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If you view $\sum_{i} \langle i |$ as a (non-properly normalized) (bra) state $\sqrt{d}\langle \psi |$, where $|\psi\rangle = \frac{1}{\sqrt{d}}\sum_{x \in \{0,1\}^{n}}|x\rangle$, the quantity $c$ becomes just the inner product of $\sqrt{d}\langle \psi |U|j\rangle$. Here, $d =2^{n}$ is the proper normalization constant.

Without imposing any other constraints on $U$, $U|j\rangle$ just becomes some random state $|\phi\rangle$. Thus, $c = \sqrt{d}\langle \psi|\phi\rangle$ and the only thing we can conclude is:

$$
0 \leq |c| \leq \sqrt{d}
$$

Are you interested in just the inner products or their squares? – Hasan Iqbal – 2020-11-24T22:04:52.813

@HasanIqbal I am only interested in the inner product given above, not the squares – Oli – 2020-11-25T11:08:50.813