## How to write the map $\mathbb C\ni z\mapsto zv$ in bra-ket notation?

1

As part of a course, I've been asked to write a map $$C\rightarrow H,z \rightarrow zv$$ for $$v \in H=C^3\otimes C^2$$, $$v=[1, 0, 0, 1, 0, 1]$$ in bra-ket notation.

However, I never written such a map before, and must have missed the lecture. I would very much appreciate if someone show me some example of how to write this kind of thing.

you can find explanations of braket notation in here and links therein. Could you spell out what specifically (if anything) you find unclear in what is said there?

– glS – 2019-02-26T20:11:19.347

## Answers

2

This map takes a complex number and returns a ket vector. In your case, this would simply mean writing $$f(z) = z | v \rangle$$.

Maybe a part of the exercise is to decipher $$v$$, then I would also expand it in ket notation: $$| v \rangle = |0 \rangle \otimes |0 \rangle + |1\rangle \otimes |0 \rangle + |2\rangle \otimes |1 \rangle$$.

It is somewhat tricky at first to think about vectors as maps from complex numbers to the vector space. If you want to describe a linear map $$\mathbb{C} \rightarrow H$$, then it's sufficient to specify $$f(1)$$, which is $$v$$.