The Optimal Uncertainty Relation

Abstract

Employing the lattice theory on majorization, the universal quantum uncertainty relation for any number of observables and general measurement is investigated. It is found that 1) the least bounds of the universal uncertainty relations can only be properly defined in the lattice theory; 2) contrary to variance and entropy, the metric induced by the majorization lattice implies an intrinsic structure of the quantum uncertainty; and 3) the lattice theory correlates the optimization of uncertainty relation with the entanglement transformation under local quantum operation and classical communication. Interestingly, the optimality of the universal uncertainty relation found can be mimicked by the Lorenz curve, initially introduced in economics to measure the wealth concentration degree of a society.

Experimental investigation of the uncertainty relations with coherent light

“…While equations (21,22) are directly verifiable by measuring the quantities on both sides of the inequalities, the…”

“…To test inequalities (21)(22)(23)(24), we choose a series of states |ψ(θ, 0) and |ψ( π 3 , φ) , where θ or φ = nπ/12. Those system states can be generated by tuning the setting angles of Q1 and HWP, and Q2 is set to be 45 • .…”

“…In spite of the great effort devoted to the subject, getting the optimal lower bounds of the entropic uncertainty relations remains a challenging task [20] and a hot topic [21]. Contrary to the variance-based and entropic uncertainty relations, the optimal bound problem of the majorization uncertainty relations has just been solved by exploring the lattice theory [22].…”

“…However, measuring the Schur-concave function is not sufficient for the majorization relation between distributions, e.g., we have p ≺ q ⇒ H( p ) > H( q ) for the Schur-concave function of Shannon entropy, but H( p ) > H( q ) p ≺ q [30]. A faithful representation of the majorization uncertainty relation using the Lorenz curve has been introduced in [22], where p ≺ q if and only if the Lorenz curve of p is everywhere below (enclosed by) that of q.…”

Optimal common resource in majorization-based resource theories

“…Before doing that, we stress that the lattice theoretical aspects of majorization theory have not been sufficiently exploited in comparison with other features of it in the area of quantum information. Indeed, the first applications were given in [30,31]; only recently, new applications of the majorization lattice have been found [32][33][34][35][36][37][38][39].…”