# 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: $$|\psi⟩ = cos(\frac{\theta}{2}) |0⟩ + e^{i \phi} sin(\frac{\theta}{2}) |1⟩$$
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.