# Is qubit superposition a basis-dependent concept?

The qubit $$a\left|0\right>+b\left|1\right>$$ is a superposition w.r.t. the basis $$\left\lbrace\left|0\right>,\, \left|1\right>\right\rbrace$$ because it may collapse to $$\left|0\right>$$ or $$\left|1\right>$$ when measured w.r.t. the above basis. However the same qubit, when measured w.r.t the basis $$\left\lbrace a\left|0\right>+b\left|1\right>, b^*\left|0\right>−a^*\left|1\right>\right\rbrace$$, always collapses to the first vector.

Can I therefore conclude that superposition is a basis-dependent concept?

Yes. You already have the reasoning for why. Sometimes the basis is implied but not explicitly stated though. So caution when reading something like that.

• Thanks! I thought it was a little bit strange and not sure if I misunderstand the concept. Thanks for the confirmation. – Mathlusiverse Jan 14 '19 at 5:34
• It might help to emphasise that in real physical systems, there is often a "natural" basis. For example, if we encode the levels 0 and 1 as two distinct energy levels of an atom, then there is typically an error mechanism called relaxation (aka amplitude damping) that performs an equivalent function to the measurement in the 0/1 basis. – DaftWullie Jan 14 '19 at 7:51