Dirac notation: what if we display tensor products vertically? Ok hear me out.
One thing I have always liked about Dirac notation is that it visually displays where expressions expect inputs/outputs. For example $\langle\psi|$ expects an input to the right to form a complete expression like $\langle\psi|\phi\rangle$. The angle brackets indicate the expression is 'complete'. Things like $|\psi\rangle\langle\phi|$ require input both on the left and the right as indicated by the vertical bars. Now for tensor products this behaviour kind of breaks down. Expressions like $|n\rangle|\phi\rangle\equiv|n\rangle\otimes|\phi\rangle$ have two inputs but the input of the second ket is stuck behind the first ket. If you calculate something like $\langle m|\langle\psi|\langle\alpha||n\rangle|\phi\rangle|\beta\rangle=\langle m| n\rangle \langle\psi |\phi\rangle \langle\alpha |\beta\rangle$ the corresponding inputs and outputs are separated really far.
This might not sound like a big a deal (it isn't a big deal) but what if we stacked tensor products vertically? Like
$$|n\rangle|\phi\rangle\equiv\begin{array}{c}|n\rangle\\|\phi\rangle\end{array}$$
Then the triple dot product from earlier becomes
$$\begin{array}{c}\langle m|\\\langle\psi|\\\langle\alpha|\end{array}\cdot\begin{array}{c}|n\rangle\\|\phi\rangle\\|\beta\rangle\end{array}=
\begin{array}{c}\langle m| n\rangle\\ \langle\psi |\phi\rangle \\\langle\alpha |\beta\rangle \end{array}=
\langle m| n\rangle \langle\psi |\phi\rangle \langle\alpha |\beta\rangle$$
and products of general operators become
$$(A\otimes B)\cdot(C\otimes D)=\begin{array}{c}A\\B\end{array}\cdot \begin{array}{c}C\\D\end{array}=\begin{array}{c}A\cdot C\\B\cdot D\end{array}=(A\cdot B)\otimes(C\cdot D)$$
So there are two benefits:

*

*Inputs/outputs are visually closer. So this is subjectively easier to understand.

*Expression with tensor products tend to work 'in parallel', that is each term in a tensor product often works separately from the others. The $|n\rangle$ and $|\phi\rangle$ in my example could be from entirely different Hilbert spaces. This vertical notation displays this visually because each tensor product term gets a separate row and the rows never talk to eachother.

Any thoughts? Improvements? I don't expect this notation will ever catch on because it is quite unwieldy but I hope this question gives you a fresh view on tensor products
 A: Well, "de gustibus et de coloribus non est disputandum"... As indicated in my comment above, there is a red alert weakness, and steering users away from the evident enticing misunderstanding is tough. But you never know.
Suppose the dimension of $|n\rangle$ is a, take it to be 3, and that of $|\phi\rangle$ is b, take it to be 4.
Then the dimension of the vector $|n\rangle|\phi\rangle$ is ab, so, 12.
However, the proposed notation,
$$|n\rangle|\phi\rangle\equiv\begin{array}{c}|n\rangle\\|\phi\rangle\end{array}$$
all but invites the reader to think of the r.h.s. as the pileup of the two vectors, i.e. an a+b dimensional vector, here 7-d, (so $|n\rangle \oplus|\phi\rangle$), and not a 12-d vector.
You might beat your readers to not misunderstand the obvious, but somebody always will. (I am reminded of the trick-question calculus wag, who'd use the $x(f)$, $~~dx/df$,  $~~d^2x/df^2$,... notation in trivial questions and stump his graders.)  After virtually a century of Clebsching, it is a tall order to rescind collective public expectations.
