# 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

## 3 Answers

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