While the Bloch sphere is a commonly used representation for visualizing the pure state space of a single qubit, it becomes impractical for systems with more than one qubit. As the number of qubits increases, the dimensionality of the quantum state space grows exponentially, making it challenging to visualize using a simple geometric representation.
There isn't a widely adopted alternative geometrical representation for the pure state space of multiple qubits that is as intuitive as the Bloch sphere for a single qubit. The reason is that the complexity and richness of quantum state spaces increase significantly with more qubits, and it becomes difficult to visualize them in a straightforward geometric form.
However, there are alternative mathematical representations used for describing multi-qubit systems, such as the density matrix formalism and the tensor product notation. These representations capture the full quantum state information of multiple qubits, including entanglement, superposition, and correlations between qubits.
Creating a new geometrical representation specifically tailored for multi-qubit systems could potentially be advantageous for visualization purposes and gaining a deeper understanding of quantum states. Such a representation could help in studying entanglement patterns, quantum correlations, and other properties of multi-qubit systems.
Researchers are continuously exploring new techniques and representations to gain insights into quantum systems with multiple qubits. While a comprehensive and widely accepted alternative geometrical representation for multi-qubit systems does not currently exist, ongoing research in this area could lead to the development of novel visualization methods in the future.