Continuous majorization in quantum phase space

Abstract

We explore the role of majorization theory in quantum phase space. To this purpose, we restrict ourselves to quantum states with positive Wigner functions and show that the continuous version of majorization theory provides an elegant and very natural approach to exploring the information-theoretic properties of Wigner functions in phase space. After identifying all Gaussian pure states as equivalent in the precise sense of continuous majorization, which can be understood in light of Hudson's theorem, we conjecture a fundamental majorization relation: any positive Wigner function is majorized by the Wigner function of a Gaussian pure state (especially, the bosonic vacuum state or ground state of the harmonic oscillator). As a consequence, any Schur-concave function of the Wigner function is lower bounded by the value it takes for the vacuum state. This implies in turn that the Wigner entropy is lower bounded by its value for the vacuum state, while the converse is notably not true. Our main result is then to prove this fundamental majorization relation for a relevant subset of Wigner-positive quantum states which are mixtures of the three lowest eigenstates of the harmonic oscillator. Beyond that, the conjecture is also supported by numerical evidence. We conclude by discussing some implications of this conjecture in the context of entropic uncertainty relations in phase space.

19 pages, 9 figures; v2: several improvements and addition of Appendix C (version accepted in Quantum)

References in corpus (3)

• Majorization criterion for distillability of a bipartite quantum state
• Majorization approach to entropic uncertainty relations for coarse-grained observables
• Quantum Wigner entropy

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• Wigner non-negative states that verify the Wigner entropy conjecture
• Entropy estimation in a spin-1 Bose-Einstein condensate
• Wigner entropy conjecture and the interference formula in quantum phase space
• The Kirkwood-Dirac representation associated to the Fourier transform for finite abelian groups: positivity
• Information and majorization theory for fermionic phase-space distributions
• Upper bounds on the purity of Wigner positive quantum states that verify the Wigner entropy conjecture