A quantum uncertainty relation based on Fisher's information

We explore quantum uncertainty relations involving the Fisher information functionals I-x and I-p evaluated, respectively, on a wavefunction Psi(x) defined on a D-dimensional configuration space and the concomitant wavefunction (Psi) over tilde (p) on the conjugate momentum space. We prove that the associated Fisher functionals obey the uncertainty relation IxIp >= 4D(2) when either Psi(x) or (Psi) over tilde (p) is real. On the other hand, there is no lower bound to the above product for arbitrary complex wavefunctions. We give explicit examples of complex wavefunctions not obeying the above bound. In particular, we provide a parametrized wavefunction for which the product IxIp can be made arbitrarily small.