In the expression $|+\rangle|+\rangle \otimes(\cdot)$, what does the centered dot represent? I have come across a definition of the centered dot previously, similar to the one given here, but I am just querying to see if this means anything different in this context.


This comes from the three qubit phase flip code in Nielsen and Chaung, section 10.1.2, page 431. For this specific example, the measurement of $X_1X_2$ gives $+1$ for the state $|+\rangle|+\rangle \otimes(\cdot)$.

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    $\begingroup$ Does not seem possible to infer without a context. Perhaps $(\cdot)$ here just means anything (in the brackets)? $\endgroup$ Sep 13, 2022 at 19:56

1 Answer 1


Read in the context of Nielsen and Chuang it's clear that $(\cdot)$ is acting as a wildcard for extra, dummy or unspecified qubits, as @NikitaNemkov hinted.

Quoting from my copy of N+C:

...Measurement of the observables $X_1X_2$ and $X_2X_3$ corresponds to comparing the sign of qubits one and two, and two and three, respectively, in the sense that measurement of $X_1X_2$, for example, gives $+1$ for states like $|+\rangle|+\rangle\otimes(\cdot)$ or $|-\rangle|-\rangle$, and $-1$ for states like $|+\rangle|-\rangle\otimes(\cdot)$ or $|-\rangle|+\rangle\otimes(\cdot)$.

The first two qubits are $X_1X_2$ while the other qubits (e.g. $X_3$) could be represented by $(\cdot)$.


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