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This is a very basic question, but I am wondering how does one determine if the state is in a superposition in the standard basis? What I know is a state is in superposition iff ⍺ and β are both ≠ 0, else pure-- so superposition is basis-dependent. But I don't know how to apply that knowledge when looking at the following state which is using Hadamard basis:

Is this state a superposition in the standard basis:

$$\frac{\sqrt{3}}{2}|+\rangle − \frac{1}{2}|−\rangle$$

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    $\begingroup$ Welcome to QCSE! Do you know how to express $|+\rangle$ and $|-\rangle$ in the computational basis? What happens if you substitute that into your formula? $\endgroup$ Oct 14, 2022 at 4:03

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In order to see what this is in a superposition of the standard basis is to recognize what the standard basis is: $\{|0\rangle, |1\rangle\}$ Secondly, you need to understand what $|+\rangle$ and $|-\rangle$ represent. Standard notation states that $|+\rangle = \frac1{\sqrt{2}}(|0\rangle+|1\rangle)$ and $|-\rangle = \frac1{\sqrt{2}}(|0\rangle-|1\rangle)$

Using these definitions you can rewrite your equation to see if it is in a superposition once written in terms of $|0\rangle$ and $|1\rangle$.

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Generally, any basis can be transformed to another one. This means that any state can be after a transformation expressed in computational basis. This comes from linear algebra where you can also freely switch vectors from one basis to another. By the way, what we call computational basis in quantum computing is called standard basis of vector space $\mathbb{C}^n$ in linear algebra.

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