Identifying Quantum Correlations Using Explicit SO(3) to SU(2) Maps

TL;DR

This paper presents an explicit formula for mapping SO(3) rotations to SU(2) unitaries, simplifying the manipulation and diagonalization of two-qubit correlation matrices crucial for quantum control.

Contribution

It introduces a direct, programmable method to relate SO(3) rotations to SU(2) unitaries, enabling easier quantum state manipulation and correlation matrix diagonalization.

Findings

Explicit formula for SO(3) to SU(2) mapping provided

Simplifies correlation matrix manipulation in two-qubit systems

Facilitates time-optimal control of quantum states

Abstract

Quantum state manipulation of two-qubits on the local systems by special unitaries induces special orthogonal rotations on the Bloch spheres. An exact formula is given for determining the local unitaries for some given rotation on the Bloch sphere. The solution allows for easy manipulation of two-qubit quantum states with a single definition that is programmable. With this explicit formula, modifications to the correlation matrix are made simple. Using our solution, it is possible to diagonalize the correlation matrix without solving for the parameters in SU(2) that define the local unitary that induces the special orthogonal rotation in SO(3). Since diagonalization of the correlation matrix is equivalent to diagonalization of the interaction Hamiltonian, manipulating the correlation matrix is important in time-optimal control of a two-qubit state. The relationship between orthogonality conditions on SU(2) and SO(3) are given and manipulating the correlation matrix when only one qubit can be accessed is discussed.