Why are the + and - states for a Qubit represented as |0>+|1> and |0>-|1> respectively and not the other way around?

Is this only a matter of convention or is there a formula to arrive at each vector in Hilbert space only from |0> and |1> states?


1 Answer 1


It's a matter of convention. Remember that what you write inside the ket is just a label. Labels can be anything you want. So it is a convenience that helps us remember that $$ |\pm\rangle=|0\rangle\pm|1\rangle. $$ We'd all be making mistakes all the time if it were the other way around!

I've just argued that the notation makes a lot of sense in terms of helping us to remember the correct way of writing the state. It also makes a lot of sense in terms of the Pauli $X$ matrix: $$ \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) $$ This has two eigenvalues $\pm 1$ with corresponding eigenvectors $|\pm\rangle$, so there's also a convenient correspondence between the label and the eigenvalue.


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