This question is mostly about group theory but I would like to understand it in the context of qubits rotating in a Bloch Sphere.(adsbygoogle = window.adsbygoogle || []).push({});

What my understanding of things are right now:

In the rotation Lie Group ##SO(3)##, we have three free parameters (##\frac{n(n-1)}{2}##), and this is also why we end up with three generators for the group. The two dimensional complex matrix representation of ##SO(3)## (which is also a representation to the ##SU(2)## group) also has three generators, namely the Pauli matrices.

What the problem is about:

I can't understand how these facts relate to the number of free parameters in the 'representation space' on which these matrix representations act. For example, the representation space for the 3x3 matrices representing ##SO(3)## would be the column vectors spanned by the standard basis ##e_1, e_2, e_3##.

How it relates to qubits:

If we consider a general complex 2-vector (in Dirac notation):

\begin{equation}

r_{1}e^{\iota\phi_{1}}{|{0}>} + r_{2}e^{\iota\phi_{2}}{|{1}>}

\end{equation}

and use the idea that the global phase of the state is of no significance, we end up with a three parameter representation:

\begin{equation}

r_{1}{|{0}>} + r_{2}e^{\iota\phi^{'}}{|{1}>}

\end{equation}

the free parameters being ##r_1, r_2## and ##\phi^{'}##. Now, if we add the condition of the normalization of the qubit, we get another constraint on these three parameters, and we end up with inly two free parameters. But this is enough to place the qubit exactly on the Bloch Sphere - ##\theta## is the azimuthal angle and ##\phi## is the polar angle. (the surface of the Bloch Sphere is fixed, so we only need two angular spherical coordinates and no ##r##).

What the problem is:

All single qubit quantum gates are unitary and thus belong to the group ##U(2)## which has three free parameters. If the quantum gates are to perform rotations on these qubits, shouldn't the qubits - being the representation space of the group of quantum gates - have three parameters (not two)? Is it possible that the Bloch Sphere can't representallpossible qubits?

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# Rotations in Bloch Sphere, and Free Parameters of a Qubit

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