Extending the Bloch sphere model to an N-qubit system
Abstract
The Bloch sphere is an elegant tool for representing single-qubit states. However, a widely accepted generalization for multi-qubit systems with entanglement remains absent. We propose a novel geometric model extending the Bloch sphere representation to arbitrary N-qubit systems using 2N−1 spheres. We demonstrate that any pure 2-qubit state is uniquely described by three spheres: two for individual qubits and a third encapsulating bipartite entanglement. Generalizing this, we establish an N-qubit parameterization through the hierarchical application of controlled rotation gates along the Z and Y axes. We formally prove a strict bijection between the standard state vector representation and our model's angular parameters. This framework provides an intuitive visualization of multiple entanglement, offering potential computational advantages for quantum simulators and new analytical perspectives on quantum gates.
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BibTeX
@misc{pinero2026extending,
title = {{Extending the Bloch sphere model to an N-qubit system}},
author = {Francisco Piñero and Cristian Franco and Hernán I. de la Cruz and Fernando L. Pelayo and Vicente Pascual and Mauro Mezzini and Jose Javier Paulet and Fernando Cuartero},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.04979},
eprint = {2607.04979},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04979},
}
RIS
TY - UNPB
TI - Extending the Bloch sphere model to an N-qubit system
AU - Francisco Piñero
AU - Cristian Franco
AU - Hernán I. de la Cruz
AU - Fernando L. Pelayo
AU - Vicente Pascual
AU - Mauro Mezzini
AU - Jose Javier Paulet
AU - Fernando Cuartero
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2607.04979
UR - https://arxiv.org/abs/2607.04979
AB - The Bloch sphere is an elegant tool for representing single-qubit states. However, a widely accepted generalization for multi-qubit systems with entanglement remains absent. We propose a novel geometric model extending the Bloch sphere representation to arbitrary N-qubit systems using 2^N−1 spheres. We demonstrate that any pure 2-qubit state is uniquely described by three spheres: two for individual qubits and a third encapsulating bipartite entanglement. Generalizing this, we establish an N-qubit parameterization through the hierarchical application of controlled rotation gates along the Z and Y axes. We formally prove a strict bijection between the standard state vector representation and our model's angular parameters. This framework provides an intuitive visualization of multiple entanglement, offering potential computational advantages for quantum simulators and new analytical perspectives on quantum gates.
ER -