Dicke state |D²₄⟩

The Dicke state on $n$ qubits with $k$ excitations is

$|D^n_k\rangle = \binom{n}{k}^{-1/2} \sum_{\substack{x\in\{0,1\}^n \\ |x|=k}} |x\rangle,$

the uniform superposition over every computational-basis string of Hamming weight $k$. For $n=4,\,k=2$ this is

$|D^2_4\rangle = \frac{1}{\sqrt6}\big(|0011\rangle+|0101\rangle+|0110\rangle+|1001\rangle+|1010\rangle+|1100\rangle\big).$

Symmetric-subspace structure. $|D^n_k\rangle$ is the unique (up to phase) state in the totally symmetric subspace of $(\mathbb{C}^2)^{\otimes n}$ with definite total excitation number $k$ — equivalently, it is the highest-weight-in-$m$ eigenstate of collective total angular momentum operators $J^2 = j(j+1)$ with $j=n/2$ and $J_z = k - n/2$. This makes Dicke states the natural basis for collective spin / atomic-ensemble physics, where permutation symmetry is enforced by the physical setup (indistinguishable atoms coupled identically to a shared field).

Relation to $W$. $|D^n_1\rangle$ is exactly the $n$-qubit $W$ state; Dicke states are therefore the full family of "how many excitations, spread evenly" states, with $W$ sitting at $k=1$.

Construction. The circuit here uses the deterministic Bärtschi–Eidenbenz recursive scheme: prepare $|D^{n-1}_{k}\rangle \otimes |0\rangle$ and $|D^{n-1}_{k-1}\rangle\otimes|1\rangle$ in superposition with a controlled rotation, weighted by the ratio of binomial coefficients so probabilities of adding a 0 vs. a 1 exactly match $\binom{n-1}{k}/\binom{n}{k}$ and $\binom{n-1}{k-1}/\binom{n}{k}$. The rotation angles are therefore closed-form functions of $n$ and $k$, not fitted numerically.

Significance. Dicke states are used as robust reference states for metrology (their quantum Fisher information for phase estimation scales favorably with $k$), for symmetric quantum error-detection codes, and appear as ground states of ferromagnetic Heisenberg-type Hamiltonians restricted to a fixed magnetization sector.