Strong unitary uncertainty relations

Abstract

In this paper we provide a new set of uncertainty principles for unitary operators using a sequence of inequalities with the help of the geometric-arithmetic mean inequality. As these inequalities are "fine-grained" compared with the well-known Cauchy-Schwarz inequality, our framework naturally improves the results based on the latter. As such, the unitary uncertainty relations based on our method outperform the best known bound introduced in [Phys. Rev. Lett. 120, 230402 (2018)] to some extent. Explicit examples of unitary uncertainty relations are provided to back our claims. arXiv:1908.05053v1 [quant-ph]

Improved unitary uncertainty relations

“…We use brackets and convex functions to derive variance-based unitary uncertainty relations in the products form for two unitary operators in all quantum systems, which is simple to understand and can be well presented. We show that our new bound outperforms Bong et al's [35] in the whole range as well as Yu et al's [38] and Li et al's [39] in almost the whole interval.…”

“…The lower bounds K 2 and K 0.1 2 can be calculated by Theorem 1(2). The bounds are drawn in figure 3 and it is seen that our bound is tighter than Bong et al's, Yu et al's and Li et al's [35,38,39](see figure 4).…”

“…In this case, the result shows that Another improved uncertainty relation based on [38] was given by Li et al [39]. They defined a quantity…”

“…In [38] Yu et al derived strong variance-based uncertainty relations for two unitary operators by using a sequence of "fined-grained" inequalities:…”

“…Bong's beautiful bound is expressed implicitly in terms of the Gram determinant, which is sometimes hard to extract the exact form. In [38] one of us and collaborators have proposed an improvement to evaluate the lower bound using a sequence of "fine-grained" inequalities [40].…”