Holographic fluctuations and the principle of minimal complexity

TL;DR

The work develops a framework in which bulk holographic geometry emerges from the boundary's quantum information structure via the principle of minimal complexity. It proposes two complementary constructions to assign bulk topology from a boundary unitary path in $\mathrm{SU}(\mathcal{H})$: a Euclidean-like approach using thermal correlations and a Lorentzian-like causal-set approach, with both approaches yielding a metric or causal structure on the bulk. It then introduces a Brownian-bridge model on $\mathrm{SU}(\mathcal{H})$ to describe bulk fluctuations, linking boundary perturbations to bulk dynamics through a Jacobi equation that serves as an Einstein-like constraint. The paper illustrates the ideas with simple examples (trivial background, pairwise perturbations, and quenches) and discusses outlooks toward continuum limits, tensor-network connections, and broader holographic interpretations.

Abstract

We discuss, from a quantum information perspective, recent proposals of Maldacena, Ryu, Takayanagi, van Raamsdonk, Swingle, and Susskind that spacetime is an emergent property of the quantum entanglement of an associated boundary quantum system. We review the idea that the informational principle of minimal complexity determines a dual holographic bulk spacetime from a minimal quantum circuit U preparing a given boundary state from a trivial reference state. We describe how this idea may be extended to determine the relationship between the fluctuations of the bulk holographic geometry and the fluctuations of the boundary low-energy subspace. In this way we obtain, for every quantum system, an Einstein-like equation of motion for what might be interpreted as a bulk gravity theory dual to the boundary system.

Figures (1)

• Figure 1: Example of the fluctuation in bulk spacetime $\mathcal{M}$ and bulk causal structure due to a fluctuation on the boundary. The boundary quantum system $\partial \mathcal{M}$ is comprised of $n=100$ qubits, and the boundary Hamiltonian is given by the $1D$ nearest-neighbour transverse Ising model $H = \sum_{j=0}^{100} \sigma_j^x\sigma_{j+1}^x + h\sigma_j^z$, with periodic boundary conditions. The $x$ axis is labelled by site number and the $y$ axis is holographic time $r$. The dots represent events in bulk holographic spacetime and have been chosen according to the Poisson distribution. The unitary operator $U$ studied here is $U = e^{-i50 H}$, a quench scenario. We studied the minimal geodesic $\gamma(r) = e^{-irH}$ connecting the identity $\mathbb{I}$ to $U$. The blue lines illustrate causal connections from a reference event at $(j=50,r=25)$ to the Poisson distributed events according to the criteria Eq. (\ref{['eq:causalconnect']}). We considered a fluctuation $U'= e^{-i\delta h_{50,75}}U$ which models the addition of a remote entangled pair between the distant sites $50$ and $75$ (the spacetime history of both of the involved sites are illustrated with black lines) at time $r=50$. The bulk holographic spacetime for the new geodesic $\gamma'$ connecting $\mathbb{I}$ to $U'$ was calculated according to the principle of minimal complexity by solving the Jacobi equation and the additional causal connections illustrated in red. One can readily observe the change in spacetime topology induced by the fluctuation, which might be interpreted as the creation of a wormhole between sites $50$ and $75$.