Area and volume as emergent phenomena from entangled qubits
Juan M. Romero, Emiliano Montoya-González
TL;DR
The paper investigates whether basic geometric quantities can emerge from entangled qubit states. It expresses the area and volume as multilinear polynomials, then realizes them with explicit multipartite entangled states: a $4$-qubit state for the $2D$ area, three $6$-qubit states for the $3D$ vector area, and a $9$-qubit state for the $3D$ volume, complemented by Qiskit circuits to generate these states. The key contributions are the concrete algebraic mappings $A(v_1,v_2) = x_1x_4 - x_2x_3$, $A_1=A_2=A_3$ with their respective $6$-qubit encodings, and $V(v_1,v_2,v_3)$ with the $9$-qubit encoding, validating that $raket{A_i|oldsymbol{ ight|} abla angle ext{(ψ)}}$ and $raket{V|oldsymbol{ ight|} abla angle ext{(ψ)}}$ reproduce the desired geometric quantities. This work supports the notion that spacetime-like geometry can emerge from entanglement and offers scalable quantum-information-based methods to study emergent geometry.
Abstract
Recently, a connection has been shown between certain geometric quantities and quantum information theory. In this paper, we demonstrate that geometric quantities such as area and volume can emerge directly from entangled multi-qubit states. In particular, the area of a two-dimensional parallelogram is derived from a 4-qubit entangled state, the vector area of a three-dimensional parallelogram from three 6-qubit entangled states, and the volume of a three-dimensional parallelepiped from a 9-qubit entangled state. Corresponding quantum circuits are constructed and implemented using Qiskit to generate the required entangled states. Given that parallelograms and parallelepipeds serve as elementary building blocks for more complex geometric structures, these results may offer a pathway toward exploring emergent geometry in quantum information frameworks