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

Write the state: enter image description hereas two separate qubits.

The answer has already been covered in this question here, and is defined as:

enter image description here

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:

enter image description here

Using the answer supplied:

enter image description here

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.

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    $\begingroup$ 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 $\endgroup$
    – Rammus
    Feb 21 at 14:38
  • $\begingroup$ 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. $\endgroup$ Feb 21 at 16:43
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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 $$ |xy\rangle = |x\rangle\otimes |y\rangle = \begin{pmatrix} ac \\ ad \\ bc \\ bd \end{pmatrix}. $$

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  • $\begingroup$ Right - so my error is in the first vector calculation and both should be 1/(√2) [1 i 0 0] ? $\endgroup$ Feb 21 at 16:48
  • $\begingroup$ 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 $$ |01\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} $$ $\endgroup$
    – Rammus
    Feb 21 at 16:57
  • $\begingroup$ 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 $\frac{1}{\sqrt(2)} \begin {pmatrix} 1 \\ i \\ 0 \\ 0 \end{pmatrix}$ $\endgroup$ Feb 21 at 17:04
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    $\begingroup$ Ok, yeah, exactly! $\endgroup$
    – Rammus
    Feb 21 at 19:45

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