# Extracting entanglement geometry from quantum states

## TL;DR

This paper introduces an unbiased algorithm to extract the entanglement geometry of a quantum state by transforming it into an unentangled state and analyzing the resulting unitary circuits, revealing hyperbolic geometry features.

## Contribution

The authors develop a novel method to determine the entanglement geometry directly from quantum states without fixed tensor network assumptions.

## Key findings

- Recovered signatures of scale invariance in 1D critical systems
- Identified hyperbolic geometry properties in the unitary circuits
- Demonstrated the approach on non-interacting quantum systems

## Abstract

Tensor networks impose a notion of geometry on the entanglement of a quantum system. In some cases, this geometry is found to reproduce key properties of holographic dualities, and subsequently much work has focused on using tensor networks as tractable models for holographic dualities. Conventionally, the structure of the network - and hence the geometry - is largely fixed a priori by the choice of tensor network ansatz. Here, we evade this restriction and describe an unbiased approach that allows us to extract the appropriate geometry from a given quantum state. We develop an algorithm that iteratively finds a unitary circuit that transforms a given quantum state into an unentangled product state. We then analyze the structure of the resulting unitary circuits. In the case of non-interacting, critical systems in one dimension, we recover signatures of scale invariance in the unitary network, and we show that appropriately defined geodesic paths between physical degrees of freedom exhibit known properties of a hyperbolic geometry.