bibkey: baiochruscinskihorodeckimessinasarbicki2019fixedmarginals
authors: Giuseppe Baio; Dariusz Chruściński; Paweł Horodecki; Antonino Messina; Gniewomir Sarbicki
year: 2019
title: "Bounds on the entanglement of two-qutrit systems from fixed marginals"
doi: 10.1103/PhysRevA.99.062312
url: https://arxiv.org/abs/1904.07650v2
claim: "For two qutrits with fixed ordered marginals, the authors maximize the negativity (Eq. (4)) within the circulant family (22), obtain the quasidistillable candidates (27)–(29), and conjecture from numerical optimization that these candidates are the maximally entangled states with respect to the fixed marginals."
strata_touched:
  - D5/S3/Quantum/Entanglement/FixedMarginalNegativityRefutation
license: citation-only
triage: anchor

Baio, Chruściński, Horodecki, Messina and Sarbicki, entanglement from fixed marginals

G. Baio, D. Chruściński, P. Horodecki, A. Messina and G. Sarbicki, Bounds on the entanglement of two-qutrit systems from fixed marginals, Phys. Rev. A 99, 062312 (2019), DOI 10.1103/PhysRevA.99.062312; arXiv:1904.07650 [quant-ph] (v1 16 April 2019, v2 18 April 2019; arXiv title Bounds on bipartite entanglement from fixed marginals).

Verified locator

DOI: 10.1103/PhysRevA.99.062312

Source: https://arxiv.org/abs/1904.07650v2; the text was read from the PDF.

• page 1: the negativity, Eq. (4);
• pages 2–3: the two-qubit maximally entangled marginally mixed states, Eqs. (5)–(13);
• pages 4–5: the family (22), the marginals (24), the negativity formula (25)–(26), the candidates (27)–(29) and the matrix (30);
• pages 6–7: Section IV, the numerical comparison and the conjecture.

Source statements

Eq. (4): “N(ρ_AB) ≡ (1/2)(‖ρ^τ_AB‖₁ − 1)”.

Two-qubit case: “Two-qubit MEMMS states are thus achieved maximizing concurrence or negativity of states in the form (7).”

Eq. (24): “ρ_A = diag{1−λ₁−λ₂, λ₁, λ₂}, ρ_B = diag{1−μ₁−μ₂, μ₁, μ₂}, where λ₁ ≥ λ₂, μ₁ ≥ μ₂ correspond to the decreasingly ordered local eigenvalues and, thus, λ₂, μ₂ ≤ 1/3. Without loss of generality we also assume λ₁ + λ₂ ≥ μ₁ + μ₂.”

Eq. (27): “ρ̃⁽¹⁾_AB = (1−p₁₀−p₂₀)|Ψ⁽¹⁾_mc⟩⟨Ψ⁽¹⁾_mc| + p₁₀|10⟩⟨10| + p₂₀|20⟩⟨20|, p₁₀ = λ₁ − μ₁, p₂₀ = λ₂ − μ₂ valid when λ₁ > μ₁ and λ₂ > μ₂”. The matrix (30) gives the entries 1−λ₁−λ₂, μ₁, μ₂ on |00⟩, |11⟩, |22⟩ with the coherences √(μ₁(1−λ₁−λ₂)), √(μ₂(1−λ₁−λ₂)) and √(μ₁μ₂).

Section III: “Maximum negativity within our family is then attained by the following three states”.

Section IV: “To summarize, all the above results strongly support our conjecture that our quasidistillable states (27, 28, 29) are legitimate candidates for two-qutrits MEMS with respect to fixed marginals and motivate the search for analytical proofs in further studies.”

Reading used by the repository

A MEMS with respect to fixed marginals maximizes the negativity over the two-qutrit states with those marginals, as in the two-qubit case and the numerical comparison of Section IV. In the region of Eq. (27), the repository states the conjecture as: every two-qutrit density matrix with marginals (24) has negativity at most that of the state (27), and refutes it at λ = (47, 4)/100, μ = (5, 1)/100.