bibkey: girard2017binegativity authors: Mark W. Girard; Gilad Gour year: 2017 title: “The binegativity of two qubits” doi: 10.48550/arXiv.1701.02724 url: https://arxiv.org/abs/1701.02724v3 claim: “For a two-qubit state sigma with partial transpose sigma^Gamma, the negativity is N(sigma) = 2 Tr[(sigma^Gamma)-] and the binegativity is N_2(sigma) = Tr[(sigma^Gamma)-] + 2 Tr[(((sigma^Gamma)-)^Gamma)-], where X_+ and X_- are the positive and negative components of a self-adjoint X. Based on randomly generated states, the paper conjectures that nu (c + nu)(nu + 1)/((c + nu)^2 + 2c(1 - c)) <= N_2(sigma) <= (nu/2)(c + nu)^2/(c^2 + nu^2) for all states with negativity nu and concurrence c.” strata_touched:

• D5/S3/Quantum/Entanglement/TwoQubitBinegativityUpperBoundRefutation license: citation-only triage: anchor

The binegativity of two qubits

Mark W. Girard, Gilad Gour, arXiv:1701.02724v3 (quant-ph), 2017. No journal version is listed on arXiv. Quotations are from the arXiv v3 source, with its macros kept as printed (\Gammaminus is ${\Gamma }_{-}$).

The negativity:

The negativity can be written as N(\sigma) = 2\Tr[\sigma^{\Gammaminus}], where we use the notation \sigma^{\Gammaplus}:=(\sigma^\Gamma)_+ \quad\text{ and }\quad \sigma^{\Gammaminus}:=(\sigma^\Gamma)_- to denote the positive and negative components of the partially transposed $\sigma$, respectively.

The binegativity:

In this paper we introduce a computable quantity that we call \emph{binegativity} given by N_2(\sigma):=\Tr[\sigma^{\Gammaminus}] + 2\Tr[\sigma^{\Gammaminus\Gammaminus}], where \sigma^{\Gammaminus\Gammaminus} = ((\sigma^\Gammaminus)^\Gamma)_-.

The conjecture (Eq. (9)):

In general, based on numerical evidence from randomly generated states, we conjecture that the the binegativity of two-qubit states is bounded by $\nu \frac{(c+\nu )(\nu +1)}{(c+\nu {)}^2+2c(1-c)}\le N_2(\sigma )\le \frac{\nu }{2}\frac{(c+\nu {)}^2}{c^2+{\nu }^2}$ for all states $\sigma$ with fixed negativity $N(\sigma )=\nu$ and concurrence $C(\sigma )=c$.

The concurrence is the standard two-qubit concurrence (the paper cites Hill–Wootters and Wootters): the infimum of ${\sum }_i{p}_{i}C({\psi }_i)$ over the decompositions $\sigma ={\sum }_i{p}_i|{\psi }_i⟩⟨{\psi }_i|$, with $C(\psi )=2|{\psi }_{00}{\psi }_{11}-{\psi }_{01}{\psi }_{10}|$.

The encoding indexes two-qubit matrices by Fin 2 × Fin 2, takes the partial transposition on the second qubit (the binegativity does not depend on which qubit is transposed), uses Mathlib’s negative part of a self-adjoint matrix, and writes the negativity as in the paper, N(\sigma)=2\Tr[\sigma^{\Gamma}_-].

Verified locator

• DOI: https://doi.org/10.48550/arXiv.1701.02724 (the arXiv DOI; no journal version is listed).
• URL: https://arxiv.org/abs/1701.02724v3 (v3, 2017-02-10, the latest version; arXiv comment “v3: previous versions contained an erroneous proof claiming that the binegativity is a monotone”; source binegativity_v9.tex, md5 377c42f23d1a50af36324a4ea5bfd221): the negativity (l. 178–186), the binegativity (l. 188–192) and the conjecture (l. 277–281).