bibkey: gurvitsbarnum2002largest authors: Leonid Gurvits and Howard Barnum year: 2002 title: “Largest separable balls around the maximally mixed bipartite quantum state” doi: 10.1103/PhysRevA.66.062311 url: https://doi.org/10.1103/PhysRevA.66.062311 claim: “Theorem 1: I + Δ is separable for every Hermitian Δ with Frobenius norm at most one. Proposition 1: the block matrix with identity diagonal blocks and off-diagonal block X is separable when the operator norm of X is at most one.” strata_touched:
- D5/S3/Quantum/Entanglement/AbsoluteSeparability/GurvitsBarnumBall
- D5/S3/Quantum/Entanglement/AbsoluteSeparability/ContractionBlocks license: citation-only triage: anchor
Verified locator
Source: https://doi.org/10.1103/PhysRevA.66.062311
DOI: 10.1103/PhysRevA.66.062311 (Phys. Rev. A 66, 062311). Text read from https://arxiv.org/abs/quant-ph/0204159v2.
Theorem 1 (printed p. 4) reads:
The matrix I + ∆ is separable for all hermitian ∆ with ||∆||₂ ≤ 1.
Proposition 1 (printed p. 2) reads:
If ||X|| ≤ 1, the block matrix is separable.
Corollary 2 (scaling, printed p. 5) reads:
Let A be an (unnormalized) density matrix of a bipartite system with total dimension d = NM and λ = (λ₁,…,λ_d) be the vector of eigenvalues of A. If S(λ) =: d − ||λ||₁²/||λ||₂² ≤ 1 then A is separable.
Its proof writes A = b(I + ∆) with b > 0 and ||∆||₂² ≤ 1. The paper proves Theorem 1 through positive unital maps and the separability criterion by positive maps.
S. J. Szarek, E. Werner and K. Życzkowski, Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive, J. Math. Phys. 49, 032113 (2008), https://doi.org/10.1063/1.2841325 (text read from https://arxiv.org/abs/0710.1571v2, printed p. 18), state the dual inequality and ask for a direct proof:
If M = (M_jk) is a block-positive matrix, then Tr (M²) ≤ (Tr M)². It would be nice to have a simple direct proof of the above inequality, as it would yield (via Lemma 1 and (21)) an alternate derivation of the result from [13] concerning the in-radius of the set of separable states in the bivariate case.