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Entanglement-entropy count conjecture for Dicke-state orbits

Let ∣D1N⟩\ket{D^N_1} denote the NN-qubit Dicke state with one excitation, and let (HC)1,2(HC)_{1,2} denote circuits generated by Hadamard and controlled-not gates acting on qubits 11 and 22. Write ∣sN∣|s_N| for the number of unique entanglement entropies comprising all entropy vectors in the (HC)1,2(HC)_{1,2} orbit of ∣D1N⟩\ket{D^N_1}. Entanglement-entropy count conjecture. For N≥2N\geq 2, this number is

∣sN∣=⌊5N−72⌋.|s_N|=\left\lfloor\frac{5N-7}{2}\right\rfloor.

The conjecture predicts how the variety of individual entanglement entropies grows with the number of qubits, even though the orbit contains at most five distinct entropy vectors. Its status is not resolved in the supplied source context.

Sources & referencesView supporting material

Primary source

William Munizzi and Howard J. Schnitzer, “Entropy Cones and Entanglement Evolution for Dicke States”, arXiv:2306.13146 (2023).

dicke-statesentanglement-entropyquantum-information

Progress summary

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Open

The proposed formula remains an unproved conjecture: computations support it, but no proof or counterexample was found.

A 2023 preprint formulates the conjecture for the number of distinct entanglement entropies in the (HC)1,2(HC)_{1,2} orbit of ∣D1N⟩\lvert D^N_1\rangle, predicting ∣sN∣=⌊(5N−7)/2⌋\lvert s_N\rvert=\left\lfloor(5N-7)/2\right\rfloor for N≥2N\geq 2. It also observes that, for N≥3N\geq 3, the orbit has five entropy vectors even though their individual entropy values proliferate.

Known results

  • Explicit calculations in the relevant analysis cover N≤10N\leq 10 and support the stated pattern.
  • For N≥3N\geq 3, five distinct entropy vectors are reported in the orbit.
  • The conjecture is stated without a proof or counterexample.

2023 Clifford-double-coset follow-up

A later study derives upper bounds for entropy-vector reachability graphs and discusses Dicke-state symmetry, but it does not prove, disprove, or otherwise resolve the individual-entropy count conjecture.

Current status (as of August 2026): The formula is supported by finite computations but remains an open conjecture, with no publicly reported proof, counterexample, or verification found.

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Solutions 0

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