# How do I represent my 3-qubit state in the computational basis?

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 $$\{|0\rangle, |1\rangle\}$$?

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}

• Thank you very much, appreciate the help Feb 22, 2021 at 21:51