Welcome to the Quantum Uncertainty Pages. After nearly 90 years since Heisenberg's discovery of what has become known as the Uncertainty Principle, there has remained a cloud of vagueness over the subject of quantum indeterminacy (perhaps not unfitting for the subject).
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To be sure, there is the standard uncertainty relation for pairs of quantum observable A, B that you find in every textbook, whose meaning is is well understood as a relation concerning the standard deviations of the distributions of values of A and B obtained in separate, accurate measurements of these quantities if performed on a large number of systems prepared in the same state (we will review this in more detail in these pages). However, this very relation is often interpreted as a limitation to making joint measurements of A and B if these observables do not commute This interpretation takes the standard deviations as a reflection and measure of measurement errors.
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This conflation of standard deviations (of statistics of accurate, separate measurements) and error (in approximate joint measurements) is rooted in the fact that in Heisenberg's intuitive, semi-classical examples, these quantities happened to coincide. Heisenberg's concern was clearly with the inevitable disturbance caused by quantum measurements - both disturbance of the object and mutual disturbance of the measurement operations. He envisaged a limitation of the joint measurability of two conjugate quantities and captured it intuitively in the form of an uncertainty relation.
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As it happened, the standard uncertainty relation quickly and easily got formalised mathematically and proven (in the greatest generality allowed by the famous Cauchy-Schwarz inequality for Hilbert space vectors) in work of Kennard, Weyl, Robertson, and Schrödinger. It is now considered to be the canonical expression of preparation uncertainty - the first and better known face of Heisenberg's Principle.
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In contrast, a quantum mechanical theory of approximate joint measurements for two or more noncommuting observables has remained elusive for many decades and has only become available, and gradually more widely accepted, since the 1960s. In the early 2000s, the study of measures of approximation error for quantum measurements commenced, leading to rigorous formulations of error (and error-disturbance) trade-off relations for joint measurements, expressing measurement uncertainty - the second face of the uncertainty principle.