Rényi-Wehrl entropies as measures of localization in phase space

Abstract

Abstract. We generalize the concept of the Wehrl entropy of quantum states which gives a basis-independent measure of their localization in phase space. We discuss the minimal values and the typical values of these Rényi-Wehrl entropies for pure states for spin systems. According to Lieb's conjecture the minimal values are provided by the spin coherent states. Though Lieb's conjecture remains unproven, we give new proofs of partial results that may be generalized for other systems. We also investigate random pure states and calculate the mean Rényi-Wehrl entropies averaged over the natural measure in the space of pure quantum states.

A Lower Bound for the Wehrl Entropy of Quantum Spin with Sharp High-Spin Asymptotics

“…A related conjecture has recently been discussed in the literature in the context of measures of localization in phase-space, see [GŻ01] and references therein. It appears there as a conjectured lower bound for the so-called Rényi-Wehrl entropy.…”

“…We henceforth refer to the left-hand side of the above inequality as an entropy of Rényi-Wehrl type of index q/2, but caution the reader that it differs from the definition of the usual Rényi-Wehrl entropy in the literature [GŻ01,Sug02] by the explicit q-dependence contained in the norm |||f ||| q . A virtue of the normalization used here is that neither (9) nor (11) are explicitly j-dependent.…”

Formulation of the uncertainty relations in terms of the Rényi entropies

“…By taking directly the logarithm of the inequality (16), and using the relations between α and β, we arrive at…”

Note on semiclassical uncertainty relations

“…We have advanced three original expressions, i.e., Eqs. (14), (22), (25) and (29). We have observed the invariance of the Fisher uncertainties under the escort transformation and, finally, we have shown that with the help of the escort distributions one is able to reobtain the canonical Heisenberg's uncertainties at the semiclassical level, thus clearly exhibiting the power of the escort-concept.…”

“…[21], in particular with reference to information instruments expressed in phase-space parlance. A main tool is that called the semi-classical Wehrl entropy W , a measure of phasespace localization [1,22] expressed via coherent states |z [2,23]. Coherent states are eigenstates of a general annihilation operatorâ, appropriate for the problem at hand [23][24][25], i.e.,â…”