bibkey: choikiemkye2020maximalbirank authors: J. Choi, Y.-H. Kiem, and S.-H. Kye year: 2020 title: Entangled edge states of corank one with positive partial transposes doi: 10.1063/1.5122836 url: https://arxiv.org/abs/1903.10745v2 claim: Conjecture 6.4 states that for every n ≥ 3 the open set of parameters yielding PPT entangled edge states of bi-rank (n²−1,n²−2n+3) is nonempty. strata_touched:

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PPT edge states of maximal bi-rank

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DOI: 10.1063/1.5122836 Source: https://arxiv.org/abs/1903.10745v2

Section 6 defines the matrices by items (1)–(7), the unit-modulus parameter conditions, and the largest real determinant root. Proposition 6.1 gives the simple-root and kernel-vector criterion for an edge state of corank one. Conjecture 6.4, p. 13, states:

“For any n ≥ 3, the open set of parameters (α₁, ⋯, αₙ, β₁, ⋯, βₙ) which give PPT entangled edge states of bi-rank (n² − 1, n² − 2n + 3) is nonempty.”

The encoding uses the first-factor partial transpose and the source-consistent Section 6 expressions recorded in preregistration issue #13859. A nonzero edge operator expresses entanglement; the matrices are unnormalized positive operators.

The source defines the first-factor convention on p. 1:

“The partial transpose (x ⊗ y)Γ of x ⊗ y ∈ Mₘ ⊗ Mₙ is given by xᵗ ⊗ y with the usual transpose xᵗ.”

The edge condition on p. 1 is:

“There exists no nonzero product vector |ξᵢ⟩ ⊗ |ηᵢ⟩ ∈ Im ϱ such that |ξ̄ᵢ⟩ ⊗ |ηᵢ⟩ ∈ Im ϱΓ. Such states are called PPT entangled edge states [19], in short, edge states in this paper.”