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I have taken the tensor product of $|0\rangle \otimes |-\rangle \otimes |+\rangle$ which resulted in the matrix

$$\begin{bmatrix} 1/2\\ 1/2 \\ -1/2 \\ -1/2 \\ 0 \\ 0\\ 0\\ 0\\ \end{bmatrix}.$$

How would I represent this in the computational basis basis $\{|0\rangle, |1\rangle\}$?

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The elements of your vector are coefficients of the state in the computational basis

$$ \begin{array}{cc} \begin{bmatrix} 1/2\\ 1/2 \\ -1/2 \\ -1/2 \\ 0 \\ 0\\ 0\\ 0\\ \end{bmatrix} &\begin{matrix} |000\rangle \\ |001\rangle \\ |010\rangle \\ |011\rangle \\ |100\rangle \\ |101\rangle \\ |110\rangle \\ |111\rangle \end{matrix} \end{array} $$

so

$$ |0\rangle\otimes|-\rangle\otimes|+\rangle = \frac{|000\rangle + |001\rangle - |010\rangle - |011\rangle}{2}. $$

We can confirm by direct calculation in Dirac notation

$$ \begin{align} |0\rangle\otimes|-\rangle\otimes|+\rangle &= |0\rangle\otimes\frac{|0\rangle-|1\rangle}{\sqrt{2}} \otimes \frac{|0\rangle+|1\rangle}{\sqrt{2}} \\ &= \frac{|0\rangle\otimes (|00\rangle + |01\rangle - |10\rangle - |11\rangle)}{2} \\ &= \frac{|000\rangle + |001\rangle - |010\rangle - |011\rangle}{2}. \end{align} $$

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  • $\begingroup$ Thank you very much, appreciate the help $\endgroup$
    – lambda
    Feb 22 at 21:51

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