A precise theory of the simultaneous meaSllrement of a pair of conjugate observables is necessary for obtaining the classical limit from the quantum theory, for determining the limitations of coherent quantum mechanical amplifiers, etc. The uncertainty principle, of course, does not directly address this problem, since it is a statement about the variances of two hypothetical ideal measurements. We will adopt the approach that there exist instantaneous inexplicable ideal measurements of a single observable. Just as von Neumann1uses an ideal measurements together with all interaction to explain an indirect observation, we use ideal measurements together with iuteractions to explain the simultaneous measurements of an observable and its conjugate.
A precise theory of the simultaneous meaSllrement of a pair of conjugate observables is necessary for obtaining the classical limit from the quantum theory, for determining the limitations of coherent quantum mechanical amplifiers, etc. The uncertainty principle, of course, does not directly address this problem, since it is a statement about the variances of two hypothetical ideal measurements. We will adopt the approach that there exist instantaneous inexplicable ideal measurements of a single observable. Just as von Neumann1uses an ideal measurements together with all interaction to explain an indirect observation, we use ideal measurements together with iuteractions to explain the simultaneous measurements of an observable and its conjugate.
In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
The state of a quantum-mechanical system after a simultaneous measurement of conjugate variables such as coordinate and momentum is derived. In the case of the minimum-uncertainty, or ideal, simultaneous measurement, the state of the system after the measurement is a coherent state corresponding to that representing a minimum-uncertainty wave packet. An operator formalism to describe simultaneous measurements of noncommuting observables is introduced which parallels that familiar for a single measurement, and an expression for the joint probability distribution predicting the results of an ideal simultaneous measurement is derived and applied to the example of a harmonic oscillator. It is found that the minimum uncertainty in a simultaneous measurement of noncommuting variables has two causes: (1) the unavoidable perturbations introduced by the measuring process, and (2) the unavoidable lack of precision in the state of the system itself. For position and momentum measurements, these two independent uncertainties contribute to give a net minimum uncertainty of $\ensuremath{\Delta}{q}^{\ensuremath{'}}\ensuremath{\Delta}{p}^{\ensuremath{'}}=\ensuremath{\hbar}$. It is also shown that the formalism of simultaneous measurement leads to a well-defined quantum-mechanical phase space.
Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the accuracy of simultaneous measurement but assuming an obsolete postulate for quantum mechanics. This assumption, known as the repeatability hypothesis, formulated explicitly by von Neumann and Schrodinger, was broadly accepted until the 1970s, but abandoned in the 1980s, when completely general quantum measurement theory was established. We also survey the author's recent proposal for a universally valid reformulation of Heisenberg's uncertainty principle under the most general assumption on quantum measurement.
Abstract In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
One of the central features of quantum theory is that there are pairs of quantum observables that cannot be measured simultaneously. This incompatibility of quantum observables is a necessary ingredient in several quantum phenomena, such as measurement uncertainty relations, violation of Bell inequalities, and steering. Two quantum observables that admit a simultaneous measurement are, in this respect, classical. A finer classification of classicality can be made by formulating four symmetric relations on the set of observables that are stronger than compatibility; they are broadcastability, one-side broadcastability, mutual nondisturbance, and nondisturbance. It is proven that the five relations form a hierarchy and their differences in terms of the required devices needed in a simultaneous measurement are explained. All four relations that are stronger than compatibility are completely characterized in the case of qubit observables.
In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.