Metadata
bibkey: louvet2025symmetricppt
authors: J. Louvet, E. Serrano-Ensástiga, T. Bastin, J. Martin
year: 2025
title: Nonequivalence between absolute separability and positive partial transposition in the symmetric subspace
doi: 10.1103/PhysRevA.111.042418
url: https://arxiv.org/abs/2411.16461v2
claim: Conjecture 1 states that symmetric N-qudit states with the specified p-spectrum are symmetric absolutely PPT for p between the conjectured threshold and one.
strata_touched:
- D5/S3/Quantum/Entanglement/AbsoluteSeparability/LouvetSymmetricAbsolutelyPPT
license: citation-only
triage: anchor
Verified locator
DOI: https://doi.org/10.1103/PhysRevA.111.042418
Source: https://arxiv.org/abs/2411.16461v2
Conjecture 1, Section II, p. 3: “Any symmetric N-qudit state ρ with a spectrum composed of D = binom(N+d−1, d−1) non-zero eigenvalues of the form (1 − (D−1)p/D, p/D, …, p/D), with p ∈ [p_min, 1] where p_min = 1/(1 + 2[D binom(N, ⌊N/2⌋)]^{−1}), is SAPPT.”
Here SAPPT means that every unitary acting on the symmetric subspace preserves positivity of every partial transpose. On the full tensor product, the characteristic polynomial includes the remaining d^N − D zero eigenvalues.
The proof combines a positive partial-transpose floor for the permutation symmetrizer with a lower bound of minus one half for the partial transpose of a unit rank-one state. Exact minimum eigenvalues and sharpness below the threshold in arbitrary qudit dimensions are separate questions.
The source states on p. 1: “They satisfy ρ = πρπ′ with π and π′ any pair of permutation operators.” It states on p. 2: “A symmetric state ρ is said to be symmetric absolutely PPT (SAPPT) [20] if UρU† is PPT for any U ∈ SU(N + 1) [24].” The full tensor-space unitary encoding preserves the symmetric subspace; the conjecture concerns d local levels.
The source states on p. 1: “In this work, we are interested in states that are PPT for any bipartition, and we will call them just as PPT states.”