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Is there a particular name for a quantum state of the form (up to the normalization):

$$\sum_{i_1+\ldots+i_n = d-1} |i_1\rangle |i_2\rangle \ldots |i_n\rangle$$

or was it studied is some papers?

This state is a permutation symmetric state of $n$ qudits (i.e. $d$-level particles), and defined as a superposition (with equal amplitudes) of all compoments for which particle indices sum up to $d-1$.

For qudits ($d=2$) it is a well known W state (or here), i.e. a permutation symmetric sum of one $|1\rangle$ among $n-1$ $|0\rangle$s.

For example, for 3 qutrits ($d=3$) it is $$|002\rangle + |020\rangle + |200\rangle + |011\rangle + |101\rangle + |110\rangle$$ and for $n=3$, $d=4$: $$|003\rangle + |030\rangle + |300\rangle + |012\rangle + |021\rangle + |102\rangle + |120\rangle + |201\rangle + |210\rangle + |111\rangle.$$

(I am writing a paper using such state, and would like to reference it; or at least - not reinvent its name.

Yes, it looks like a state for $n$ harmonic oscillators energy summing up to a particular value; however, here this "equal amplitudes" part is crucial, summing to $d-1$ is also of a big importance (albeit less crucial, as one can cut qudits to qu($c<d$)its).)

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So, out of lack an already established name, I called it excitation states in:

P. Migdał, J. Rodriguez-Laguna, M. Lewenstein,
Entanglement classes of permutation-symmetric qudit states: symmetric operations suffice, arXiv:1305.1506.

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