Dicke State

Dicke state is a symmetric superposition of all $n$-qubit computational basis states with exactly $k$ qubits in the $|1\rangle$ state. Dicke states generalize W states (the special case $k=1$) and are resources for quantum metrology and quantum error correction.

For $n$ qubits, the Dicke state $D_n^{(k)}$ is:

$$D_n^{(k)} = \frac{1}{\sqrt{\binom{n}{k}}} \sum_{\text{all states with } k \text{ ones}} |x_1 x_2 \cdots x_n\rangle$$

The normalization factor is the binomial coefficient $\binom{n}{k}$, ensuring the state is normalized.

• $D_3^{(1)}$ (three qubits, one 1): $\frac{1}{\sqrt{3}}(|100\rangle + |010\rangle + |001\rangle)$ — the W state
• $D_3^{(2)}$ (three qubits, two 1s): $\frac{1}{\sqrt{3}}(|110\rangle + |101\rangle + |011\rangle)$
• $D_3^{(0)}$ (three qubits, zero 1s): $|000\rangle$ — the ground state
• $D_3^{(3)}$ (three qubits, three 1s): $|111\rangle$ — all qubits excited

• Permutation symmetry: invariant under permutations of qubits; measurement outcome is the Hamming weight, not which qubits are in $|1\rangle$
• Entanglement: maximally entangled for fixed Hamming weight; measurement collapses all remaining qubits
• Superposition structure: equal-amplitude superposition ensures all resource phases are uniform
• Edge states: $D_n^{(0)} = |0\cdots0\rangle$ and $D_n^{(n)} = |1\cdots1\rangle$ are product states, not entangled

Measuring a Dicke state in the computational basis yields one of the $\binom{n}{k}$ basis states with equal probability $1/\binom{n}{k}$. The outcome is a fixed Hamming weight, a key distinction from arbitrary superpositions.

• Quantum metrology: Dicke states enable beating the shot-noise limit in phase estimation; $k=n/2$ is optimal for many metrological tasks
• Quantum error correction: symmetric entanglement aids in detecting particle-loss errors
• QAOA and variational algorithms: Dicke states are natural outputs of certain ansätze
• Quantum networks: robust entanglement for distributed quantum sensing

• W state: special case where $k=1$, one qubit excited
• GHZ state: different entanglement structure ($\frac{1}{\sqrt{2}}(|0\cdots0\rangle + |1\cdots1\rangle)$), all or nothing rather than fixed weight
• Permutation-invariant states: Dicke states are the basis for all fully symmetric multi-qubit states

Create $D_n^{(k)}$ via:

1. Prepare all qubits in $|0\rangle$
2. Apply Hadamards to $k$ designated qubits to create equal superposition
3. Apply controlled-X gates to enforce the symmetric superposition and fixed Hamming weight

Efficient preparation requires $O(n \log n)$ gates; for arbitrary $k$ in fixed-size systems, direct state preparation is often easier than gate synthesis.

As $n$ grows, Dicke states with $k \approx n/2$ become increasingly entangled and resource-intensive to prepare. Classical simulation of Dicke states with $k$ far from 0 or $n$ requires exponential resources.