Implications and applications of the variance-based uncertainty equalities

Abstract

In quantum mechanics, the variance-based Heisenberg-type uncertainty relations are a series of mathematical inequalities posing the fundamental limits on the achievable accuracy of the state preparations. In contrast, we construct and formulate two quantum uncertainty equalities, which hold for all pairs of incompatible observables and indicate the new uncertainty relations recently introduced by L. Maccone and A. K. Pati [Phys. Rev. Lett. 113, 260401 (2014)]. Furthermore, we present an explicit interpretation lying behind the derivations and relate these relations to the so-called intelligent states. As an illustration, we investigate the properties of these uncertainty inequalities in the qubit system and a state-independent bound is obtained for the sum of variances. Finally, we apply these inequalities to the spin squeezing scenario and its implication in interferometric sensitivity is also discussed.

Neutrinoless Double-Beta Decay: Status and Prospects

“…The energy-density functional (EDF) and the generator coordinate methods (GCM) mix many mean fields with different properties (88, 89,90), whereas the other methods use simple mean fields that the states and orbitals feel. Minimization of the energy functional finds the ground states.…”

Uncertainty relations for general unitary operators

“…[16], one may note that the minimum uncertainty states of such unitary operators are the ground states of the Harper Hamiltonian [22,32], which will be discussed in more detail in section V below. We have taken the ground states of the Harper Hamiltonian for dimensions 3,5,8,12, and found that our bound is not saturated for d ≥ 3. It is clear from the Table I that though the uncertainty relations are not tight, but they are tighter than that given in Ref.…”

“…Recently, stronger uncertainty relations are proved which go beyond the Robertson uncertainty relations and they capture the notion of incompatible observables [4]. Uncertainty relations are at the center stage of current research in quantum theory and quantum information [5][6][7][8]. The uncertainty relations are useful from the point of view of foundational aspects and have applications in quantum technology as well [9][10][11][12][13][14][15].…”

Experimental investigation of multi-observable uncertainty relations

“…After these, more state-dependent [15][16][17][18][19][20] and state-independent [21][22][23][24][25][26] lower bounds of uncertainty relations were also obtained. The generalization and improvement mainly focused on the uncertainty relation capable of dealing with more than two incompatible observables, i.e., the Heisenberg-type uncertainty relation for three canonical observables [27], uncertainty relations for angular momentum [28], and arbitrary incompatible observables [17].…”