Wigner non-negative states that verify the Wigner entropy conjecture

Abstract

We present further progress, in the form of analytical results, on the Wigner entropy conjecture set forth in https://link.aps.org/doi/10.1103/PhysRevA.104.042211 and https://iopscience.iop.org/article/10.1088/1751-8121/aa852f/meta. Said conjecture asserts that the differential entropy defined for non-negative, yet physical, Wigner functions is minimized by pure Gaussian states while the minimum entropy is equal to $1+\ln \pi$. We prove this conjecture for the qubits formed by Fock states $|0⟩$ and $|1⟩$ that correspond to non-negative Wigner functions. In particular, we derive an explicit form of the Wigner entropy for those states lying on the boundary of the set of Wigner non-negative qubits. We then consider general mixed states and derive a sufficient condition for Wigner non-negativity. For states satisfying our condition we verify that the conjecture is true. Lastly, we elaborate on the states of the set which is in accordance with our condition.

References in corpus (10)

• Entropic Uncertainty Relations and their Applications
• Continuous-variable entropic uncertainty relations
• Quantum Wigner entropy
• Entropy-power uncertainty relations : towards a tight inequality for all Gaussian pure states
• Continuous majorization in quantum phase space
• Detecting continuous variable entanglement in phase space with the $Q$-distribution
• Complex-valued Wigner entropy of a quantum state
• Quantum to classical parton dynamics in QCD media
• Quantum Thermodynamic Uncertainty Relations, Generalized Current Fluctuations and Nonequilibrium Fluctuation-Dissipation Inequalities
• Thermodynamics of quantum information in noisy polarizers

Cited by in corpus (2)

• Wigner entropy conjecture and the interference formula in quantum phase space
• Upper bounds on the purity of Wigner positive quantum states that verify the Wigner entropy conjecture