# What does the notation $|+\rangle,|−\rangle,|±i \rangle$ mean in Bloch sphere?

The axis in a a 2D diagram like the following, usually represent 2 quantities. Eg in pic below, $$x$$ represents time and $$y$$ represents velocity What gets measured along each axis of a Bloch sphere? In other words what do the different axis of Bloch sphere represent? I read that $$z$$ is $$|0⟩$$ or $$|1⟩$$ (spin), $$x$$ is $$|+⟩$$ or $$|-⟩$$ and $$y$$ is $$|i⟩$$ or $$|-i⟩$$. But what do they mean? • are you asking what the notation $|+\rangle,|-\rangle,|\pm i\rangle$ means?
– glS
Jun 6 at 19:19
• yes please. Could you explain? Jun 10 at 7:15

The standard definition of $$|\pm\rangle$$ and $$|\pm i\rangle$$ is: $$|+\rangle \equiv \frac{1}{\sqrt2}(|0\rangle+|1\rangle), \qquad |-\rangle \equiv \frac{1}{\sqrt2}(|0\rangle-|1\rangle).\\ \qquad |+i\rangle \equiv \frac{1}{\sqrt2}(|0\rangle+i|1\rangle), \qquad |-i\rangle \equiv \frac{1}{\sqrt2}(|0\rangle-i|1\rangle).$$ These are the eigenstates of the Pauli operators $$X$$ and $$Y$$, respectively, and thus are represented in the Bloch sphere on the corresponding axes.

I think its best if we start with the state $$|\psi⟩ = \alpha |0⟩ + \beta |1⟩$$. Since we know that the total probability is 1 ($$|\alpha|^2+|\beta|^2 = 1$$), we can plot $$\alpha$$ and $$\beta$$ as a circle with radius 1 centered at (0,0) (In the figure below let $$x$$ be $$\alpha$$ and $$y$$ be $$\beta$$ or vice versa). Therefore, all the possible combination for $$\alpha$$ and $$\beta$$ lies in the perimeter of this circle. Now, instead of using $$(x,y)$$ coordinates to plot the probabilities of $$\alpha$$ and $$\beta$$, we can instead use the angle formed using polar coordinates (In the figure below that would be $$(r,\theta)=(1,\theta)$$). In quantum computing however, operators may change more than the probability of the basis but also other stuff such as the phase. Therefore, in general, it is easier to represent a state in the following form: $$|\psi⟩ = cos(\frac{\theta}{2}) |0⟩ + e^{i \phi} sin(\frac{\theta}{2}) |1⟩$$

Notice that there is a second angle in play now. This means to plot the possible states, a 3D plot of the surface of a ball is needed which is the Bloch sphere. To read more about the derivation of the Bloch sphere, look at this answer.

Now, to answer the question, in my understanding, the $$(x,y,z)$$ axis of the plot is merely an arbitrary axis. The main information held in the plot lies in the polar coordinates $$(1, \theta, \phi)$$.

It is not that $$z$$ measures $$|0⟩$$ or $$|1⟩$$, but instead the states $$|0⟩$$ and $$|1⟩$$ happen to lie in the $$z$$ axis. Similarly for the other 2 axes.