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In Nielsen and Chuang, chapter 1.3.3 is named as "Measurements in bases other than the computational basis". This name confuses me - after the measurement is done on a new base, doesn't this new base become the computational basis?

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The computational basis is just a convention for the $Z$ basis, as its orthogonal basis is $\{|0\rangle, |1\rangle\}$; which is analog to the bit in classical computation, hence the name.

So in theory, yes, you can call computational basis any basis you want as long as you clarify what convention are you following, but the most common convention (and, in reality, the only one I've seen) is to call the $Z$ basis this.

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The "computational basis" really doesn't refer to anything more than that basis you choose to label with the symbols 0 and 1. It is an arbitrary designation and independent of what operations (eg. measurement) you are actually doing on the qubits.

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  • $\begingroup$ Thank you for the reply! However, doesn't the measurement give a result that collapses onto either one of the computational basis, and hence defines what's actually the computational basis? $\endgroup$
    – Alexia.
    Aug 5 at 14:44
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    $\begingroup$ That is true only if you perform the measurement in the computational basis! If you perform the measurement in a different basis, the result collapses to an eigenvalue of that different basis. $\endgroup$
    – jecado
    Aug 5 at 18:50
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Each base you are working with is a "computational basis", hence as it was mentioned before the standard one, is a conventional one in reference to the classical. The result of your measurements will be expressed in different amplitudes by will express the same phenomenon.

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