For a concrete example, we construct two MERA toy models of de Sitter space-time and explain how the exponential expansion in global de Sitter can be spoofed with many fewer quantum degrees of freedom and that local physics may be approximately preserved for an exceedingly long time before breaking down. We highlight how approximate overlapping qubits are conceptually connected to Hilbert space dimension verification, degree-of-freedom counting in black holes and holography, and approximate locality in quantum gravity.

Figures (15)

• Figure 1: Modifications to the Hilbert space of effective field theory (EFT) by quantum gravity. (a) Two EFT operators $\mathcal{O}_j$ and $\mathcal{O}_k$ will, as a result of gravity, lose exact commutativity even on the perturbative level, introducing small non-locality (dashed line). (b) Any operator $\mathcal{O}_\text{BH}$ producing a black hole state restricts the semi-classical EFT background, leading to a Hilbert space truncation.
• Figure 2: The non-isometric map $V$ maps nominally exact qubits (circles) in $\mathcal{H}_N$ onto approximate overlapping qubits (jagged circles) in a lower-dim. $\mathcal{H}_n$ where $\{X_p,Z_p\}\approx 0$.
• Figure 3: For $V$ being a low energy truncation of a 1d critical Ising CFT, (a) shows the eigenvalue distribution of $T_p^{[X_{i},Z_{i+\Delta}]}$ as a histogram, where $X_{i},Z_{i+\Delta}$ are local Pauli operators separated by a fixed spatial distance $\Delta$. (b) shows a correlation between the amount of microcausality violation and the "energy" $\omega$ of each state $|\phi_{k}\rangle$ labelled by different $k$s, such that more energetic states generally incur greater violations, consistent with our intuition. The energy $\omega$ is defined by a truncated critical Ising Hamiltonian. See Appendix \ref{['app:tij']} for detailed definitions.
• Figure 4: Global De Sitter spacetime as a hyperboloid in a higher-dimensional Minkowski embedding. The volume of time-slices increases exponentially with global time $t$, with a minimum effective Hilbert space $\mathcal{H}_n$ at $t=0$. We propose a projection of effective Hilbert spaces $\mathcal{H}_N$ at $t>0$ into the "fundamental" Hilbert $\mathcal{H}_n$. Two causal patches with constant volume are shaded in red and blue, emanating from two complementary regions $A$ and $B$ at $t=0$.
• Figure 5: Discretization oft two-dimensional de Sitter spacetime dS$_2$ with the MERA tensor network consisting of "disentangling" unitaries $U$ (square) and isometric tensors $I$ (triangles) whose properties are shown on the right. The left and right edges of the the tensor network (dashed lines) are periodically identified. The "entanglement renormalization" direction of the MERA is identified with global dS time $t$. The two complementary static patches from Fig. \ref{['fig:ds_slices']} are shown here as well, again shaded in red and blue.

Theorems & Definitions (3)

• Remark 1
• Proposition 1
• Proposition 2