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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?

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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.

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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.

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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.

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    $\begingroup$ 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 $\endgroup$
    – glS
    May 31, 2022 at 10:33

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