Quantum entanglement as an extremal Kirkwood-Dirac nonreality

TL;DR

This work links bipartite entanglement to anomalous nonreal values of Kirkwood-Dirac quasiprobabilities by constructing an entanglement monotone E_KD^{NRe} based on the nonreality of Pr_KD over a product basis for A and an optimized AB basis. For pure states, E_KD^{NRe} has a closed-form, E_KD^{NRe}(|ψ⟩⟨ψ|_{AB}) = Tr_A{(ρ_A−ρ_A^2)^{1/2}}, making it a Schur-concave function of the Schmidt spectrum and a Vidal-type monotone; when extended to mixed states via convex roof, the normalized measure upper-bounds the concurrence and equals it for two-qubit states. The authors derive upper and lower bounds in terms of subsystem uncertainty, via a nonadditive entropy S_KD^{NRe}(ρ_A) and a trace-norm asymmetry, and provide a geometric interpretation as minimum disturbance from nonselective local measurements. They also propose weak-value-based estimation schemes and discuss connections to strange weak values and quantum contextuality, showing that nonzero KD-nonreality entanglement implies contextuality in a broad sense. Overall, the work offers a quantitative, operational bridge between entanglement and generalized nonclassicality captured by KD nonreality, with potential applications in quantum foundations and near-term quantum information processing.

Abstract

Understanding the relationship between various different forms of nonclassicality and their resource character is of great importance in quantum foundation and quantum information. Here, we discuss a quantitative link between quantum entanglement and the anomalous or nonclassical nonreal values of Kirkwood-Dirac (KD) quasiprobability, in a bipartite setting. We first construct an entanglement monotone for a pure bipartite state based on the nonreality of the KD quasiprobability defined over a pair of orthonormal bases in which one of them is a product, and optimizations over these bases. It admits a closed expression as a Schur-concave function of the state of the subsystem having a form of nonadditive quantum entropy. We then construct a bipartite entanglement monotone for generic quantum states using the convex roof extension. Its normalized value is upper bounded by the concurrence of formation, and for two-qubit systems, they are equal. We also derive lower and upper bounds in terms of different forms of uncertainty in the subsystem quantified respectively by an extremal trace-norm asymmetry and a nonadditive quantum entropy. The entanglement monotone can be expressed as the minimum total state disturbance due to a nonselective local binary measurement. Finally, we discuss its estimation using weak value measurement and classical optimization, and its connection with strange weak value and quantum contextuality.