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 informationtheoretic properties of Wigner functions in phase space. After identifying all Gaussian pure states of a harmonic oscillator as equivalent in the precise sense of continuous majorization, which can be well understood in light of Hudson's theorem, we conjecture a fundamental majorization relation: any positive Wigner function is majorized by the Wigner function of the ground state (or vacuum state). As a consequence, any Shur-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 -hence also the Wehrl 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 the 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 the implications of the conjecture in the context of entropic uncertainty relations in phase space.

Quantum Wigner entropy

“…In this context, the theory of majorization [27] has proved to be another powerful tool, and it notably allows to formulate a generalization of Wehrl conjecture [7]. In a forthcoming paper [28], we use the theory of majorization to state a stronger conjecture on the uncertainty content of Wigner functions. This enables us, for instance, to demonstrate analytically the lower bound on h(W ) for all phase-invariant Wigner-positive states in S 2 + , including the dark blue region.…”

“…Let us also define the usual radial parameter r = x 2 + p 2 , so that each value of t corresponds to a specific value of r through the relation t = 2r 2 . When condition (28) is fulfilled for some t, the Wigner function is non-negative at r = √ t/2. We call S + the restriction of S satisfying the Wigner-positivity conditions (28), so that any vector p in S + is associated with a unique phase-invariant Wigner-positive state in Q + .…”

“…When condition (28) is fulfilled for some t, the Wigner function is non-negative at r = √ t/2. We call S + the restriction of S satisfying the Wigner-positivity conditions (28), so that any vector p in S + is associated with a unique phase-invariant Wigner-positive state in Q + .…”

“…Each value of t in Eq. (28) gives the equation of a hyperplane dividing S in two halves [p must be located on one side of the hyperplane to guarantee that W (r) 0 for the corresponding r]. Two hyperplanes associated respectively to t and t + dt intersect in a (lower-dimensional) hyperplane which is at the boundary of the convex set S + .…”

“…Two of them correspond to the physicality conditions (27), i.e., p 1 0 and p 2 0. The third one corresponds to condition (28) where we have set t = 0, which FIG. 7.…”