1212.4242 (Alexey E. Rastegin)

Uncertainty and certainty relations for complementary qubit observables
in terms of Tsallis' entropies    [PDF]

Alexey E. Rastegin

We present novel uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Tsallis $\alpha$-entropies. For all real $\alpha\in(0;1]$ and integer $\alpha\geq2$, lower bounds on the sum of three $\alpha$-entropies are obtained. These bounds are tight in the sense that they are always reached with certain pure states. The necessary and sufficient condition for equality is that the qubit state is an eigenstate of one of the Pauli observables. Using concavity with respect to the parameter $\alpha$, we further derive approximate lower bounds for non-integer $\alpha\in(1;+\infty)$. In the case of pure states, the developed method also allows to obtain upper bounds on the entropic sum for real $\alpha\in(0;1]$ and integer $\alpha\geq2$. For applied purposes, entropic bounds are often used with averaging over the individual entropies. Combining the obtained lower and upper bounds leads to a band, in which the rescaled average $\alpha$-entropy ranges in the pure-state case.

View original: http://arxiv.org/abs/1212.4242