Uncertainty from Heisenberg to Today

Abstract

We explore the different meanings of "quantum uncertainty" contained in Heisenberg's seminal paper from 1927, and also some of the precise definitions that were explored later. We recount the controversy about "Anschaulichkeit", visualizability of the theory, which Heisenberg claims to resolve. Moreover, we consider Heisenberg's programme of operational analysis of concepts, in which he sees himself as following Einstein. Heisenberg's work is marked by the tensions between semiclassical arguments and the emerging modern quantum theory, between intuition and rigour, and between shaky arguments and overarching claims. Nevertheless, the main message can be taken into the new quantum theory, and can be brought into the form of general theorems. They come in two kinds, not distinguished by Heisenberg. These are, on one hand, constraints on preparations, like the usual textbook uncertainty relation, and, on the other, constraints on joint measurability, including trade-offs between accuracy and disturbance.

Entropic measurement uncertainty relations for all the infinite components of a spin vector

“…Therefore the strict positivity in (28) holds. The second property in (28) follows from (A.6) and the symmetry of the angles in the discretization (12).…”

“…where q is the set of angles satisfying the discretization conditions (12) and involved in the expression (27) of the qcoefficients. Moreover, the following bounds hold:…”

“…In the last twenty years the idea of quantum uncertainty relations has been deeply developed and formalized by introducing different related notions. Measurement uncertainty relations (MURs) for joint measurements quantify to which extent one can approximate a set of measurements of incompatible observables by means of a single joint measurement [1][2][3][4][5][6][7][8][9][10][11][12][13]. On the other side, MURs of noise/disturbance type quantify the total uncertainty generated by an approximate measurement of a first observable disturbing the measurement of a second one [10][11][12][13][14][15][16][17].…”

“…Measurement uncertainty relations (MURs) for joint measurements quantify to which extent one can approximate a set of measurements of incompatible observables by means of a single joint measurement [1][2][3][4][5][6][7][8][9][10][11][12][13]. On the other side, MURs of noise/disturbance type quantify the total uncertainty generated by an approximate measurement of a first observable disturbing the measurement of a second one [10][11][12][13][14][15][16][17]. Finally, one speaks of preparation uncertainty relations (PURs) when some lower bound is given on the 'spreads' of the distributions of some observables measured in the same state [8][9][10][11][12][18][19][20][21][22][23][24]].…”

“…On the other side, MURs of noise/disturbance type quantify the total uncertainty generated by an approximate measurement of a first observable disturbing the measurement of a second one [10][11][12][13][14][15][16][17]. Finally, one speaks of preparation uncertainty relations (PURs) when some lower bound is given on the 'spreads' of the distributions of some observables measured in the same state [8][9][10][11][12][18][19][20][21][22][23][24]]. An important point in MURs for joint measurements and PURs is to arrive to formulate them for more than two observables [5,8,11,[20][21][22][23][24].…”