= Dicke state
{c}
{title2=$|D_m^n\rangle$}
The $n$-<qubit> Dicke state with $m$ excitations is the normalized equal superposition
$$
|D_m^n\rangle=\binom nm^{-1/2}\sum_{|x|=m}|x\rangle,\qquad 0\leq m\leq n,
$$
where $|x|$ is the <Hamming weight> in the <computational basis>. The <binomial coefficient> counts the mutually orthogonal summands, proving normalization. Every permutation of the qubits leaves the vector unchanged. Splitting the sum according to the first bit and using $\binom{n-1}{m}/\binom nm=(n-m)/n$ gives
$$
|D_m^n\rangle=\sqrt{\frac{n-m}{n}}|0\rangle|D_m^{n-1}\rangle+\sqrt{\frac mn}|1\rangle|D_{m-1}^{n-1}\rangle.
$$
Terms with zero coefficient are omitted at the endpoints. This recursion is useful for <partial traces> and for distributing a logical <qubit> among many symmetric physical qubits.
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