# What is the difference between $|+\rangle$ and $|-\rangle$?

What is the difference between $$|+\rangle$$ and $$|-\rangle$$. I was reading about quantum states and found that $$|+\rangle$$ and $$|-\rangle$$ represent state in superposition of other states. I was just wondering what is the difference between both of them, being in the state of superposition means qubit is 0 and 1 at the same time. So why do we need two different representation of the 'same' state called superposition?

• Please do not feel ofended but I would suggest you to read some basics on quantum computing. I think this would be good start: qiskit.org/textbook/ch-states/introduction.html May 18, 2022 at 17:24
• Sure I will do it.. @Martin May 19, 2022 at 0:38

States with different complex amplitudes are simply different states with different observable consequences. They are not the same. We need them because they describe the outcomes of measurements.

To give a little insight, I like to think of an experiment using a two-level system. Let's arbitrarily label these two levels as $$|0\rangle$$ and $$|1\rangle$$. Let's say I start in level $$|0\rangle$$, but I want to create level $$|1\rangle$$. There'll be a physical process that lets me do this. It might, for example, involve shining a laser on the system for a specific time $$t_0$$. So, we have a process that goes $$|0\rangle\xrightarrow{\quad t_0\quad} |1\rangle.$$ Similarly, had I started with $$|1\rangle$$, I would have to achieve $$|1\rangle\xrightarrow{\quad t_0\quad} |0\rangle.$$

Now, let me ask what happens if I start from $$|0\rangle$$, shine the laser for time $$t_0/2$$, stop for a bit, then shine the laser for another $$t_0/2$$?

Well, the total evolution time is $$t_0$$, so you must achieve the $$0\rightarrow 1$$ transition. But, between those to $$t_0/2$$ pulses, what state was the atom in? In some sense, it's half way between $$0\rangle$$ and $$|1\rangle$$. But the same could be said for having started from $$|1\rangle$$ and evolved with the laser for $$t_0/2$$. So, how does the atom know that it started in $$|0\rangle$$ and is heading towards $$|1\rangle$$ and not the other way around?

Its state must contain some sort of record beyond simply "half way between 0 and 1" so that it "knows" where it's got to get to. That is exactly what the phase information on the complex amplitudes is giving you in this instance: $$(|0\rangle\pm|1\rangle)/\sqrt{2}$$ are both half way between 0 and 1 but are different states because they contain that extra information.

By superposition of the two states $$|0\rangle$$ and $$|1\rangle$$ we mean the state $$\alpha|0\rangle + \beta|1\rangle$$ where $$\alpha$$ and $$\beta$$ are complex numbers. So, superposition is a physical jargon for linear combination.

$$|+\rangle$$ refers to the case when $$\alpha = 1/\sqrt 2$$ and $$\beta = 1/\sqrt 2$$, and

$$|-\rangle$$ refers to the case when $$\alpha = 1/\sqrt 2$$ and $$\beta = -1/\sqrt 2$$

We say that $$|+\rangle$$ and $$|-\rangle$$ differ in the relative phase.

In terms of measurement probability distribution, there is no difference between ∣+⟩ and ∣−⟩ BUT even with the same measurement probability distributions there is a difference in how they interact with quantum gates. For example, the Hadamard—gate takes the state ∣0⟩ to the state ∣+⟩ and the state ∣1⟩ to the state ∣−⟩. Thus, the H gate sends the state ∣+⟩ back to the state ∣0⟩ and sends the state ∣−⟩ back to the state ∣1⟩. so we can say there is difference how ∣+⟩ and ∣−⟩ interact with quantum gates.

• it is not correct that "In terms of measurement probability distribution, there is no difference between ∣+⟩ and ∣−⟩". There is no difference only when you measure in the computational basis. You'll see a lot of difference measuring in the eigenbasis of $\sigma_x$, for example. See also quantumcomputing.stackexchange.com/a/1476/55 about this point
– glS
May 31, 2022 at 10:33