A note on Majorana representation of quantum states

By the Majorana representation, for any \(d > 1\) there is a one-one correspondence between a quantum state of dimension d and \(d-1\) qubits represented as \(d-1\) points in the Bloch sphere. Using the theory of symmetry class of tensors, we present a simple scheme for constructing \(d-1\) points on the Bloch sphere and the corresponding \(d-1\) qubits representing a d-dimensional quantum state. Additionally, we demonstrate how the inner product of two d-dimensional quantum states can be expressed as a permanent of a matrix related to their \((d-1)\)-qubit state representations. Extension of the result to mixed states is also considered.