# Qiskit textbook - exercise 2.2 - why doesn't the correct answer give the same statevector?

In the Qiskit textbook, I'm stuck on one of the exercises in section 2.2. The question is:

Write the state: as two separate qubits.

The answer has already been covered in this question here, and is defined as: This makes sense to me - we're basically factoring out the first qubit. However, for my own benefit, I tried multiplying out the vectors and I get different results:

Using the original form:  Why don't these give the same values?

n.b. Apologies for the use of images - I only had MS Word at my disposal and couldn't find an easy way to export the equations bar screen-shotting.

• You do not need additional software to write mathematics on this site, for a quick overview of how to do this please see this page Feb 21 at 14:38
• Thanks - I already had the equations in Word, so I was trying (unsuccessfully) to convert them without downloading any other tools. Next time, I'll start by writing them directly in the question. Feb 21 at 16:43

In your answer you are treating $$|xy\rangle$$ as a direct sum of vectors $$|x\rangle$$ and $$|y\rangle$$. However, $$|xy\rangle = |x\rangle \otimes |y\rangle$$ is actually shorthand for the tensor (Kronecker) product of $$|x\rangle$$ and $$|y\rangle$$ (see wiki for more details on the Kronecker product).
In particular if $$|x\rangle = \begin{pmatrix} a \\ b\end{pmatrix}$$ and $$|y\rangle = \begin{pmatrix} c \\ d \end{pmatrix}$$ then $$|xy\rangle = |x\rangle\otimes |y\rangle = \begin{pmatrix} ac \\ ad \\ bc \\ bd \end{pmatrix}.$$
• Not sure what you mean by both. $$|00\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}$$ and $$|01\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}$$ Feb 21 at 16:57
• By "both", I meant the vector of the factorised answer and the vector from the original question. I think both should have come out as $\frac{1}{\sqrt(2)} \begin {pmatrix} 1 \\ i \\ 0 \\ 0 \end{pmatrix}$ Feb 21 at 17:04