Area and volume as emergent phenomena from entangled qubits

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.