Can you show a complete code example to reproduce this including imports? What should the correct answer be?

It might be a bug in Sum rather than qapply.

The quantum module is currently unmaintained. Feel free to submit a PR to fix this.

Following are the imports I used to reproduce this issue:

import sympy as sp
from sympy.abc import k,m,J

import sympy.physics.quantum as qt

import sympy.concrete.summations as sm

The OP's statement state = sm.Sum((2**m)*qt.spin.JzKet(J, m), (m, -J, J)) produces the output to be:

which, while "looks" correct to our expectation, is not, as is indicated by:

This shows that the object is not behaving as a quantum state, but instead the $\sum_m 2^m$ chunk of the expression is being interpreted by sympy as a separate, geometric summation. The kets $|J,m\rangle$ ought to be part of the summation as well.

Hence, I do believe that it's an inadequacy of Sum, as suggested—an incompatibility of Sum and Quantum objects' implementations.

Above,state is a <class 'sympy.concrete.summations.Sum'> object, while one would expect a correct implementation to return an object of perhaps some subclass of <class 'sympy.physics.quantum.state'>.

These seem to be the reasons (or more) for the OP's issue, hereby seen reproduced with a slight deviation (suspected to be from some update ever since):

Attempting to calculate OP's desired norm expression by alternatively using the innerproduct method, hints that we should perhaps somehow try to implement returning a sympy.physics.quantum.state.KetBase object.

I am wondering if this issue can be solved, because I also confront the same problem when I want to define a thermal state in Fock basis.

The problem here is that the summation function cannot recognize the dummy index inside the braket notation like sum over k for |k><k|.