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This research introduces a novel geometric framework that extends the single-qubit Bloch sphere model to arbitrary N-qubit systems, addressing a long-standing gap in visualizing multi-qubit entanglement. By establishing a strict bijection between standard state vectors and a hierarchical arrangement of spheres, the study offers new analytical tools for quantum simulators and gate analysis.
Extending the Bloch sphere model to an N-qubit system
arXiv:2607.04979v2 Announce Type: replace 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 $2^N-1$ 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.
Authors:Francisco Piñero (1), Cristian Franco (2), Hernán I. de la Cruz (3), Fernando L. Pelayo (1), Vicente Pascual (1), Mauro Mezzini (4), Jose Javier Paulet (5), Fernando Cuartero (1) ((1) Universidad de Castilla-La Mancha, Albacete, Spain, (2) Universidad de La Rioja, Logroño, Spain, (3) Universidad Complutense de Madrid, Madrid, Spain, (4) Roma Tre University, Roma, Italy, (5) QSimov, Talavera de la Reina, Spain)
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 $2^N-1$ 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.
arXiv:2607.04979 [quant-ph]
(or arXiv:2607.04979v2 [quant-ph] for this version)
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