Generalizations of the Heisenberg and Schrödinger uncertainty relations

Abstract

In this paper, releasing the restriction on operators which are self-adjoint, we propose a Heisenberg-type uncertainty relation and a Schrödinger-type uncertainty relation with any pair of operators on a Hilbert space. A generalization of Luo's theorem [S. Luo, “Heisenberg uncertainty relation for mixed states,” Phys. Rev. A 72, 042110 (2005)] is investigated.

Uncertainty Relations: Curiosities and Inconsistencies

“…As it is seen, the inconsistencies of this type and others discussed in previous Sections are integrated into inequality (4). For this reason, attempts are being made to improve and refine the Heisenberg as well as Robertson and Schrodinger uncertainty relations (see, e.g., [14][15][16][17]20]). From the analysis presented in Section 5 it follows that a status and role of the uncertainty relations (1), (2) and (4) in P T -symmetric quantum theory seems to be unclear.…”

“…As it was mentioned, there is still a discussion on how to interpret inequalities (4) and (2) and how to improve them (see, e.g., [10] and references therein, [11,12,[14][15][16][17][18][19][20] and many other papers). From the derivation of the formula (4) it follows that the standard deviations ∆ φ A and ∆ φ B characterize the statistical distribution of the most probable values of A and B in the state |φ .…”

Time operators and time crystals: self-adjointness by topology change

“…This contradiction is avoided with generalized uncertainty relations. The Robertson uncertainty relation [40] between two Hermitian operators is given by [41] proposed the following generalization (which is written here with a different notation)…”

“…We note that the uncertainty relations (38), (40) and (41) are not unique but other equivalent or inequivalent forms can be defined if necessary. Adding (40) to (41) and using the triangle inequality we obtain (37). Moreover, by adding (38) to the inequivalent form (39) we obtain (36).…”

A generalization of Schrödinger’s uncertainty relation described by the Wigner–Yanase skew information

“…With the increasing interests in quantum information processing [2] and the foundations of quantum mechanics, the notion of skew information and its various derivations have been discussed by many authors [3][4][5][6][7]. Recall that for a density ρ ∈ D(H ) and an observable A, the WignerYanase skew information [8] is defined as…”