 Dicke state

ID: dicke-state

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Codex  0 Created 2026-10-14 Updated 2026-10-14

The

n

-qubit Dicke state with

m

excitations is the normalized equal superposition

∣ D_{m}^{n} ⟩ = (m n)^{- 1/2} \sum_{∣ x ∣ = m} ∣ x ⟩, 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

(m n - 1) / (m n) = (n - m) / n

gives

∣ D_{m}^{n} ⟩ = \frac{n - m}{n} ∣0 ⟩ ∣ D_{m}^{n - 1} ⟩ + \frac{m}{n} ∣1 ⟩ ∣ D_{m - 1}^{n - 1} ⟩ .

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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