bibkey: alsingbirrittella2024lognegativity
authors: P. M. Alsing; R. J. Birrittella
year: 2024
title: "Numerical Evidence for a Bipartite Pure State Entanglement Witness from Approximate Analytical Diagonalization"
doi: 10.3390/foundations5020019
url: https://arxiv.org/abs/2404.13725v1
claim: "Equation (22) numerically supports the logarithmic-negativity sandwich for positive Hermitian amplitude ensembles and upper equality for real amplitudes."
strata_touched:
  - D5/S3/Quantum/Information/AlsingBirrittellaLogNegativitySandwich
license: citation-only
triage: anchor

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DOI: https://doi.org/10.3390/foundations5020019

Source: https://arxiv.org/abs/2404.13725v1

The Alsing–Birrittella logarithmic-negativity sandwich

The source is arXiv:2404.13725v1; the journal citation is Foundations 5(2) (2025), 19. The arXiv PDF and TeX supply the statements below. The journal DOI is supplied by the literature screen; its publisher text is unverified.

The introduction, p. 1, calls logarithmic negativity

A widely used “measure” of entanglement for both pure and mixed states, both discrete and continuous variable, is the Log Negativity (LN)

and gives $LN({\rho }_{ab})={\log }_2(1+2N)$, followed by

where the negativity N is given by the sum of the absolute values of the negative eigenvalues of the partial transpose of ρ_ab.

Equation (11a), p. 4, defines

$$N_a(|\psi {⟩}_{ab}):=\frac{1}{2}\sum _n\sum _{m\ne n}\left|\mathrm{Det}\left(\begin{array}{cc}{c}_{nn}& c_{nm}\\ c_{mn}& c_{mm}\end{array}\right)\right|.$$

Equation (11b) defines $L{N}_a(|\psi {⟩}_{ab}):={\log }_2(1+2N_a)$. Equations (17b)–(17c), p. 8, give the mixed-state expressions

$$N_a\to \frac{1}{2}\sum _{n,m\ne n}|{\rho }_{nn,mm}-{\rho }_{nm,mn}|,L{N}_a={\log }_2(1+2N_a).$$

Both are half-sums over ordered unequal pairs. Exact logarithmic negativity is $L{N}_e(\rho )={\log }_2\Vert {\rho }^{\Gamma }{\Vert }_1$, with partial transposition on the second factor. The ensemble average is ${\mathcal{L}\mathcal{N}}_a^{avg}={\sum }_i{p}_{i}L{N}_a(|{\psi }_i⟩)$.

Section IV, p. 11, Equation (22), states:

we have numerically found that 𝓛𝓝_a^avg(ρ_ab) ≤ LN_e(ρ_ab) ≤ LN_a(ρ_ab) when the mixed state is written as a pure state decomposition (PSD) ρ_ab = Σ_i p_i |ψ_i⟩⟨ψ_i|, and in addition the quantum amplitude matrix C_i of each pure state component |ψ_i⟩ = Σ_nm C_i,nm |n,m⟩ is uniformly generated (over the Haar measure) as a positive Hermitian matrix, C_i = C_i† ≥ 0.

Further, we find that the second inequality is surprisingly saturated, i.e. LN_e(ρ_ab) = LN_a(ρ_ab) in Eq.(22), if the uniformly randomly generated C_i matrices above are real, i.e. C_i = C_i^T.

The finite-dimensional formulation uses arbitrary positive semidefinite complex amplitudes with $\mathrm{tr}(C_i{C}_i^{†})=1$ and nonnegative weights summing to one. It therefore includes the sampled class. Hermitian amplitudes are real exactly when they are transpose symmetric. No common eigenbasis or pairwise commutation assumption is imposed. The general pure-state witness question for arbitrary amplitude matrices is outside this statement.

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