Contradictory uncertainty relations

Uncertainty relations play a significant role in quantum physics, primarily discussed in terms of the product of variances of the corresponding Hermitian operators representing observables. However, circumstances arise where such formulation may not produce satisfactory results, necessitating alternative methods. Hence, the introduction of alternative measures of fluctuations and uncertainty relations.

In this blog, we delve into the concept that different assessments of fluctuations could lead to uncertainty relations resulting in contradictory conclusions. Even within the same family of uncertainty measures such as Tsallis and Rényi entropies, these contradictions exist, and are quite crucial given the importance of uncertainty relations in quantum physics.

Tsallis and Rényi entropies

Mainly we look into Tsallis entropies, where Sq(A) is always nonnegative making it an ideal measure of uncertainty. Maximum uncertainty occurs when all outcomes are equal probabilities. This family includes the Shannon entropy in the limit and is also closely related to Rényi entropy.

These measures enter uncertainty relations for two observables, A, B via nontrivial lower bounds on different combinations of these entropies. We're concerned with the contradictions between the conclusions derived from different choices of q for the same family of entropy combinations.

Two-dimensional observables

To make the contradiction apparent, we consider a two-dimensional system with two observables A, B with outcomes and probabilities. The states will not be orthogonal, adding complexity.

For complementary observables where θ is a variable, we plotted Πq, Uq, and Σq as functions of θ and it can be seen that there is an exchange from maximum to minimum depending on the value of q.

Not Fully Complementary Observables

The behaviour is displayed even when considering observables that are not completely complementary. A local maximum becomes an absolute minimum for larger q. The maximum for low q also becomes a local minimum for larger q.

Conclusion

It's clearly seen that maximum uncertainty states can transform into minimum uncertainty states and vice versa, based on the measure of uncertainty employed. Despite stemming from the same family of measures, the conclusions may still be contradictory to the extent of exchanging maxima and minima.

While uncertainty relations have a long history, they continue to offer surprising results that warrant further investigation. For instance, recent discussions argue that there are fluctuation measures that seemingly lead to no uncertainty relation for complementary observables.