A note on Majorana representation of quantum states

TL;DR

This paper explores the Majorana representation of quantum states, providing a scheme to construct qubit representations for any dimension and expressing inner products as matrix permanents, with extensions to mixed states.

Contribution

It introduces a simple construction method for representing high-dimensional quantum states as points on the Bloch sphere using symmetry class of tensors.

Findings

Constructed $d-1$ qubit representations for $d$-dimensional states.

Expressed inner products as permanents of related matrices.

Extended the framework to include mixed 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.