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I am studying the basics of quantum computing math and am confused about qubit associativity. As I understand it, in quantum math, multiple qubits are represented as the tensor product of the qubits and the tensor product is defined as the Kronecker product.

Wikipedia tells me that the Kronecker product is associative, and elsewhere I have seen statements that the tensor product is associative. However, I cannot get a worked example to demonstrate this. For example:

$$|101\rangle = |1\rangle\otimes|0\rangle\otimes|1\rangle = \begin{pmatrix} 0\\ 1\\ \end{pmatrix} \otimes \begin{pmatrix} 1\\ 0\\ \end{pmatrix} \otimes \begin{pmatrix} 0\\ 1\\ \end{pmatrix} $$

If the Kronecker product were associative then we would have:

$$ \left( \begin{pmatrix} 0\\ 1\\ \end{pmatrix} \otimes \begin{pmatrix} 1\\ 0\\ \end{pmatrix} \right) \otimes \begin{pmatrix} 0\\ 1\\ \end{pmatrix} = \begin{pmatrix} 0\\ 1\\ \end{pmatrix} \otimes \left( \begin{pmatrix} 1\\ 0\\ \end{pmatrix} \otimes \begin{pmatrix} 0\\ 1\\ \end{pmatrix} \right) $$

But $$ \left( \begin{pmatrix} 0\\ 1\\ \end{pmatrix} \otimes \begin{pmatrix} 1\\ 0\\ \end{pmatrix} \right) \otimes \begin{pmatrix} 0\\ 1\\ \end{pmatrix} = \begin{pmatrix} 0\\ 0\\ 1\\ 0\\ \end{pmatrix} \otimes \begin{pmatrix} 0\\ 1\\ \end{pmatrix} = \begin{pmatrix} 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ \end{pmatrix} $$

whereas $$ \begin{pmatrix} 0\\ 1\\ \end{pmatrix} \otimes \left( \begin{pmatrix} 1\\ 0\\ \end{pmatrix} \otimes \begin{pmatrix} 0\\ 1\\ \end{pmatrix} \right) = \begin{pmatrix} 0\\ 1\\ \end{pmatrix} \otimes \begin{pmatrix} 0\\ 1\\ 0\\ 0\\ \end{pmatrix} = \begin{pmatrix} 0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ \end{pmatrix} $$ which looks correct because $101_2 = 5$. What am I missing? Is it necessary to apply the tensor product to qubits right to left?

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1 Answer 1

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Looks like a simple error in the left-associative calculation:

$$ \left( \begin{pmatrix} 0\\ 1\\ \end{pmatrix} \otimes \begin{pmatrix} 1\\ 0\\ \end{pmatrix} \right) \otimes \begin{pmatrix} 0\\ 1\\ \end{pmatrix} = \begin{pmatrix} 0\\ 0\\ 1\\ 0\\ \end{pmatrix} \otimes \begin{pmatrix} 0\\ 1\\ \end{pmatrix} = \begin{pmatrix} 0\\ 0\\ 0\\ 0\\ 0\\ \textbf{1}\\ \textbf{0}\\ 0\\ \end{pmatrix} $$

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