Uncertainty principle and quantum Fisher information. II.

Abstract

Heisenberg and Schrödinger uncertainty principles give lower bounds for the product of variances 2000 Mathematics Subject Classification. Primary 62B10, 94A17; Secondary 46L30, 46L60.

Quantum Covariance, Quantum Fisher Information, and the Uncertainty Relations

“…In the case N = 2 the inequality was proved by Luo, Q. Zhang and Z. Zhang [16,17,15], by Kosaki [11] and by Yanagi, Furuichi and Kuriyama [27] for some special functions f. The general case is due to Gibilisco, Imparato and Isola [7,4]. Gibilisco and Isola emphasized the geometric aspects of the inequality (10) and conjectured it for general quantum Fisher information [4].…”

“…Identity (12) is easy to check but it is not obvious that for a standard f the functionf is operator monotone. It is indeed true thatf is a standard function as well, see Propositions 5.2 and 6.3 in [7]. Note that the left-hand-side of (12) was called (metric adjusted) skew information in [8].…”

“…The general case was proved by Hansen in [8] and shortly after by Gibilisco, Imparato and Isola with a di erent technique in [7].…”

Born–Jordan quantization and the uncertainty principle

“…The study of uncertainties for non-standard situations has been tackled (from a very different point of view) by Korn [24]; see also the review paper [10] by Folland and Sitaram, which however unfortunately deliberately ignores the fundamental issue of covariance. Gibilisco and his collaborators [11][12][13] give highly nontrivial refinements of uncertainty relations using convexity properties, and study the notion of statistical covariance in depth.…”

Uncertainty Relations for Some Central Potentials in N-Dimensional Space

“…Such variances do not necessarily exist, and if they do, they describe the quantum probability distribution relative to a specific point of the probability domain. Therefore, various alternative formulations have been suggested by the use of information-theoretic uncertainty measures like the Shannon entropy [26] [27], Renyi entropies [28] [29], Tsallis entropies [30], entropic moments [31] [32] and Fisher information [32]- [36]. During the past years, the generalization of three dimensional quantum problems to higher space dimensions receives a considerable development in theoretical and mathematical physics.…”