How many parameters do we need to characterize a pure state?

Question

Suppose I have a pure qubit. I can think of starting with the state $\vert 0\rangle$ and apply some unitary to it. Such a unitary has three parameters according to this link. In $d$ dimensions, the link above claims that a unitary matrix is defined with $d^2 - 1$ parameters. But I thought $U^\dagger U = I$ requires $d^2$ real independent parameters - see for example this post. So there's an "off by one" problem with my understanding - what happened there?

Now, let's go back to the qubit and look at the Bloch sphere. Since the qubit is pure, I only need two parameters to get anywhere on the sphere. So now, it seems like we only need 2 parameters!

I suspect that this has something to do with the Bloch sphere ignoring the phase added to a qubit i.e. $\vert\psi\rangle$ is the same as $e^{i\theta}\vert\psi\rangle$ but I am not sure which of the three possible answers are correct!

How many parameters do we need for a 1-qubit pure state? Also, how does this generalize for higher dimensions?

Answer 1

For your first point: in quantum physics, the evolution of a quantum system is defined with an unitary, up to a global phase. So both unitary $U$ and $e^{i\theta} U$ have the same meaning. This allow you to remove one real parameter from the parametrization.

For you qubit question, this is related to the distinction between a state, and a unitary that act on this state. Indeed, a single qubit state can be represented with 2 parameters using the Bloch sphere. But for a transformation of a state to another state on the Bloch sphere, you need more: first you need the direction of the rotation (which will be another 2 parameter to have the direction) and another real parameter to encode the angle. Thus you get 3 parameters to map the Bloch sphere to Bloch Sphere state using a unitary.

Answer 2

The way that I view the distinction between the number of parameters for a unitary and a state is as follows:

• For a one-qubit state, you need two parameters.
• To define a unitary, you need to specify what arbitrary state $|0\rangle$ is mapped to, so you need two parameters for that.
• However, you also need to specify what state $|1\rangle$ is mapped to. Now, in fact, it must be mapped to a state that is orthogonal to the one that $|0\rangle$ mapped to, so there's actually very little freedom. But there is a single parameter, its overall phase.

For example, if I required $$ |0\rangle\rightarrow \frac{1}{\sqrt{2}}(|0\rangle+|1\rangle), $$ then all I know from orthogonality is that $$ |1\rangle\rightarrow e^{i\phi}\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle) $$ so I need one extra parameter to fix the $\phi$.

You might ask why it appears that I can have a global phase on this second state when I don't have it on the first state. The answer is that, overall, there is one global phase that you can absorb into both states, or the overall unitary. The $e^{i\phi}$ that I have left behind sort of looks like an overall multiplying phase, but the important thing is that it isn't actually global. It's a relative phase between the $|0\rangle$ and $|1\rangle$ components that has a meaningful impact if your input is a superposition.

Answer 3

Perhaps this will help in geometrically identifying the extra parameter. Two pure-state qubits are related by a unitary that spins the Bloch sphere about some axis by some angle. To specify that unitary, one needs three parameters, even though the location of each state is specified by two parameters. The extra parameter in the unitary is there because there is a whole family of different unitaries that will take the first state to the second one.

How can we interpret this redundancy? Let's think about the family of unitaries $U(\theta,\mathbf{n})$ that rotate a Bloch sphere by angle $\theta$ about axis $\mathbf{n}$ such that $U(\theta,\mathbf{n})|\psi\rangle=|\phi\rangle$ for a pair of states $|\psi\rangle$ and $|\phi\rangle$. If $\mathbf{n}$ pointed to the north pole, the two states $|\psi\rangle$ and $|\phi\rangle$ would have to lie on the same lattitude (polar angle); for example, two states on the equator could be transformed into each other. One can always change their viewpoint of any coordinate system to pretend $\mathbf{n}$ points to the north pole, so this picture will always be valid. The question now becomes: given a pair of states $|\psi\rangle$ and $|\phi\rangle$, what possible rotation axes $\mathbf{n}$ will allow the two to be connected; i.e., what are the axes for which $|\psi\rangle$ and $|\phi\rangle$ are at the same polar angle?

The answer comes directly from symmetry: given two points on the Bloch sphere, find the shortest path between them; i.e., the great circle that connects the two, which is unique. Take the point halfway between the two points and draw the perpendicular great circle. You can visualize this as orienting the coordinate system such that one point is $\varphi$ degrees above the equator and the other $\varphi$ degrees below the equator. Then this perpendicular great circle, like the equator in the visualized coordinate system, defines the set of rotation axes $\mathbf{n}$ that can connect $|\psi\rangle$ and $|\phi\rangle$.

The magic is done! We have reduced the two parameters of the unitary $\mathbf{n}$ to a single parameter specifying a coordinate on the "perpendicular great circle." Once that is specified, there is only one possible rotation angle $\theta$, so the problem is complete. We see directly that the redundancy in the unitary is caused by choosing which point on the perpendicular great circle to choose as the rotation axis. Of course, these transformations are only the same in that they all enact $U|\psi\rangle=|\phi\rangle$ by construction, but they are different unitaries in general and will not all perform the same action on some other state.