The Bloch vector of a qubit is a vector in three-dimensional space that represents the state of the qubit. It is defined as: ⃗ v = (vx, vy , vz ) where vx, vy , and vz are the components of the vector along the x, y, and z axes, respectively. Rotations of a qubit in the Bloch sphere can be represented by unitary matrices, and the same unitary matrices can be used to rotate the Bloch vector. Specifically, if we have a unitary matrix U that represents a rotation of the qubit state, we can apply this rotation to the Bloch vector by multiplying it by the matrix U †, where U † is the conjugate transpose of U . This is because the action of a unitary matrix on a quantum state is equivalent to a change of basis, and the conjugate transpose of a unitary matrix is its inverse, which reverses the change of basis. For example, suppose we have a qubit in the state |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers such that |α|2 + |β|2 = 1. The corresponding Bloch vector is: ⃗ v = (Re(β), Im(β), Re(α)) Now suppose we want to rotate the qubit around the z-axis by an angle θ. The corresponding unitary matrix is: U = ( e )−iθ/2 0 0eiθ/2 The conjugate transpose of U is: U † = ( e )iθ/2 0 0e−iθ/2 To rotate the Bloch vector, we multiply it by U †, giving: ⃗ v ′ = U †⃗v = ( e )iθ/2 0 0e−iθ/2 ( ) Re(β) Im(β) Re(α) = ( e )iθ/2 Re(β) e−iθ/2Im(β) Re(α) This new vector represents the state of the qubit after the rotation around the z-axis.]Introduction Yes, we can rotate Bloch vectors for qubits just like we do with qubits in the Bloch sphere. 1

The Bloch vector of a qubit is a vector in three-dimensional space that represents the state of the qubit. It is defined as: ⃗ v = (vx, vy , vz ) where vx, vy , and vz are the components of the vector along the x, y, and z axes, respectively. Rotations of a qubit in the Bloch sphere can be represented by unitary matrices, and the same unitary matrices can be used to rotate the Bloch vector. Specifically, if we have a unitary matrix U that represents a rotation of the qubit state, we can apply this rotation to the Bloch vector by multiplying it by the matrix U †, where U † is the conjugate transpose of U . This is because the action of a unitary matrix on a quantum state is equivalent to a change of basis, and the conjugate transpose of a unitary matrix is its inverse, which reverses the change of basis. For example, suppose we have a qubit in the state |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers such that |α|2 + |β|2 = 1. The corresponding Bloch vector is: ⃗ v = (Re(β), Im(β), Re(α)) Now suppose we want to rotate the qubit around the z-axis by an angle θ. The corresponding unitary matrix is: U = ( e )−iθ/2 0 0eiθ/2 The conjugate transpose of U is: U † = ( e )iθ/2 0 0e−iθ/2 To rotate the Bloch vector, we multiply it by U †, giving: ⃗ v ′ = U †⃗v = ( e )iθ/2 0 0e−iθ/2 ( ) Re(β) Im(β) Re(α) = ( e )iθ/2 Re(β) e−iθ/2Im(β) Re(α) This new vector represents the state of the qubit after the rotation around the z-axis.