time): The two simultaneous measurements on A and B are necessarily unsharp or weak. It is also possible to derive an uncertainty relation that, as the Ozawa's one, combines both the statistical and systematic error components, but keeps a form very close to the Heisenberg original inequality. By adding Robertson and Ozawa relations we obtain The four terms can be written as: Defining: as the inaccuracy in the measured values of the variable A and as the resulting fluctuation in the conjugate variable B, Fujikawa established an uncertainty relation similar to the Heisenberg original one, but valid both for systematic and statistical errors: Quantum entropic uncertainty principle For many distributions, the standard deviation is not a particularly natural way of quantifying the structure. For example, uncertainty relations in which one of the observables is an angle has little physical meaning for fluctuations larger than one period. Other examples include highly bimodal distributions, or unimodal distributions with divergent variance. A solution that overcomes these issues is an uncertainty based on entropic uncertainty instead of the product of variances. While formulating the many-worlds interpretation of quantum mechanics in 1957, Hugh Everett III conjectured a stronger extension of the uncertainty principle based on entropic certainty. This conjecture, also studied by Hirschman and proven in 1975 by Beckner and by Iwo Bialynicki-Birula and Jerzy Mycielski is that, for two normalized, dimensionless Fourier transform pairs and where and the Shannon information entropies and are subject to the following constraint, where the logarithms may be in any base. The probability distribution functions associated with the position wave function and the momentum wave function have dimensions of inverse length and momentum respectively, but the entropies may be rendered dimensionless by where and are some arbitrarily chosen length and momentum respectively, which render the arguments of the logarithms dimensionless. Note that the entropies will be functions of these chosen parameters. Due to the Fourier transform relation between the position wave function and the momentum wavefunction , the above constraint can be written for the corresponding entropies as where is Planck's constant. Depending on one's choice of the product, the expression may be written in many ways. If is chosen to be , then If, instead, is chosen to be , then If and are chosen to be unity in whatever system of units are being used, then where is interpreted as a dimensionless number equal to the value