A note on Majorana representation of quantum states

Abstract: 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.